QUATERNIONIC GRASSMANNIANS AND PONTRYAGIN CLASSES IN ALGEBRAIC GEOMETRY

IVAN PANIN AND CHARLES WALTER

Abstract. The quaternionic Grassmannian HGr(r, n) is the affine open subscheme of the ordinary Grassmannian parametrizing those 2r-dimensional subspaces of a 2n-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular there is HP^{n} = HGr(1, n+1). For a symplectically oriented cohomology theory A, including oriented theories but also hermitian K-theory, Witt groups and algebraic symplectic cobordism, we have A(HP^{n}) = A(pt)[p]/(p^{n+1}). We define Pontryagin classes for symplectic bundles. They satisfy the splitting principle and the Cartan sum formula, and we use them to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Pontryagin classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Pontryagin classes.

The cell structure of the HGr(r, n) exists in the cohomology, but it is difficult to see more than part of it geometrically. The exception is HP^{n} where the cell of codimension 2i is a quasi-affine quotient of 𝔸^{4n−2i+1} by a nonlinear action of 𝔾_{a}.

1. Introduction

The quaternionic projective spaces ℍℙ^{n} and Grassmannians ℍ𝔾r(r, n) are important spaces in topology. They have cell structures like real and complex projective spaces and Grassmannians, but the dimensions of the cells are multiples of 4. In symplectically oriented cohomology theories E^{∗}, including oriented theories but also KO^{∗}, MSp^{∗} and MSU^{∗}, there is a quaternionic projective bundle theorem E^{∗}(ℍℙ^{n}) = E^{∗}(pt)[p]/(p^{n+1}) which leads to a theory of Pontryagin classes of quaternionic bundles satisfying the Cartan sum formula and the Splitting Principle. These classes are used to prove a number of well-known theorems, including Conner and Floyd’s description of KO^{∗} as a quotient of MSp^{∗} [6, Theorem 10.2]. Infinitedimensional quaternionic Grassmannians provide models for the classifying spaces BSp_{2r} and BSp and for symplectic K-theory. For symplectically oriented cohomology theories we have E^{∗}(BSp_{2r}) = E^{∗}(pt)[[p_{1}, …, p_{r}]] and E^{∗}(BSp) = E^{∗}(pt)[[p_{1}, p_{2}, …]] where the p_{i} are the Pontryagin classes of the universal bundle.

In this paper we lay the foundations for a similar theory in motivic algebraic geometry. Since quaternionic Grassmannians are quotients of compact Lie groups ℍ𝔾r(r, n) = U_{n}(ℍ)/(U_{r}(ℍ) × U_{n−r}(ℍ)), we take as our models the corresponding quotients of algebraic groups HGr(r, n) = Sp_{2n} /(Sp_{2r} × Sp_{2n−2r}). Then HGr(r, n)(ℂ) has the same homotopy type as ℍ𝔾r(r, n) but twice the dimension and a significantly more complicated geometry.

Date: September 20, 2010.

The first author thanks for the support and excellent working conditions Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice Sophia Antipolis, the DFG–RFFI project 09-01-91333-NNIO-a, the Presidium of RAS Program “Fundamental Research in modern mathematics” and the RFFI-grant 10-01-00551.

An alternative description of HGr(r, n) is that if we equip the trivial vector bundle on the base 𝒪^{2n} with the standard symplectic form ψ_{2n}, then HGr(r, n) is the open subscheme of Gr(2r, 2n) parametrizing the subspaces U ⊂ 𝒪^{2n} of dimension 2r on which ψ_{2n} is nonsingular. The HGr(r, n) are smooth and affine of dimension 4r(n − r) over the base scheme with the same global units and the same Picard group as the base.

Our first results concern HP^{n} = HGr(1, n + 1). In Theorems 3.1, 3.2 and 3.4 we prove the following facts.

Theorem 1.1. The scheme HP^{n} is smooth of dimension 4n over the base scheme. It has a decomposition into locally closed strata

HP^{n} = ⨆_{i=0}^{n} X_{2i} = X_{0} ⊔ X_{2} ⊔ ⋯ ⊔ X_{2n},  (3.3)

such that each X_{2i} is of codimension 2i, smooth and quasi-affine, but X_{0}, …, X_{2n−2} are not affine. The closure \overline{X}_{2i} = X_{2i} ⊔ X_{2i+2} ⊔ ⋯ ⊔ X_{2n} is a vector bundle of rank 2i over HP^{n−i}. Each X_{2i} is the quotient of a free action of 𝔾_{a} on 𝔸^{4n−2i+1}.

One can study ordinary Grassmannians inductively by considering the closed embedding Gr(r, n−1) ↪ Gr(r, n). The complement of the image is a vector bundle over Gr(r−1, n−1). The normal bundle of the embedding is isomorphic to the dual 𝒰_{r,n−1}^{∨} of the tautological subbundle on Gr(r, n − 1), and it embeds as an open subvariety of Gr(r, n). This gives a long exact sequence

⋯ → A_{Gr(r,n−1)}(𝒰_{r,n−1}^{∨}) → A(Gr(r, n)) → A(Gr(r − 1, n − 1)) → ⋯

in any cohomology theory.

There is an analogous long exact sequence for quaternionic Grassmannians, but one has to wade through a more complicated geometry to reach it. We take a symplectic bundle (E, ϕ) of rank 2n over a scheme S. We then let (F, ψ) be the symplectic bundle of rank 2n with

F = 𝒪_{S} ⊕ E ⊕ 𝒪_{S},  ψ = (\begin{matrix}0 & 0 & 1 \\ 0 & ϕ & 0 \\ −1 & 0 & 0 \end{matrix}) .  (4.2)

We have the quaternionic Grassmannian bundles HGr(E) = HGr_{S}(r; E, ϕ) and HGr(F) = HGr(r; F, ψ). Earlier, the complement of the open stratum X_{0} ⊂ HP^{n} was a vector bundle \overline{X}_{2} over HP^{n−1} rather than HP^{n−1} itself. The same thing happens now. Let

N^{+} = HGr(F) ∩ Gr_{S}(2r; 𝒪_{S} ⊕ E),  N^{−} = HGr(F) ∩ Gr_{S}(2r; E ⊕ 𝒪_{S}).

For any cohomology theory A we have a long exact sequence

⋯ → A_{N^{+}}(HGr(F)) → A(HGr(F)) → A(HGr(F) ∖ N^{+}) → · · · .  (5.4)

Theorem. (a) The loci N^{+} and N^{−} are vector bundles over HGr(E) isomorphic to the tautological rank 2r symplectic subbundle 𝒰_{r,E}.

(b) The vector bundle N^{+}⊕N^{−} is the normal bundle of HGr(E) ⊂ HGr(F), and it naturally embeds as an open subscheme of the Grassmannian bundle Gr_{S}(2r; F).

(c) N^{+} and N^{−} are the loci in HGr(F) where certain sections s_{+} and s_{−} of the tautological bundle 𝒰_{r,F} intersect the zero section transversally.

This is part of Theorem 4.1. Since HGr(F) ⊂ Gr_{S}(2r; F) is also an open subscheme, we deduce natural isomorphisms

A_{N^{+}}(HGr(F)) ≅ A_{N^{+}}(N^{+} ⊕ N^{−}) ≅ A_{HGr(r;E,ϕ)}(𝒰_{r,E}).

The study of the open subscheme Y = HGr(F) ∖ N^{+} in §5 is crucial to this paper. The main result is:

Theorem. We have morphisms

Y ⟵^{g_{1}} Y_{1} ⟵^{g_{2}} Y_{2} ⟶^{q} HGr_{S}(r − 1, E, ϕ)

over S with g_{1} an 𝔸^{2r−1}-bundle, g_{2} an 𝔸^{2r−2}-bundle and q an 𝔸^{4n−3}-bundle. Moreover, there is an explicit symplectic automorphism of the pullback of (F, ψ) to Y_{2} which induces an isometry

\xymatrix{g_{2}^{∗}g_{1}^{∗}(𝒰_{r}, ϕ_{r}) \ar[r]^{≅} & (𝒪 ⊕ q^{∗}𝒰_{r−1} ⊕ 𝒪, ( \begin{matrix}0 & 0 & 1 \\ 0 & ϕ_{r−1} & 0 \\ −1 & 0 & 0 \end{matrix} )) ,}

where (𝒰_{r}, ϕ_{r}) and (𝒰_{r−1}, ϕ_{r−1}) are the tautological symplectic subbundles on HGr(r, F, ψ) and HGr(r − 1, E, ϕ), respectively.

This is a simplified version of Theorem 5.1. From it we deduce that the natural closed embedding HGr_{S}(r − 1, E, ϕ) ↪ Y induces isomorphisms A(Y) ≅ A(HGr_{S}(r − 1, E, ϕ)) in any cohomology theory. For r = 1 this is an isomorphism A(Y) ≅ A(S). We thus get our long exact sequence of cohomology groups

⋯ → A_{HGr(r,E,ϕ)}(𝒰_{r,E}) → A(HGr(r, F, ψ)) → A(HGr(r − 1, E, ϕ)) → · · · .  (5.5)

The geometry of Theorems 4.1 and 5.1 and the decomposition of HGr_{S}(r, F, ψ) into the vector bundle N^{+} and the complementary open locus Y is recurrent throughout the paper. This seems to be a fundamental geometry of quaternionic Grasssmannian bundles.

After the initial description of the geometry, we begin to introduce symplectically oriented cohomology theories. We follow the point of view used for oriented cohomology theories in [14]. In the end (Definition 14.3) a symplectic orientation on a ring cohomology theory is a family of Thom isomorphisms th_{X}^{E,ϕ}: A(X) ≅ A_{X}(E) for every scheme X and every symplectic bundle (E, ϕ) over X. There are also isomorphisms th_{Z}^{E,ϕ}: A_{Z}(X) ≅ A_{Z}(E) for closed subsets Z ⊂ X. These isomorphisms satisfy several axioms of functoriality, of compatibility with the ring structure, and of compatibility with orthogonal direct sums (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}).

There are five ways of presenting a symplectic orientation. The Thom class th(E, ϕ) ∈ A_{X}(E) is the image of 1_{X} ∈ A(X). One can present a symplectic orientation by giving either the Thom isomorphisms, the Thom classes or the Pontryagin classes of all symplectic bundles or by giving the Thom classes or the Pontryagin classes of rank 2 symplectic bundles only. In each case the classes are supposed to obey a certain list of axioms. In the end (Theorem 14.4) the five ways of presenting a symplectic orientation on a ring cohomology theory are equivalent.

We start in §7 by presenting the version where one gives the Thom classes for rank 2 symplectic bundles only. This is a symplectic Thom structure. With the Thom classes one can define Thom isomorphisms A(X) ≅ A_{X}(E) for rank 2 symplectic bundles and direct image maps i_{A,♭}: A(X) → A_{X}(Y) and i_{A,♮}: A(X) → A(Y) for regular embeddings i: X ↪ Y of codimension 2 whose normal bundle is equipped with a symplectic form (N_{X/Y} , ϕ). The Pontryagin class of a rank 2 symplectic bundle (E, ϕ) on X is p(E, ϕ) = −z^{A}e^{A} th(E, ϕ) where e^{A}: A_{X}(E) → A(E) is extension of supports and z^{A}: A(E) ≅ A(X) is the restriction to the zero section. The main formula is that if a section of E intersects the zero section transversally in a subscheme Z, then for the inclusion i: Z ↪ X and for all b ∈ A(X) we have

i_{A,♮}i^{A}b = −b ∪ p(E, ϕ).  (7.5)

With the geometry of Theorems 4.1 and 5.1, the long exact sequence of cohomology (5.4) and formula (7.5) we prove one of the main results of the paper.

Theorem 8.2. (Quaternionic projective bundle theorem). Let A be a ring cohomology theory with a symplectic Thom structure. Let (E, ϕ) be a rank 2n symplectic bundle over a scheme S, let (𝒰, ϕ|_{𝒰}) be the tautological rank 2 symplectic subbundle over the quaternionic projective bundle HP_{S}(E, ϕ), and let ζ = p(𝒰, ϕ|_{𝒰}) be its Pontryagin class. Write π: HP_{S}(E, ϕ) → S for the projection. Then for any closed subset Z ⊂ X we have an isomorphism of two-sided A(S)-modules (1, ζ, …, ζ^{n−1}): A_{Z}(S)^{⊕n} ⟶^{≅} A_{π^{−1}(Z)}(HP_{S}(E, ϕ)), and we have unique classes p_{i}(E, ϕ) ∈ A(S) for 1 ≤ i ≤ n such that there is a relation

ζ^{n} − p_{1}(E, ϕ) ∪ ζ^{n−1} + p_{2}(E, ϕ) ∪ ζ^{n−2} − ⋯ + (−1)^{n}p_{n}(E, ϕ) = 0.

If (E, ϕ) is trivial, then p_{i}(E, ϕ) = 0 for 1 ≤ i ≤ n.

The classes p_{i}(E, ϕ) are the Pontryagin classes of (E, ϕ) with respect to the symplectic Thom structure on A. One also sets p_{0}(E, ϕ) = 1 and p_{i}(E, ϕ) = 0 for i > n. Pontryagin classes are 𝔸^{1}-deformation invariant and nilpotent.

The Pontryagin classes of an orthogonal direct sum of symplectic bundles (F, ψ) ≅ (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}) satisfy the Cartan sum formula

p_{i}(F, ψ) = p_{i}(E_{1}, ϕ_{1}) + ∑_{j=1}^{i−1} p_{i−j}(E_{1}, ϕ_{1})p_{j}(E_{2}, ϕ_{2}) + p_{i}(E_{2}, ϕ_{2}).  (10.6)

The p_{1} is additive, and the top Pontryagin classes are multiplicative. The corresponding formula for Chern classes is usually proven with a geometric argument shortly after a proof of the projective bundle theorem. Unfortunately we have not found a geometric proof of the Cartan sum formula for Pontryagin classes. The geometry of quaternionic projective bundles (as we have defined them) is just not as nice as the geometry of projective bundles. The best we have done geometrically is to show that if (E, ϕ) is an orthogonal direct summand of (F, ψ) then the Pontryagin polynomial P_{E,ϕ}(t) divides the Pontryagin polynomial P_{F,ψ}(t) (Lemma 9.3).

So we give a roundabout cohomological proof of the Cartan sum formula. The first step is to consider the scheme HFlag(1^{r}; n) = Sp_{2n} /(Sp_{2}^{×r} × Sp_{2n−2r}). It classifies decompositions

(𝒪^{2n}, ψ_{2n}) ≅ (𝒰_{n}^{(1)}, ϕ_{n}^{(1)}) ⊥ ⋯ ⊥ (𝒰_{n}^{(r)}, ϕ_{n}^{(r)}) ⊥ (𝒱_{r,n}, ψ_{r,n})  (9.1)

of the trivial symplectic bundle of rank 2n into the orthogonal direct sum of r symplectic subbundles of rank 2 plus a symplectic subbundle of rank 2n−2r. It is an iterated quaternionic projective bundle over both the base scheme and HGr(r, n). The weak substitute for the Cartan sum formula (Lemma 9.3) is good enough for us to show that we have

\underleftarrow{lim}_{n→∞} A(HFlag(1^{r}; n)) ≅ A(k)[[y_{1}, …, y_{r}]]  (9.2)

with the indeterminate y_{i} corresponding to the element given by the system (p(𝒰_{n}^{(i)}, ϕ_{n}^{(i)}))_{n≥r}. Using the universality of the families of schemes HFlag(1^{r}; n) and HGr(r, n) and the fact that the y_{i} − y_{j} are not zero divisors in A(k)[[y_{1}, …, y_{r}]], we are able to deduce the Symplectic Splitting Principle (Theorem 10.2) from Lemma 9.3. The Cartan sum formula then follows (Theorem 10.5).

We then calculate the cohomology of quaternionic Grassmannians for any ring cohomology theory with a symplectic Thom structure. We get

A(HFlag(1^{r}, n)) = A(k)[y_{1}, …, y_{r}]/(h_{n−r+1}, …, h_{n}),  (11.6)
A(HGr(r, n)) = A(k)[p_{1}, …, p_{r}]/(h_{n−r+1}, …, h_{n}),  (11.7)

where the y_{i} are the Pontryagin classes of the r tautological rank 2 symplectic subbundles on HFlag(1^{r}, n), the p_{i} are the Pontryagin classes of the tautological rank 2r symplectic subbundle on HGr(r, n) and are the elementary symmetric polynomials in the y_{i}, and the h_{i} are the complete symmetric polynomials. It follows that for the standard embeddings the restrictions A(HGr(r, n + 1)) → A(HGr(r, n)) and A(HGr(r + 1, n + 1)) → A(HGr(r, n)) are surjective, and we have isomorphisms

\underleftarrow{lim}_{n→∞} A(HGr(r, n)) = A(k)[[p_{1}, p_{2}, …, p_{r}]],  (11.12)
\underleftarrow{lim}_{n→∞} A(HGr(n, 2n)) = A(k)[[p_{1}, p_{2}, p_{3}, …]].  (11.13)

In §§12–14 we prove the equivalence of the five ways of presenting a symplectic orientation on a ring cohomology theory. Above all, this involves using the exact sequence (5.5) and the calculations of the cohomology of Grassmannian bundles to recover the Thom classes from the Pontryagin classes.

The isomorphism A(Y) ≅ A(S) when Y = X_{0} × S ⊂ HP^{n} ×S is the open stratum is absolutely critical to the entire paper. Without it there is no proof of the quaternionic projective bundle theorem and no definition of Pontryagin classes. Therefore in §15 we give a second proof of the isomorphism using a completely different geometric argument. However, the geometry of Theorem 5.1 seems to be more generally useful than that of Theorem 15.1.

We stop here. This is already a long paper because the theory of Thom and Pontryagin classes of symplectic bundles and the calculation of the cohomology of quaternionic Grassmannian bundles are intertwined, and we do not see how to separate one from the other. We leave for other papers the discussion of symplectic orientations on specific nonoriented cohomology theories like hermitian K-theory, derived Witt groups and symplectic algebraic cobordism, as well as discussion of the 2 and 4-valued formal group laws obeyed by the symplectic Pontryagin classes.

Acknowledgements. The authors are grateful for the hospitality of Paul Balmer, Max-Albert Knus and ETH Zürich where the project began. They warmly thank Alexander Nenashev for his interest: §12 is joint work.

2. Cohomology theories

We review the notions of a cohomology theory and a ring cohomology theory as used in [14].

We fix a base scheme k. We will study ring cohomology theories on some category 𝒱 of k-schemes. The category could be nonsingular quasi-projective varieties over k = Spec F with F a field, it could be nonsingular quasi-affine varieties over k = Spec F, it could be quasicompact semi-separated schemes with an ample set of line bundles over k, it could be regular noetherian separated schemes of finite Krull dimension over k, or it could be something else.

Which it is is unimportant as long as the definitions make sense and the constructions work. There are only some minor complications concerning deformation to the normal bundle when we go beyond smooth varieties over a field. Our theorems all hold if 𝒱 has the following properties (partly borrowed from Levine-Morel [9, (1.1.1)]):

(1) All schemes in 𝒱 are quasi-compact and quasi-separated and have an ample family of line bundles.

(2) The schemes k and ∅ are in 𝒱.

(3) If X and Y are in 𝒱, then so are X ⊔ Y and X ×_{k} Y.

(4) If X is in 𝒱, and U ⊂ X is a quasi-compact open subscheme, then U is in 𝒱.

(5) If X is in 𝒱, and Y is a quasi-compact open subscheme of a Grassmannian bundle over X for which the projection map Y → X is an affine morphism, then Y is in 𝒱.

Some of our discussions of deformation to the normal bundle and of direct images make more sense if a sixth property is also true. Recall that a regular embedding Z → X is a closed embedding such that locally Z is cut out by a regular sequence.

(6) For a regular embedding Z → X with X and Z in 𝒱 the deformation to the normal bundle space D(Z, X) of §6 is in 𝒱.

We will use our language imprecisely as if all schemes were noetherian. Thus when we write “open subscheme” we really mean a quasi-compact open subscheme, and “closed subset” means the complement of a quasi-compact open subscheme.

The condition that every scheme in 𝒱 have an ample family of line bundles is used in the proof of the symplectic splitting principle (Theorem 10.2).

Definition 2.1 ([14, Definition 2.1]). A cohomology theory on a category 𝒱 of k-schemes is a pair (A, ∂) with A a functor assigning an abelian group A_{Z}(X) to every scheme X and closed subset Z ⊂ X with X and X ∖ Z in 𝒱 and assigning a morphism of abelian groups f^{A}: A_{Z}(X) → A_{Z′}(X′) to every map f: X′ → X such that Z′ ⊃ f^{−1}(Z). One writes A(X) = A_{X}(X). In addition one has a morphism of functors ∂: A(X ∖ Z) → A_{Z}(X). Together they have the following properties.

(1) Localization: The functorial sequences

A(X ∖ Z) ⟶^{∂} A_{Z}(X) ⟶^{e^{A}} A(X) ⟶^{j^{A}} A(X ∖ Z) ⟶^{∂} A_{Z}(X),

with e^{A} and j^{A} the appropriate maps of A, are exact.

(2) Etale excision: f^{A}: A_{Z}(X) → A_{Z′}(X′) is an isomorphism if f: X′ → X is étale, Z′ = f^{−1}(Z), and f|_{Z′}: Z′ → Z is an isomorphism,

(3) Homotopy invariance: the maps pr_{1}^{A}: A(X) → A(X × 𝔸^{1}) are isomorphisms.

Cohomology theories have Mayer-Vietoris sequences and satisfy A_{∅}(X) = 0. They have homotopy invariance for 𝔸^{n}-bundles (torsors for vector bundles). Deformation to the normal bundle isomorphisms will be discussed in §6. See [14, §2.2] for these and other properties.

Zariski excision suffices for the main results of the paper. Excision is used mainly for Mayer-Vietoris and for direct images, but we use direct images only for one Grassmannian embedded in another, and then there are global Zariski tubular neighborhoods.

Definition 2.2 ([14, Definition 2.13]). A ring cohomology theory is a cohomology theory in the sense of Definition 2.1 with cup products

∪: A_{Z}(X) × A_{W} (X) → A_{Z∩W} (X)

which are functorial, bilinear and associative and have two other properties:

(1) There exists an element 1 ∈ A(k) such that for every scheme π_{X}: X → k in 𝒱 and every closed subset Z ⊂ X, the pullback 1_{X} = π_{X}^{A}(1) ∈ A(X) satisfies 1_{X} ∪ a = a ∪ 1_{X} = a for all a ∈ A_{Z}(X).

(2) For the maps ∂: A(X ∖ Z) → A_{Z}(X) one has ∂(a ∪ b) = ∂a ∪ b for all a ∈ A(X ∖ Z) and all b ∈ A(X).

3. Basic geometry of HP^{n}

We define HP^{n} and discuss a stratification resembling the cell decomposition of the topological HP^{n}. We also present quaternionic projective bundles, Grassmannians and flag varieties.

Let (V, ϕ) be a trivial symplectic bundle of rank 2n + 2 over the base scheme k. The symplectic group Sp_{2n+2} = Sp(V, ϕ) acts on the Grassmannian Gr(2, V) with (i) a closed orbit GrSp(2, V, ϕ) parametrizing 2-dimensional subspaces U ⊂ V with ϕ|_{U} ≡ 0, and (ii) a complementary open orbit parametrizing 2-dimensional subspaces U ⊂ V with ϕ|_{U} nondegenerate which we will call the quaternionic projective space HP^{n}. We will use this object as a motivic analogue of the topological ℍℙ^{n}. Determining the stabilizer of a point of HP^{n} yields an identification HP^{n} = Sp_{2n+2} /(Sp_{2} × Sp_{2n}), which compares well with ℍℙ^{n} = U_{n+1}(ℍ)/ U_{1}(ℍ) × U_{n}(ℍ). So the manifold HP^{n}(ℂ) of complex points is the complexification of the quotient of compact Lie groups ℍℙ^{n} and has the same homotopy type. It does not have the homotopy type of a complex projective manifold.

The topological ℍℙ^{n} is the union of cells of dimensions 0, 4, 8, …, 4n. A related decomposition of the space HP^{n} may be defined. Fix a flag

0 = E_{0} ⊂ E_{1} ⊂ ⋯ ⊂ E_{n+1} = E_{n+1}^{⊥} ⊂ ⋯ ⊂ E_{1}^{⊥} ⊂ E_{0}^{⊥} = V  (3.1)

of subbundles of (V, ϕ) with the E_{i} ≅ 𝒪_{k}^{⊕i} totally isotropic and satisfying dim E_{i} = i and dim E_{i}^{⊥} = 2n + 2 − i. Set

\overline{X}_{2i} = Gr(2, E_{i}^{⊥}) ∩ HP^{n},  X_{2i} = \overline{X}_{2i} ∖ \overline{X}_{2i+2}.  (3.2)

Choose for convenience a lagrangian supplementary to the lagrangian E_{n+1}, and let GL_{n+1} ⊂ Sp_{2n+2} be the subgroup fixing the two lagrangians.

Theorem 3.1. The scheme HP^{n} is the disjoint union of the locally closed strata

HP^{n} = ⨆_{i=0}^{n} X_{2i} = X_{0} ⊔ X_{2} ⊔ ⋯ ⊔ X_{2n},  (3.3)

which have the following properties:

(a) The scheme X_{2i} and its closure \overline{X}_{2i} = X_{2i} ⊔ X_{2i+2} ⊔ ⋯ ⊔ X_{2n} are smooth of relative dimension 4n − 2i over the base scheme k. The \overline{X}_{2i} and X_{2n} = \overline{X}_{2n} are affine over the base, but X_{0}, …, X_{2n−2} are not. We have Pic(\overline{X}_{2i}) = Pic(k) and 𝒪(\overline{X}_{2i})^{×} = 𝒪(k)^{×}.

(b) Each \overline{X}_{2i} is the transversal intersection of i translates of \overline{X}_{2} under the action of the subgroup GL_{n+1} ⊂ Sp_{2n+2}.

(c) The intersection of n + 1 general translates of \overline{X}_{2} under the action of GL_{n+1} is empty.

In particular, since \overline{X}_{0} = HP^{n} we have Pic(HP^{n}) = Pic(k) and 𝒪(HP^{n})^{×} = 𝒪(k)^{×}.

Proof. (a) Let 𝒰_{Gr} be the tautological rank 2 subbundle on Gr = Gr(2, V). Then GrSp(2, V, ϕ) is the zero locus of a section of Λ^{2}𝒰_{Gr}^{∨} ≅ 𝒪_{Gr}(1) induced by ϕ. So HP^{n} is the complement in the smooth projective scheme Gr of an irreducible ample divisor whose class generates Pic(Gr)/ Pic(k) ≅ ℤ. It follows that HP^{n} is smooth and affine of the same dimension 4n as the Grassmannian, and that it satisfies Pic(HP^{n})/ Pic(k) = 0 and 𝒪(HP^{n})^{×} = 𝒪(k)^{×}.

The same argument applied to Gr(2, E_{i}^{⊥}) shows that the nonempty \overline{X}_{2i} are smooth and affine of the same dimension 4n − 2i as Gr(2, E_{i}^{⊥}) and satisfy Pic(\overline{X}_{2i})/ Pic(k) = 0 and 𝒪(\overline{X}_{2i})^{×} = 𝒪(k)^{×}. Moreover, \overline{X}_{2i} is empty if and only if E_{i}^{⊥} is totally isotropic, and this is true only for i = n + 1. This implies \overline{X}_{2n+2} = ∅ and HP^{n} = ⨆_{i=0}^{n} X_{2i} and that X_{2n} = \overline{X}_{2n} is affine. But the other X_{2i} are obtained by removing a nonempty closed subscheme of codimension 2 from a smooth affine scheme, and such schemes are never affine.

(b) If one chooses g_{1}, …, g_{i} ∈ GL_{n+1} ⊂ Sp_{2n+2} satisfying ⨁_{j=1}^{i} g_{j}(E_{1}) = E_{i} then one has \overline{X}_{2i} = ⋂_{j=1}^{i} g_{j}(\overline{X}_{2}) because both parametrize subspaces U ⊂ ⋂ g_{j}(E_{1})^{⊥} = E_{i}^{⊥} with ϕ|_{U} ≢ 0. Moreover, the intersection is transversal because the map ⨁_{j=1}^{i} N_{g_{j}(\overline{X}_{1})/ HP^{n},[U]}^{∨} → T_{HP^{n},[U]}^{∨}

which needs to be injective can be identified with the natural map ⨁_{j=1}^{i}(U ⊗ g_{j}(E_{1})) → U ⊗ U^{⊥}.

(c) Idem with i = n + 1. □

We have two other results which make the stratification (3.3) of HP^{n} resemble the cell decomposition of ℍℙ^{n}. We will prove them later, but we state them now so they don’t get lost amid more general results about quaternionic Grassmannian bundles.

Theorem 3.2. There is a natural map q: \overline{X}_{2i} → HP^{n−i} = HP(E_{i}^{⊥}/E_{i}, \overline{ϕ}) which is an 𝔸^{2i}- bundle. The stratification \overline{X}_{2i} = X_{2i} ⊔ X_{2i+2} ⊔ ⋯ ⊔ X_{2n} is the inverse image under q of the stratification HP^{n−i} = X′_{0} ⊔ X′_{2} ⊔ ⋯ ⊔ X′_{2n−2i} associated to the flag of subspaces of E_{i}^{⊥}/E_{i} induced by (3.1). The pullbacks to \overline{X}_{2i} of the two tautological symplectic subbundles are naturally isometric q^{∗}(𝒰_{HP}_{n−i}, \overline{ϕ}|_{𝒰_{_{HP}n−i}}) ≅ (𝒰_{HP}^{n}, ϕ|_{𝒰_{HP}^{n}})|_{\overline{X}_{2i}}.

Corollary 3.3. There is a natural isomorphism \overline{X}_{2n} = X_{2n} ≅ 𝔸^{2n}.

Theorem 3.4. (a) The stratum X_{2i} is the quotient of the free action of 𝔾_{a} on 𝔸^{4n−2i+1} = 𝔸^{i} × 𝔸^{2n−2i} × 𝔸^{i} × 𝔸^{2n−2i} × 𝔸^{1} given by

t · (α, a, β, b, r) = (α, a, β + tα, b + ta, r + t(1 − ϕ(a, b)))  (3.4)

where ϕ: 𝔸^{2n−2i} × 𝔸^{2n−2i} → 𝔸^{1} is the standard symplectic form.

(b) For any scheme S, pullback along the structural map t: X_{2i} → k induces isomorphisms (t × 1_{S})^{A}: A(S) ⟶^{≅} A(X_{2i} × S) for any cohomology theory A.

Theorem 3.2 is essentially a special case of Theorem 4.1. We will prove Theorem 3.4 in §5. Some curious things are beginning to happen. The strata in our decomposition are coho-

mological cells but not affine spaces or even affine schemes. They are of the right codimension but the wrong dimension, for dim_{ℝ} HP^{n}(ℂ) = 8n but dim_{ℝ} ℍℙ^{n} = 4n. The \overline{X}_{2i} are not copies of the subspaces HP^{n−i} but vector bundles over copies of the HP^{n−i}, and the difference in dimensions is worse: dim_{ℝ} HP^{n}(ℂ) = 8n − 4i but dim_{ℝ} ℍℙ^{n−i} = 4n − 4i. There is no problem cohomologically, but it is curious geometrically.

For a rank 2n + 2 symplectic bundle (E, ϕ) over a scheme S there is a quaternionic projective bundle HP_{S}(E, ϕ). It is the open subscheme of the Grassmannian bundle Gr_{S}(2, E) whose points over s ∈ S correspond to those U ⊂ E_{s} on which ϕ is nondegenerate. Let π: HP_{S}(E, ϕ) → S be its structure map, i.e. the natural projection to S. Then the symplectic bundle π^{∗}(E, ϕ) splits as the orthogonal direct sum

π^{∗}(E, ϕ) ≅ (𝒰, ϕ|_{𝒰}) ⊥ (𝒰^{⊥}, ϕ|_{𝒰^{⊥}})  (3.5)

of the tautological rank 2 symplectic subbundle and its orthogonal complement. This decomposition is universal in the sense that for any a morphism g: X → S any orthogonal direct sum decomposition g^{∗}(E, ϕ) ≅ (F_{1}, ψ_{1}) ⊥ (F_{2}, ψ_{2}) with rk F_{1} = 2 is the pullback along a unique morphism f: X → HP_{S}(E, ϕ) of the universal decomposition (3.5). The map f is said to classify either the decomposition or the rank 2 symplectic subbundle (F_{1}, ψ_{1}) ⊂ g^{∗}(E, ϕ).

We define the quaternionic Grassmannians and partial flag varieties as the quotient varieties

HGr(r, n) = Sp_{2n} /(Sp_{2r} × Sp_{2n−2r}),
HFlag(a_{1}, …, a_{r}; n) = Sp_{2n} /(Sp_{2a_{1}} × ⋯ × Sp_{2a_{r}} × Sp_{2n−∑ 2a_{i}}).

The second family of schemes includes the first, so we discuss it. Over HFlag(a_{1}, …, a_{r}; n) there are r + 1 universal symplectic subbundles (𝒰_{1}, ϕ_{1}), …, (𝒰_{r}, ϕ_{r}), (𝒱, ψ) with rk 𝒰_{i} = 2a_{i} and rk 𝒱 = 2n − ∑ 2a_{i} and a canonical decomposition of the trivial symplectic bundle of rank 2n into their orthogonal direct sum

(V, ϕ) ⊗ 𝒪 ≅ (𝒰_{1}, ϕ_{1}) ⊥ ⋯ ⊥ (𝒰_{r}, ϕ_{r}) ⊥ (𝒱, ψ)

Moreover, any decomposition over a scheme S of the trivial symplectic bundle (V, ϕ)⊗𝒪_{S} into an orthogonal direct sum of symplectic subbundles of the appropriate ranks is the pullback of this universal decomposition along a unique morphism S → HFlag(a_{1}, …, a_{r}; n).

Now write ℱ_{i} = ⨁_{j=1}^{i} 𝒰_{i}. We then have a filtration

0 ⊂ ℱ_{1} ⊂ ℱ_{2} ⊂ ⋯ ⊂ ℱ_{r} ⊂ V ⊗ 𝒪  (3.6)

Clearly HFlag(a_{1}, …, a_{r}; n) parametrizes flags of subspaces 0 ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} ⊂ V of the appropriate dimensions such that ϕ|_{F_{i}} is nondegenerate for all i. So it is an open subscheme of a flag variety.

Theorem 3.5. The quaternionic flag varieties are dense open subschemes of the flag varieties

HFlag(a_{1}, …, a_{r}; n) ⊂ Flag(2m_{1}, 2m_{2}, …, 2m_{r}; 2n).

with m_{i} = ∑_{j=1}^{i} a_{j}. They are smooth and affine of relative dimension 4n ∑_{i} a_{i} − 4 ∑_{i≤j} a_{i}a_{j} over the base k, and they satisfy H^{0}(HFlag, 𝒪^{×}) = 𝒪(k)^{×} and Pic(HFlag) = Pic(k).

The proof is essentially the same as that of Theorem 3.1.

4. Normal bundles of sub-Grassmannians

For ordinary Grassmannians, there are closed embeddings Gr(r, n − 1) ↪ Gr(r, n), and the complement of the image is isomorphic to a vector bundle over Gr(r − 1, n − 1). For the cohomology there is then a long exact sequence

⋯ → A_{Gr(r,n−1)}(Gr(r, n)) → A(Gr(r, n)) → A(Gr(r − 1, n − 1)) → · · · .  (4.1)

Moreover, the normal bundle N of Gr(r, n − 1) in Gr(r, n) embeds as an open subvariety of Gr(r, n), and so excision gives an isomorphism A_{Gr(r,n−1)}(Gr(r, n)) ≅ A_{Gr(r,n−1)}(N).

We will show that something similar happens for the cohomology of quaternionic Grassmannians but with many differences in the details.

We work in a relative situation. Our basic setup is as follows. Suppose (E, ϕ) is a symplectic bundle of rank 2n − 2 over S, and let (F, ψ) be the symplectic bundle of rank 2n with

F = 𝒪_{S} ⊕ E ⊕ 𝒪_{S},  ψ = (\begin{matrix}0 & 0 & 1 \\ 0 & ϕ & 0 \\ −1 & 0 & 0 \end{matrix}) .  (4.2)

We will consider the natural embedding of HGr_{S}(r; E, ϕ) in HGr_{S}(r; F, ψ). Let (𝒰_{E}, ϕ_{𝒰_{E}}) and (𝒰_{F} , ψ_{𝒰_{F}}) be the tautological symplectic subbundles of rank 2r. We will abbreviate

HGr(E) = HGr_{S}(r; E, ϕ)  HGr(F) = HGr_{S}(r; F, ψ).  (4.3)

We now study the embedding of HGr(E) in HGr(F).

Theorem 4.1. (a) The normal bundle N of HGr(E) in HGr(F) has a canonical embedding as an open subscheme of Gr_{S}(2r; F) overlapping the open subscheme HGr_{S}(r; F, ψ).

(b) The subschemes N^{+} = HGr(F)∩Gr_{S}(2r; 𝒪_{S} ⊕E) and N^{−} = HGr(F)∩Gr_{S}(2r; E ⊕𝒪_{S}) are closed in HGr(F) and are subbundles of N with N = N^{+} ⊕ N^{−}. We have N^{+} ∩ N^{−} = HGr(E).

(c) There are natural vector bundle isomorphisms N^{+} ≅ 𝒰_{E} and N^{−} ≅ 𝒰_{E}.

(d) There is a natural section s_{+} of 𝒰_{F} intersecting the zero section transversally in N^{+} and similarly for N^{−}.

(e) Let π_{+} : N^{+} → HGr(E) be the structural map. Then π_{+}^{∗}(𝒰_{E}, ϕ|_{𝒰_{E}}) is isometric to (𝒰_{F} , ψ|_{𝒰_{F}})|_{N^{+}} and similarly for N^{−}.

Actually in (b) and (c) the truly natural isomorphisms are N^{+} ≅ 𝒰_{E}^{∨} and N^{−} ≅ 𝒰_{E}^{∨}, while s_{+} is naturally a section of 𝒰_{F}^{∨}. But since 𝒰_{E} and 𝒰_{F} are symplectic, this does not matter.

When S is k, and (E, ϕ) is the trivial hyperbolic symplectic bundle h(𝒪_{k}^{⊕n−1}), then we are in the situation of Theorem 3.2 with HGr(F) = HP^{n−1} and N^{+} = \overline{X}_{2} and HGr(E) = X′_{0}.

Proof. (a) The normal bundle N of HGr(E) in HGr(F) is isomorphic to 𝒰_{E}^{∨}⊗F/E ≅ 𝒰_{E}^{∨}⊗𝒪^{⊕2}. Therefore N has a universal property: to give a map T → N one gives a map t: T → S plus a symplectic subbundle i: U ⊂ t^{∗}E of rank 2r plus a morphism (α_{1}, α_{2}): U → 𝒪_{T}^{⊕2}. Giving such data is the same as giving t: T → S plus a rank 2r subbundle (α_{1}, i, α_{2}): U ↪ 𝒪_{T} ⊕ t^{∗}E ⊕ 𝒪_{T} = t^{∗}F such i^{∨}ϕi is everywhere of maximal rank. So N is naturally isomorphic to an open subscheme of Gr_{S}(2r; F).

(b)(e) To give a morphism T → N^{+} one gives a map t: T → S plus a subbundle of rank 2r of the form (α_{1}, i, 0): U ↪ 𝒪_{T} ⊕t^{∗}E ⊕𝒪_{T} = t^{∗}F such that (α_{1}, i, 0)^{∨}ψ(α_{1}, i, 0) is nonsingular. That is equivalent to giving t: T → S and i: U ↪ t^{∗}E such that i^{∨}ϕi is nonsingular plus α_{1} : U → 𝒪_{T} ⊕ 0. So N^{+} is the subbundle 𝒰_{E}^{∨} ⊗ (𝒪_{S} ⊕ 0) of N = 𝒰_{E}^{∨} ⊗ 𝒪_{S}^{⊕2}. Similarly N^{−} = 𝒰_{E}^{∨} ⊗ (0 ⊕ 𝒪_{S}). The other assertions of (b) are clear. The two presentations have the same bundle U and the same form (α_{1}, i, 0)^{∨}ψ(α_{1}, i, 0) = i^{∨}ϕi. Hence (e).

(c) By construction we have N^{+} ≅ 𝒰_{E}^{∨}. But 𝒰_{E} is symplectic, so 𝒰_{E}^{∨} ≅ 𝒰_{E}.

(d) For 𝒰_{F} we have the inclusion (α_{1}, i, α_{2}): 𝒰_{F} ↪ 𝒪_{S} ⊕ E ⊕ 𝒪_{S} = F. The scheme N^{+} is the zero locus of α_{2} or equivalently of α_{2}^{∨}: 𝒪_{S} → 𝒰_{F}^{∨} ≅ 𝒰_{F}. □

In the proof of Theorem 12.6 we will use the following subtle point about the geometry of Theorem 4.1. The tautological section of π_{−}^{∗}N_{−} = N_{−} ×_{HGr(E)} N_{−} ⟶^{p_{2}} N_{−} is the diagonal Δ.

Lemma 4.2. The isomorphism 𝒰_{F} |_{N^{−}} ≅ π_{−}^{∗}𝒰_{E} of (d) and the pullback π_{−}^{∗}𝒰_{E} ≅ π_{−}^{∗}N^{−} of the isomorphism of (b) identifies the restriction s_{+}|_{N^{−}} of the section of (c) with the tautological section of π_{−}^{∗}N^{−}.

Both are α_{2}^{∨}. We leave the details to the reader.

Proposition 4.3. In the situation of Theorem 4.1 let f: N^{−} ↪ HGr(F) be the inclusion. Then for any cohomology theory A the map f^{A}: A_{N^{+}}(HGr(F)) → A_{HGr(E)}(N^{−}) is an isomorphism.

Proof. We have a direct sum N = N^{+} ⊕ N^{−} of vector bundles over HGr(E), so the pullback and its left inverse the restriction map A_{N^{+}}(N) → A_{HGr(E)}(N^{−}) are isomorphisms by homotopy invariance. Moreover, N^{+} is a closed subscheme of both N and of HGr(F), which are both open subschemes of Gr_{S}(2r; F). So the diagram

[figure: A_{N^{+}}(N ∩ HGr(F)); excision; ≅; excision; ≅; A_{N^{+}}(HGr(F)); f^{A}; A_{N^{+}}(N); v.b.; ≅; A_{HGr(E)}(N^{−})]  (4.4)

commutes, and the top and left arrows are isomorphisms by Zariski excision. Hence f^{A} is also an isomorphism. □

5. Geometry of the open stratum

In this section we study the geometry and cohomology of the open stratum in the localization sequence for quaternionic sub-Grassmannians. Our main results are Theorems 5.1 and 5.2. We include a proof of Theorem 3.4 as a special case of the latter. A second proof is Theorem 15.1.

Recall the basic setup of (4.2). We have a symplectic bundle (E, ϕ) of rank 2n − 2 over S, and let (F, ψ) is of rank 2n with F = 𝒪_{S} ⊕ E ⊕ 𝒪_{S} and ψ the orthogonal direct sum of ϕ and the hyperbolic symplectic form. We will consider the natural embedding of HGr_{S}(r; E, ϕ) in HGr_{S}(r; F, ψ). Let λ: F ↠ 𝒪_{S} be the projection onto the third factor with kernel 𝒪_{S} ⊕ E. For r ≤ n − 1 let

N^{+} = HGr_{S}(r, F, ψ) ∩ Gr_{S}(2r, 𝒪_{S} ⊕ E),  Y = HGr_{S}(r, F, ψ) ∖ N^{+}.  (5.1)

with N^{+} ⊂ HGr_{S}(r, F, ψ) closed of codimension 2r and Y open. Let (𝒰_{r}, ψ_{r}) be the tautological symplectic subbundle of rank 2r on HGr(r, F, ψ), and let (𝒰_{r−1}, ϕ_{r−1}) be the tautological symplectic subbundle of rank 2r − 2 on HGr(r − 1, E, ϕ). In this section we will investigate the open stratum Y.

Theorem 5.1. In the situation of (4.2) and (5.1) we have morphisms over S

Y ⟵^{g_{1}} Y_{1} ⟵^{g_{2}} Y_{2} ⟶^{q} HGr_{S}(r − 1, E, ϕ)

with g_{1} an 𝔸^{2r−1}-bundle, g_{2} an 𝔸^{2r−2}-bundle, and q an 𝔸^{4n−3}-bundle. Moreover, g_{2}^{∗}g_{1}^{∗}𝒰_{r} has two tautological sections e, f over Y_{2} satisfying the properties

λ(f) = 1,  λ(e) = 0,  ψ(e, f) = 1,  (5.2)

and writing π: Y_{2} → S for the projection, there is a symplectic automorphism ρ of π^{∗}(F, ψ) = (𝒪_{Y_{2}} ⊕ π^{∗}E ⊕ 𝒪_{Y_{2}}, ψ) with ρ(1, 0, 0) = e and ρ(0, 0, 1) = f and an orthogonal direct sum of symplectic subbundles of π^{∗}(F, ψ)

g_{2}^{∗}g_{1}^{∗}𝒰_{r} = ⟨e, f⟩ ⊥ ρ(q^{∗}𝒰_{r−1}) ⊂ π^{∗}F.  (5.3)

In (5.2) the bilinear form is ψ(f, e) = f^{∨}ψe for ψ: F → F^{∨}.

Theorem 5.2. In the situation of Theorem 5.1 with Y = HGr_{S}(r, F, ψ) ∖ N^{+} the open locus of (5.1), the following hold for any cohomology theory A.

(a) For r = 1 let t: Y → S be the projection map. Then t^{A}: A(S) → A(Y) is an isomorphism.

(b) For 1 ≤ r ≤ n let

HGr_{S}(r − 1, E, ϕ) ⟶^{σ} Y ⊂ HGr_{S}(r, F, ψ)

be the map classifying the rank 2r symplectic subbundle 𝒪 ⊕ 𝒰_{r−1} ⊕ 𝒪 of the pullback of F. Then the map σ^{A}: A(Y) → A(HGr_{S}(r − 1, E, ϕ)) is an isomorphism.

The localization exact sequence

⋯ → A_{N^{+}}(HGr(F)) → A(HGr(F)) → A(HGr(F) ∖ N^{+}) → ⋯  (5.4)

combines with Proposition 4.3 and Theorem 5.2 to give the following result.

Corollary 5.3. In the situation of (4.2)–(5.1) for any cohomology theory A the localization sequence for the closed embedding N^{+} ↪ HGr(r, F, ψ) and the complementary open embedding j: Y ↪ HGr(r, F, ψ) is isomorphic to

⋯ → A_{HGr(r,E,ϕ)}(𝒰_{E}) ⟶^{e^{A}(f^{A})^{−1}} A(HGr(r, F, ψ)) ⟶^{σ^{A}j^{A}} A(HGr(r − 1, E, ϕ)) ⟶^{f^{A}∂(σ^{A})^{−1}} ⋯

(5.5)

where 𝒰_{E} is the tautological rank 2r symplectic subbundle on HGr(r, E, ϕ), and f is as in Proposition 4.3, σ as in Theorem 5.2, and e^{A} is the extension of supports operator.

We prove a series of lemmas before proving the theorems.

Lemma 5.4. Let U be a vector bundle of rank m over S with a quotient line bundle λ: U ↠ 𝒪_{S} with a fixed trivialization. Then the locus

Y = {f ∈ U | λ(f) = 1} ⊂ U

is an 𝔸^{m−1}-bundle over S which has a section if and only if λ is split. Moreover, giving a morphism T → Y is equivalent to giving a pair (g, f_{T}) with g: T → S a morphism, and f_{T} a section of g^{∗}U with λ(f_{T}) = 1.

Proof. The subbundle ker λ acts on U by translation in the fibers, and this action preserves Y. Over any open subscheme S_{α} ⊂ S where λ has a local splitting σ_{α}, one has Y |_{S_{α}} = (ker λ)|_{S_{α}} + σ_{α}(1), so Y → S is a torsor for ker λ. A global section S → Y is by construction a section s of U with λ(s) = 1. It exists if and only if λ is split. Finally, giving a morphism T → U is equivalent to giving a morphism g: T → S plus a section f_{T} of g^{∗}U because of the cartesianness of

[figure: g^{∗}U; □; U; T; g; f_{T}; S]

The image of T → U lies in Y ⊂ U if and only if λ(f_{T}) = 1. □

Lemma 5.5. Suppose that (U, ψ) is a rank 2s symplectic bundle over S with a quotient line bundle λ: U ↠ 𝒪_{S} with a fixed trivialization. Let

Y_{1} = {f ∈ U | λ(f) = 1} ⊂ U,
Y_{2} = {(e, f) ∈ U ⊕ U | λ(f) = 1 and λ(e) = 0 and ψ(e, f) = 1} ⊂ U ⊕ U.

Then the projection map Y_{1} → S is an 𝔸^{2s−1}-bundle which has a section if and only if λ is split. The projection map Y_{2} → Y_{1} is an 𝔸^{2s−2}-bundle which has a section, and when λ is split the composition Y_{2} → Y_{1} → S is an 𝔸^{4s−3}-bundle with a section. Moreover, giving a morphism T → Y is equivalent to giving a triple (g, e_{T} , f_{T}) with g: T → S a morphism and e_{T} , f_{T} sections of g^{∗}U satisfying the three conditions (5.2).

Writing π: Y_{2} → S for the projection, the universal property implies the existence of tautological sections e, f of π^{∗}U satisfying the three conditions and such that the triples (g, e_{T} , f_{T}) of the lemma are the pullbacks of the tautological (π, e, f) along the classifying map T → Y_{2}.

Proof. The statements about Y_{1} follow from immediately from the previous lemma. Let i: ker λ → U be the inclusion, and h: Y_{1} → S the projection map. Over Y_{1} we have morphisms

\xymatrix{𝒪_{Y_{1}} ⊕ h^{∗}(ker λ) ⟶_{≅}^{(f,i)} h^{∗}U \ar[r]_{≅}^{ψ} & h^{∗}U^{∨} \ar@{->>}[r]^{f^{∨}} & 𝒪_{Y_{1}}.}

The map f^{∨} is surjective because f: 𝒪_{Y_{1}} → h^{∗}U is nowhere vanishing. We have f^{∨}ψf = 0 since ψ is alternating. So f^{∨}ψi: h^{∗}(ker λ) ↠ 𝒪_{Y_{1}} is surjective. Applying the previous lemma to f^{∨}ψi gives Y_{2}. The projection Y_{2} → Y_{1} is always split because by the nondegeneracy of ψ there exists a nowhere vanishing section e_{0} of U such that λ(?) = ψ(e_{0},?), and −e_{0} gives a splitting of f^{∨}ψi.

When λ is split by a section f_{0} : 𝒪_{S} → U, then the two sections e_{0}, f_{0} form a hyperbolic plane, and setting E = ⟨e_{0}, f_{0}⟩^{⊥} and ϕ = ψ|_{E}, we find that up to isometry we have U = 𝒪_{S} ⊕ E ⊕ 𝒪_{S}, with ψ as in (4.2), and with λ: U ↠ 𝒪_{S} the third projection. By construction Y_{2} ⊂ U ⊕ U is the locus of points (e, f) satisfying the three conditions (5.2). The conditions λ(f) = 1 and λ(e) = 0 imply that e and f are of the forms e = (a, u, 0) and f = (b, v, 1) in terms of the direct sum decomposition. The condition ψ(e, f) = 1 is a + ϕ(u, v) = 1, so we get e = (1 − ϕ(u, v), u, 0). So Y_{2} is the image of the vector bundle E ⊕ E ⊕ 𝒪_{S} under the map sending (u, v, b) ↦ ((1 − ϕ(u, v), u, 0), (b, v, 1)). So it is an 𝔸^{4s−3}-bundle with a section.

Finally, applying Lemma 5.4 twice we see that giving a morphism T → Y_{2} is equivalent to giving a pair (g_{1}, e) with g_{1} : T → Y_{1} a morphism and e a section of g_{1}^{∗}h^{∗}(ker λ) such that f^{∨}ψi(e) = ψ(i(e), f) = 1, and giving g_{1} is equivalent to giving (g, f) with g: T → S a morphism and f a section of g^{∗}U such that λ(f) = 1. Since in these equivalences one has g = hg_{1}, we see that giving T → Y_{2} is equivalent to giving (g, e, f) with e, f sections of g^{∗}U satisfying (5.2). □

Suppose now that we have (E, ϕ) and (F, ψ) as in (4.2). Let Sp_{S}(F, ψ) → S be the group scheme of automorphisms of (F, ψ). Its sections correspond to endomorphisms α: F → F such that α^{∨}ψα = ψ. There are two maps ρ_{+} : E ⊕ 𝒪_{S} → Sp_{S}(F, ψ) and ρ_{−}: E ⊕ 𝒪_{S} → Sp_{S}(F, ψ) with

ρ_{+}(e, s) = (\begin{matrix}1_{𝒪_{S}} & e^{∨}ϕ & s \\ 0 & 1_{E} & e \\ 0 & 0 & 1_{𝒪_{S}} \end{matrix} ) ,  ρ_{−}(e_{1}, s_{1}) = (\begin{matrix}1_{𝒪_{S}} & 0 & 0 \\ e_{1} & 1_{E} & 0 \\ s_{1} & −e_{1}^{∨}ϕ & 1_{𝒪_{S}} \end{matrix} )  (5.6)

These maps parametrize the unipotent radicals of the parabolic subgroups of Sp_{S}(F, ψ) associated with the two filtrations compatible with the splitting (4.2).

Lemma 5.6. In the above situation let λ = (0, 0, 1): F = 𝒪_{S} ⊕ E ⊕ 𝒪_{S} ↠ 𝒪_{S}. Suppose we have a morphism π: T → S and two sections e, f of π^{∗}F with λ(f) = 1 and λ(e) = 0 and ψ(e, f) = 1. Then e and f are of the form

e = (\begin{matrix}1 − ϕ(u, v) \\ u \\ 0 \end{matrix}) ,  f = (\begin{matrix}b \\ v \\ 1 \end{matrix}) .  (5.7)

Set ρ = ρ_{+}(v, b)ρ_{−}(u, 0). Then ρ is a symplectic automorphism of π^{∗}(F, ψ) satisfying

ρ (\begin{matrix}0 \\ 0 \\ 1 \end{matrix}) = f,  ρ (\begin{matrix}1 \\ 0 \\ 0 \end{matrix}) = e,  ρ(E) = ⟨e, f⟩^{⊥}.

Proof. The conditions λ(f) = 1 and λ(e) = 0 imply that e and f can be written in the forms e = (a, u, 0) and f = (b, v, 1). The condition ψ(e, f) = 1 is a + ϕ(u, v) = 1, so we get e = (1 − ϕ(u, v), u, 0). One calculates ρ(0, 0, 1) = f and ρ(1, 0, 0) = e, and since ρ preserves the symplectic form ψ, it takes ⟨(1, 0, 0), (0, 0, 1)⟩^{⊥} = E isometrically onto ⟨e, f⟩^{⊥}. □

Proof of Theorem 5.1. Applying Lemma 5.5 with U = F and λ: F ↠ 𝒪_{S} the third projection, we get maps Z_{2} → Z_{1} → S which are 𝔸^{2n−2}- and 𝔸^{2n−1}-bundles, respectively. Moreover, since λ has a splitting, the composition is an 𝔸^{4n−3}-bundle. Over Z_{2} there are tautological sections e, f of the pullback of F with a universal property.

Over Y the restriction λ|_{𝒰_{r}}: 𝒰_{r} → 𝒪_{Y} of the third projection F ↠ 𝒪 is surjective. Hence Lemma 5.5, applied with U = 𝒰_{r} over Y, gives us maps Y_{2} → Y_{1} → Y which are 𝔸^{2r−2}- and 𝔸^{2r−1}-bundles, resp., with the first one with a section, and tautological sections e, f of the pullback of 𝒰_{r} to Y_{2} with a certain universal property.

We claim that the varieties Y_{2} and Z_{2} ×_{S} HGr_{S}(r − 1, E, ϕ) are characterized by equivalent universal properties and thus represent isomorphic functors (Sm/S)^{op} → Set. They are therefore isomorphic over S.

To give a morphism T → Y_{2} one gives (g, U_{T} , e_{T} , f_{T}) with g: T → S a morphism, U_{T} ⊂ g^{∗}F a symplectic subbundle of rank 2r, and and e_{T} , f_{T} sections of U_{T} satisfying the three conditions (5.2).

To give a morphism T → Z_{2} ×_{S} HP_{S}(r − 1, E, ϕ) one gives (g, e_{T} , f_{T} , V_{T}) with g: T → S a morphism, e_{T} , f_{T} sections of g^{∗}F satisfying the three conditions (5.2), and a symplectic subbundle V_{T} ⊂ g^{∗}E of rank 2r − 2.

Lemma 5.6 gives us a symplectic automorphism ρ of g^{∗}(F, ψ) with ρ(1, 0, 0) = e_{T} and ρ(0, 0, 1) = f_{T} and ρ(E) = ⟨e_{T} , f_{T} ⟩^{⊥}. It follows that the data determining morphisms from T to the two varieties determine each other using the formulas

U_{T} = ⟨e_{T} , f_{T} , ρ(V_{T})⟩,  V_{T} = ρ^{−1}(U_{T}) ∩ E.

So Y_{2} and Z_{2} ×_{S} HP_{S}(r − 1, E, ϕ) are isomorphic over S.

The last sentence of the theorem is a description of these data in the universal case T = Y_{2} ≅ Z_{2} ×_{S} HP_{S}(r − 1, E, ϕ). □

Proof of Theorem 5.2. (b) Let π_{r−1} : HGr_{S}(r − 1, E, ϕ) → S be the projection. According to the proof we have just given of Theorem 5.1 there is a morphism

s: HGr_{S}(r − 1, E, ϕ) → Z_{2} ×_{S} HGr_{S}(r − 1, E, ϕ)

corresponding to the data (π_{r−1}, (1, 0, 0), (0, 0, 1), 𝒰_{r−1}). It is a section of the second projection. It composition with the isomorphism and projections

HGr_{S}(r − 1, E, ϕ) ⟶^{s} Z_{2} ×_{S} HGr_{S}(r − 1, E, ϕ) ≅ Y_{2} ⟶^{g_{2}} Y_{1} ⟶^{g_{1}} Y

is the map classified by the symplectic subbundle 𝒪 ⊕ 𝒰_{r−1} ⊕ 𝒪 ⊂ π_{r−1}^{∗}F, which is the σ of the statement of the theorem. By Theorem 5.1 g_{1} and g_{2} are affine bundles, and s is the section of an affine bundle. So g_{1}^{A}, g_{2}^{A}, s^{A} and their composition σ^{A} are isomorphisms in any cohomology theory A.

(a) For r = 1 we have HGr_{S}(0, E, ϕ) = S, and σ: S → Y is a section of t: Y → S. By (b) σ^{A} is an isomorphism, and we have 1_{S}^{A} = t^{A}σ^{A}, so t^{A} is an isomorphism. □

We conclude this section by proving Theorem 3.4.

Lemma 5.7. The fibration Y_{1} → Y of Lemma 5.5 is a torsor under the unipotent group scheme R_{u} ⊂ Sp(U, ψ) of automorphisms of U respecting the filtration 0 ⊂ (ker λ)^{⊥} ⊂ ker λ ⊂ U and the trivialization U/ ker λ ≅ 𝒪_{Y} and inducing the identity on ker λ/(ker λ)^{⊥}.

This is easily seen when the filtration is split with U = F = 𝒪_{S} ⊕ E ⊕ 𝒪_{S} and R_{u} is the ρ_{+}(E ⊕ 𝒪_{S}) of (5.6).

We now return to the decomposition HP^{n} = ⨆_{i=0}^{n} X_{2i} of Theorem 3.1.

Proof of Theorem 3.4. (a) We first look at the open stratum X_{0}. Let (U, ψ) be the tautological rank 2 symplectic subbundle over X_{0} ⊂ HP^{n}. By Lemmas 5.5 and 5.7 the fibration Y_{2} = Y_{1} → X_{0} is a torsor under R_{u} = 𝔾_{a}, and

Y_{2} = Y_{1} = {(e, f) | 𝔸^{2n+2} × 𝔸^{2n+2} | λ(f) = 1, λ(e) = 0, ψ(e, f) = 1},

with the action of 𝔾_{a} given by t · (e, f) = (e, f + te). Writing f = (b_{1}, …, b_{2n}, b_{2n+1}, 1) and e = (a_{1}, …, a_{2n}, 1 − ∑_{j=1}^{n}(a_{2j−1}b_{2j} − a_{2j}b_{2j−1}), 0) gives the presentation of X_{0} as a quotient of a free action of 𝔾_{a} on 𝔸^{4n+1}. The argument for the other X_{2i} is similar.

(b) This follows from (a) or directly from Theorem 5.2(a). □

6. Deformation to the normal bundle

The localization sequence for Grassmannians (4.1) exists for all cohomology theories, but it is particularly well-behaved for oriented theories. (See Balmer-Calmès [4] for what can happen in a non-oriented theory.) So we should now start discussing the class of cohomology theories for which the localization sequence for quaternionic Grassmannians of Corollary 5.3 is wellbehaved. But before doing so we discuss the deformation to the normal bundle construction. Our notation follows Nenashev [13].

The deformation space D(Z, X) for a closed embedding Z ↪ X of regular schemes is the complement of the strict transform of Z × 0 in the blowup bl_{Z×0}(X × 𝔸^{1}). The inclusion and blowup maps compose to a map D(Z, X) → X×𝔸^{1}, so D(Z, X) has projections p: D(Z, X) → X and D(Z, X) → 𝔸^{1}. The fiber of D(Z, X) over 0 ∈ 𝔸^{1} is N_{Z/X}, and its fiber over 1 ∈ 𝔸^{1} is X. The strict transform of Z × 𝔸^{1} is isomorphic to Z × 𝔸^{1}, and the restriction of p to it is the first projection Z × 𝔸^{1} → Z. We thus have maps

\xymatrix{N_{Z/X} \ar[r]^{i_{0}} & D(Z, X) \ar[r]_{p} & X, \ar[l]_{i_{1}} \\ A_{Z}(N_{Z/X}) & A_{Z×𝔸^{1}}(D(Z, X)) \ar[r]^{i_{1}^{A}} \ar[l]_{i_{0}^{A}} & A_{Z}(X).}  (6.1)

The map p^{A}: A(X) → A(D(Z, X)) makes i_{0}^{A} and i_{1}^{A} into morphisms of A(X)-bimodules. Because of the blowup geometry the A(X)-bimodule structure on A_{Z}(N_{Z/X}) factors through the restriction A(X) → A(Z) ≅ A(N_{Z/X}). When the maps i_{0}^{A} and i_{1}^{A} are isomorphisms, as they frequently are, their composition

d_{Z/X}^{A}: A_{Z}(N_{Z/X}) ⟶^{≅} A_{Z}(X)  (6.2)

is the deformation to the normal bundle isomorphism. The functoriality of the space D(Z, X) makes the deformation to the normal bundle isomorphisms functorial.

\xymatrix{Z′ = Z ×_{X} X′ \ar[r]^{g′} \ar[d]_{i′} & Z \ar[d]^{i} & N_{Z′/X′} ≅ N_{Z/X} ×_{Z} Z′ \ar[r]^{g_{N}} \ar[d] & N_{X/Z} \ar[d] \\ X′ \ar[r]^{g} & X & Z′ \ar[r]^{g′} & Z}  (6.3)

Proposition 6.1. Suppose that in (6.3) we have a pullback square of schemes with i and i′ closed embeddings and that that induces a pullback square of normal bundles. Then in any cohomology theory A for which the deformation to the normal bundle isomorphisms d_{Z′/X′}^{A} and d_{Z/X}^{A} exist, we have d_{Z′/X′}^{A} ∘ g_{N}^{A} = g^{A} ∘ d_{Z/X}^{A}.

The main existence result is the following.

Theorem 6.2 ([14, Theorem 2.2]). When Z ↪ X is a closed embedding of smooth quasiprojective varieties over a field, the deformation to the normal bundle isomorphism d_{Z/X}^{A} exists for any cohomology theory A.

Dévissage theorems such as Quillen’s for K-theory or Gille’s for Witt groups [7] produce Thom isomorphisms A(Z) ≅ A_{Z}(X) directly. The compatibility of these morphisms with the i_{0}^{A} and i_{1}^{A} of (6.1) gives deformation to the normal bundle isomorphisms for all closed embedding of regular schemes for these theories.

The proof of Theorem 6.2 uses Nisnevich tubular neighborhoods which do not always exist in mixed characteristic. But we only need a simple case.

A regular closed subscheme of a regular scheme i: Z ↪ X has a Zariski tubular neighborhood if there exists a open subscheme U ⊂ X containing Z and an open embedding U ↪ N_{Z/X} such that the composition Z ↪ U ↪ N_{Z/X} is the zero section.

Proposition 6.3. If Z ↪ X has a Zariski tubular neighborhood, then the deformation to the normal bundle isomorphism d_{Z/X}^{A} exists for any cohomology theory A.

Proof. If U ⊂ Y is an open embedding with Z closed in both U and Y, then the isomorphism d_{Z/U}^{A} exists if and only if d_{Z/Y}^{A} exists because D(Z, U) is an open subscheme of D(Z, Y) in which Z × 𝔸^{1} is closed, so Zariski excision provides isomorphisms between the sources and targets of the different maps i_{0}^{A} and i_{1}^{A}.

Hence given that for the vector bundle N_{Z/X} the deformation to the normal bundle isomorphism d_{Z/N_{Z/X}} exists and is the identity by a local calculation, d_{Z/U} and d_{Z/X} also exist. □

Proposition 6.4. In the situation of Theorem 4.1 or (5.1) the deformation to the normal bundle isomorphisms d^{A}(N^{+}, HP(F, ψ)): A_{N^{+}}(HP(F, ψ)) → A_{N^{+}}(𝒰_{F} |_{N^{+}}) exist for any cohomology theory A.

Proof. First, by Theorem 4.1(a)(b) N^{+} is a direct summand of the normal bundle N = N^{+} ⊕ N^{−} of HP_{S}(E, ϕ) ⊂ HP_{S}(F, ψ), which embeds as an open subscheme N ⊂ Gr_{S}(2, F) overlapping the open subscheme HP_{S}(F, ψ). It follows that the projection map N ↠ N^{+} is also the structural map of the normal bundle of N^{+} in HP_{S}(F, ψ) and that N ∩ HP_{S}(F, ψ) is a Zariski tubular neighborhood of N^{+} in HP_{S}(F, ψ). Therefore by Proposition 6.3 the deformation to the normal bundles isomorphism d^{B}(N^{+}, HP_{S}(F, ψ)) exists.

Second, by Theorem 4.1(d) N^{+} is the transversal intersection of a section s_{+} of 𝒰_{F} with the zero section. So the normal bundle of N^{+} in HP_{S}(F, ψ) is naturally isomorphic to 𝒰_{F} |_{N^{+}}. □

7. Symplectic Thom structures

An oriented cohomology theory in the sense of Panin and Smirnov [14, Definition 3.1] is a ring cohomology theory such that for every vector bundle E → X over a nonsingular variety and closed subset Z ⊂ X there is a distinguished Thom isomorphism

th_{Z}^{E}: A_{Z}(X) → A_{Z}(E)

of A(X)-bimodules satisfying several axioms. Using deformation to the normal bundle, Thom isomorphisms induce isomorphisms A_{Z}(X) ≅ A_{Z}(Y) and in particular A(X) ≅ A_{X}(Y) for any closed embedding of smooth varieties X ↪ Y over a field for oriented cohomology theories.

Some cohomology theories have Thom isomorphisms only for vector bundles with some extra structure. For instance Balmer’s derived Witt groups [3] have Thom isomorphisms W^{i}(X) → W_{X}^{i+n}(E) for triples (E, L, λ) with E a vector bundle of rank n, L a line bundle and λ: L ⊗ L ≅ det E an isomorphism. This is because the isomorphisms depending on E alone involve a twisted Witt group W^{i}(X, det E) ≅ W_{X}^{i+n}(E) (see for instance [5, 7, 13]), and the (L, λ) are required to specify an isomorphism W^{i}(X) ≅ W^{i}(X, det E). Isometric symmetric bilinear line bundles (L, λ) induce the same isomorphism, and the isometry classes of possible (L, λ) for a given E are a torsor under H^{1}(X_{ét}, μ_{2}). The extra structure is very reminiscent of Spin and Spin^{c} structures in differential topology (e.g. [1, 2]) even to the point of having the same group H^{1}(X, ℤ/2) parametrizing the choices of Spin structure if X is a complex projective variety. Derived Witt groups might therefore be called an SL^{c}-oriented cohomology theory.

A cohomology theory is symplectically oriented if to each symplectic bundle (E, ϕ) over a variety X and each closed subset Z ⊂ X there is an isomorphism th_{Z}^{E,ϕ}: A_{Z}(X) → A_{Z}(E) satisfying the properties which we will describe in detail in Definition 14.3.

Oriented cohomology theories are also symplectically oriented. Witt groups, Witt cohomology, hermitian K-theory and oriented Chow groups (also called Chow-Witt groups or hermitian K-cohomology) are symplectically oriented as are symplectic and special linear algebraic cobordism MSp^{∗,∗} and MSL^{∗,∗} defined along the lines of Voevodsky’s [15, §6.3] algebraic cobordism MGL^{∗,∗}.

We will actually prove our quaternionic projective bundle theorem using a structure which is weaker a priori but is equivalent in the end.

Definition 7.1. A symplectic Thom structure on a ring cohomology theory A on a category of schemes is rule which assigns to each rank 2 symplectic bundle (E, ϕ) over a scheme X in the category an A(X)-central element th(E, ϕ) ∈ A_{X}(E) with the following properties:

(1) For an isomorphism u: (E, ϕ) ≅ (E_{1}, ϕ_{1}) one has th(E, ϕ) = u^{A} th(E_{1}, ϕ_{1}).

(2) For a morphism f: Y → X with pullback map f_{E}: f^{∗}E → E one has f_{E}^{A}(th(E, ϕ)) = th(f^{∗}(E, ϕ)) ∈ A_{Y} (f^{∗}E).

(3) For the trivial rank 2 bundle X ×𝔸^{2} → X with symplectic structure given by h(𝒪_{X}) = (𝒪_{X}^{⊕2}, ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )), the map ∪ th(h(𝒪_{X})): A(X) → A_{X}(X × 𝔸^{2}) is an isomorphism.

Global nondegeneracy follows from the nondegeneracy condition above via local trivializations of the symplectic bundle and a Mayer-Vietoris argument.

Proposition 7.2. Let A be a ring cohomology theory with a symplectic Thom structure. Then for any rank 2 symplectic bundle (E, ϕ) over a scheme X the map ∪ th(E, ϕ): A(X) → A_{X}(E) is an isomorphism.

For (E, ϕ) a rank 2 symplectic bundle over a scheme X, let e^{A}: A_{X}(E) → A(E) be the extension of supports map, and let z^{A}: A(E) → A(X) be the restriction to the zero section (or to any section). Define the Pontryagin class of (E, ϕ) as

p(E, ϕ) = −z^{A}e^{A}(th(E, ϕ)) ∈ A(X).  (7.1)

The sign follows the convention in differential topology (for instance Milnor-Stasheff [11]) where one has p_{i}(ξ) = (−1)^{i}c_{2i}(ξ) for a real or quaternionic vector bundle ξ. Like the symplectic Thom classes, the Pontryagin classes are A(X)-central and are functorial with respect to isomorphisms of symplectic bundles over X and with respect to pullbacks along maps X′ → X.

Now suppose that i: X → Y is a closed embedding of codimension 2 of regular schemes with a normal bundle N = N_{X/Y} equipped with a symplectic form ϕ. We then say that i: X → Y has a symplectic normal bundle (N, ϕ). We can compose the isomorphism of Proposition 7.2 with the deformation to the normal bundle isomorphism when the latter exists

i_{A,♭}: A(X) ⟶_{≅}^{∪ th(N,ϕ)} A_{X}(N) ⟶_{≅}^{d_{X/Y}} A_{X}(Y).  (7.2)

The isomorphism i_{A,♭} is a Thom isomorphism, while its composition i_{A,♮}: A(X) → A_{X}(Y) → A(Y) with the extension of supports is a direct image map. The symbols ♭ and ♮ are placeholders indicating that the maps depend on more than i and A, namely the symplectic form ϕ and the symplectic Thom structure. The dependence on ϕ is very real in certain theories. For Witt groups, replacing ϕ by aϕ with a ∈ k^{×} multiplies the direct image map by the class ⟨a⟩ ∈ W (k).

We will need several facts about the Thom isomorphisms. For a section of a rank 2 symplectic bundle (E, ϕ) the deformation to the normal bundle map is the identity. Hence:

Proposition 7.3. Let z: X → E be the zero section of a rank 2 symplectic bundle (E, ϕ) over X with symplectic normal bundle (N_{X/E}, ϕ) = (E, ϕ), then the Thom isomorphism z_{A,♭}: A(X) → A_{X}(E) coincides with ∪ th(E, ϕ).

Now suppose we have a pullback diagram

\xymatrix{X′ = X ×_{Y} Y′ \ar[r]^{i′} \ar[d]_{g′} & Y′ \ar[d]^{g} \\ X \ar[r]^{i} & Y}  (7.3)

Proposition 6.1 and the functoriality of Thom classes give us the next lemma.

Proposition 7.4. If in the pullback square (7.3) all the schemes are regular, and i and i′ are closed embeddings of codimension 2 with symplectic normal bundles (N_{X/Y} , ϕ) and (N_{X′/Y′}, g^{′∗}ϕ) ≅ (g^{′∗}N_{X/Y} , g^{′∗}ϕ) and the deformation to the normal bundle isomorphisms exist, then we have g^{A}i_{A,♭} = i′_{A,♭}g^{′A}.

The A(X)-centrality of th(N, ϕ) means that ∪ th(N, ϕ) is an isomorphism of two-sided A(X)-modules and via i^{A} also of two-sided A(Y)-modules. The deformation to the normal bundle maps ν_{X/Y} : A_{X}(Y) → A_{X}(N) are also isomorphisms of two-sided A(Y)-modules as discussed earlier. We therefore have the next proposition.

Proposition 7.5. The Thom isomorphism i_{A,♭}: A(X) → A_{X}(Y) is a two-sided A(Y)-module isomorphism, and i_{A,♭}(1) is A(Y)-central. Thus for a ∈ A_{X}(Y) and b ∈ A(Y) we have

i_{A,♭}(a ∪ i^{A}b) = i_{A,♭}(a) ∪ b,  i_{A,♭}(i^{A}b ∪ a) = b ∪ i_{A,♭}(a),
i_{A,♭}i^{A}(b) = i_{A,♭}(1) ∪ b = b ∪ i_{A,♭}(1).

We now prove some key formulas.

Proposition 7.6. Let (E, ϕ) be a rank 2 symplectic bundle over a regular scheme Y with a section s: Y → E meeting the zero section z tranversally in X. Suppose that A is a ring cohomology theory with a symplectic Thom structure, and let \overline{e}^{A}: A_{X}(Y) → A(Y) be the extension of supports map. Then we have

p(E, ϕ) = −\overline{e}^{A}s^{A}(th(E, ϕ)),  (7.4)

Moreover, if the inclusion i: X ↪ Y has a deformation to the normal bundle isomorphism, then for all b ∈ A(Y) we have

i_{A,♮}i^{A}(b) = \overline{e}^{A}i_{A,♭}i^{A}(b) = −b ∪ p(E, ϕ).  (7.5)

Proof. In the diagram

[figure: A(Y); ∪ th(E,ϕ); = z_{A,♭}; i^{A}; A_{Y} (E); s^{A}; e^{A}; A(E); s^{A}; z^{A}; A(X); i_{A,♭}; A_{X}(Y); _{\overline{e}_{A}}; A(Y); π^{A}]

the righthand rectangle commutes by functoriality. The pullbacks along the two sections of π: E → Y are left inverses of the same isomorphism π^{A}, so they satisfy s^{A} = z^{A}. We get

p(E, ϕ) = −z^{A}e^{A}(th(E, ϕ)) = −s^{A}e^{A}(th(E, ϕ)) = −\overline{e}^{A}s^{A}(th(E, ϕ)).

The lefthand rectangle of the diagram commutes using the label z_{A,♭} by Proposition 7.4, and the equality z_{A,♭} = ∪ th(E, ϕ) is Proposition 7.3. It follows that \overline{e}^{A}i_{A,♭}i^{A}(b) = i_{A,♮}i^{A}(b) is the same as \overline{e}^{A}s^{A}(b ∪ th(E, ϕ)). Since all the maps are two-sided A(Y)-modules maps, that is the same as b ∪ \overline{e}^{A}s^{A}(th(E, ϕ)) = −b ∪ p(E, ϕ) using formula (7.4). □

Proposition 7.7. For any ring cohomology theory with a symplectic Thom structure a rank 2 symplectic bundle (E, ϕ) with a nowhere vanishing section has p(E, ϕ) = 0.

Proof. The nowhere vanishing section s: Y → E meets the zero section in ∅. Since we have th(E, ϕ) ∈ A_{Y} (E), we have s^{A}(th(E, ϕ)) ∈ A_{∅}(Y) = 0. Then formula (7.4) gives p(E, ϕ) = −\overline{e}^{A}s^{A}(th(E, ϕ)) = 0. □

8. The quaternionic projective bundle theorem

We return to the situation where (V, ϕ) is a symplectic space of dimension 2n + 2, and HP^{n} = Gr(2, V) ∖ GrSp(2, V, ϕ) is the affine scheme parametrizing 2-dimensional subspaces U ⊂ V such that ϕ|_{U} is nondegenerate.

Theorem 8.1 (Quaternionic projective bundle theorem for trivial bundles). Let A be a ring cohomology theory with a symplectic Thom structure. Let (𝒰, ϕ|_{𝒰}) be the tautological rank 2 symplectic subbundle over HP^{n} and ζ = p(𝒰, ϕ|_{𝒰}) its Pontryagin class. Then for any scheme S we have A(HP^{n} ×S) = A(S)[ζ]/(ζ^{n+1}).

Proof. It is enough to consider the case S = k.

We need to show that (1, ζ, …, ζ^{n}): A(k)^{⊕(n+1)} → A(HP^{n}) is an isomorphism of two-sided A(k)-modules and that we have ζ^{n+1} = 0.

We go by induction on n. For n = 0 we have HP^{0} = k, and the tautological rank 2 symplectic subbundle of the trivial symplectic bundle of rank 2 over k is the trivial bundle. Its Pontryagin class verifies ζ = 0 by Proposition 7.7.

For n ≥ 1 we have morphisms smooth schemes by Theorem 3.1 or 4.1.

\xymatrix{N^{+} = \overline{X}_{2} \ar[r]^{i}_{closed} \ar[d]_{𝔸^{2}-bundle}^{q} & HP^{n} \ar[dr] & Y = X_{0} \ar@{^{(}->}[l]_{open}^{j} \ar[d]^{f} \\ _{HP_{n−1}} &  & k}  (8.1)

This yields a localization exact sequence and various maps of two-sided A(k)-modules.

\xymatrix{⋯ \ar[r]^{∂ [= 0]} & A_{N^{+}}(HP^{n}) \ar[r]^{e^{A}} & A(HP^{n}) \ar[r]^{j^{A}} \ar@{-->}[dl]^{i^{A}} & A(Y) \ar[r]^{∂ [= 0]} & ⋯ \\ A(HP^{n−1}) \ar[r]^{q^{A}}_{≅} & A(N^{+}) \ar[u]^{i_{A,♭}}_{≅} &  & A(k) \ar[u]_{t^{A}}^{≅} \ar@{-->}[ul]^{∪1}} (8.2)

The map t^{A}: A(k) → A(Y) is an isomorphism by Theorem 3.4 or 5.2, so j^{A} is an epimorphism split by ∪1, the boundary map ∂ vanishes, and (1, e^{A}): A(k) ⊕ A_{N^{+}}(HP^{n}) ⟶^{≅} A(HP^{n}) is an isomorphism.

The map q^{A}: A(HP^{n−1}) → A(N^{+}) is an isomorphism because q is an 𝔸^{2}-bundle by Theorem 3.2 or 4.1. The locus N^{+} ⊂ HP^{n} is the transversal intersection of the section s_{+} of the rank 2 symplectic bundle (𝒰, ψ) and the zero section. So it has a symplectic normal bundle (𝒰, ψ)|_{N^{+}}. In addition the deformation to the normal bundle isomorphisms d(N^{+}, HP^{n}) exist by Proposition 6.4. So the Thom isomorphisms i_{A,♭}: A(N^{+}) → A_{N^{+}}(HP^{n}) are defined. Writing τ = −e^{A}i_{A,♭}q^{A}, we get an isomorphism of two-sided A(k)-modules

(1, τ): A(k) ⊕ A(HP^{n−1}) ⟶^{≅} A(HP^{n}).

Write (𝒱, \overline{ϕ}|_{𝒱}) for the tautological rank 2 symplectic subbundle on HP^{n−1} and ξ = p(𝒱, \overline{ϕ}|_{𝒱}) for its Pontryagin class. By induction we have an isomorphism of two-sided A(k)-modules (1, ξ, …, ξ^{n−1}): A(k)^{⊕n} ⟶^{≅} A(HP^{n−1}). We therefore have an isomorphism

(1, τ(1), τ(ξ), …, τ(ξ^{n−1})): A(k)^{⊕(n+1)} ⟶^{≅} A(HP^{n}).

By Theorem 3.2 or 4.1 the two symplectic bundles q^{∗}(𝒱, \overline{ϕ}|_{𝒱}) and i^{∗}(𝒰, ϕ|_{𝒰}) on N^{+} are isomorphic. By functoriality of the Pontryagin classes, this gives q^{A}ξ = i^{A}ζ and therefore also q^{A}(ξ^{ℓ}) = i^{A}(ζ^{ℓ}). We also have −e^{A}i_{A,♭}i^{A}(b) = ζ ∪ b for b ∈ A(HP^{n}) by Proposition 7.6. This gives us

τ(ξ^{ℓ}) = −e^{A}i_{A,♭}q^{A}(ξ^{ℓ}) = −e^{A}i_{A,♭}i^{A}(ζ^{ℓ}) = ζ ∪ ζ^{ℓ} = ζ^{ℓ+1}

for all ℓ. This gives the desired isomorphism

(1, ζ, ζ^{2}, …, ζ^{n}): A(k)^{⊕(n+1)} ⟶^{≅} A(HP^{n}).

Finally, by induction we have ξ^{n} = 0, which gives ζ^{n+1} = τ(ξ^{n}) = 0. □

If (E, ϕ) is a rank 2n symplectic bundle over a scheme S, we can define a quaternionic projective bundle HP_{S}(E, ϕ) = Gr_{S}(2, E)∖GrSp_{S}(2, E, ϕ). A Mayer-Vietoris argument gives the general quaternionic projective bundle theorem.

Theorem 8.2. (Quaternionic projective bundle theorem). Let A be a ring cohomology theory with a symplectic Thom structure. Let (E, ϕ) be a rank 2n symplectic bundle over a scheme S, let (𝒰, ϕ|_{𝒰}) be the tautological rank 2 symplectic subbundle over the quaternionic projective bundle HP_{S}(E, ϕ), and let ζ = p(𝒰, ϕ|_{𝒰}) be its Pontryagin class. Write π: HP_{S}(E, ϕ) → S for the projection. Then for any closed subset Z ⊂ X we have an isomorphism of two-sided A(S)-modules (1, ζ, …, ζ^{n−1}): A_{Z}(S)^{⊕n} ⟶^{≅} A_{π^{−1}(Z)}(HP_{S}(E, ϕ)), and we have unique classes p_{i}(E, ϕ) ∈ A(S) for 1 ≤ i ≤ n such that there is a relation

ζ^{n} − p_{1}(E, ϕ) ∪ ζ^{n−1} + p_{2}(E, ϕ) ∪ ζ^{n−2} − ⋯ + (−1)^{n}p_{n}(E, ϕ) = 0.

If (E, ϕ) is trivial, then p_{i}(E, ϕ) = 0 for 1 ≤ i ≤ n.

Definition 8.3. The classes p_{i}(E, ϕ) (i = 1, …, n) of Theorem 8.2 are the Pontryagin classes of (E, ϕ) with respect to the symplectic Thom structure of the ring cohomology theory A. For i > n one sets p_{i}(E, ϕ) = 0, and one sets p_{0}(E, ϕ) = 1 and p_{i}(E, ϕ) = 0 for i < 0.

The Pontryagin classes are universally A(S)-central, because they are the components of the universally A(S)-central ζ^{n} in a two-sided A(S)-module decomposition A(HP_{S}(E, ϕ)) ≅ A(S)^{⊕n} which is compatible with base change.

The Pontryagin classes are compatible with base change. This implies that they are 𝔸^{1}- deformation invariant in the following sense.

Proposition 8.4. Let (E_{0}, ϕ_{0}) and (E_{1}, ϕ_{1}) be symplectic bundles on a scheme S. Suppose there exists a symplectic bundle (E, ϕ) on S × 𝔸^{1} with (E, ϕ)|_{S×{0}} ≅ (E_{0}, ϕ_{0}) and (E, ϕ)|_{S×{1}} ≅ (E_{1}, ϕ_{1}). Then we have p_{i}(E_{0}, ϕ_{0}) = p_{i}(E_{1}, ϕ_{1}) for all i.

A vector bundle L has an associated hyperbolic symplectic bundle h(L) = (L ⊕ L^{∨}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )).

Proposition 8.5. Suppose that (E, ϕ) is a symplectic bundle over a scheme S with a sublagrangian subbundle L ⊂ E. Let (E_{0}, ϕ_{0}) = (L^{⊥}/L, \overline{ϕ}) ⊕ h(L). Then we have p_{i}(E, ϕ) = p_{i}(E_{0}, ϕ_{0}) for all i.

This is because there exists a symplectic bundle (E_{1}, ϕ_{1}) over S × 𝔸^{1} with (E_{1}, ϕ_{1})|_{S×{t}} ≅ (E, ϕ) for t ≠ 0 and (E_{1}, ϕ_{1})|_{S×{0}} ≅ (E_{0}, ϕ_{0}).

Theorem 8.6 (Nilpotence). Let (E, ϕ) be a symplectic bundle on a scheme X. Then its Pontryagin classes p_{i}(E, ϕ) ∈ A(X) are nilpotent.

Proof. Recall from §2 that we are assuming that our schemes are quasi-compact. So we may cover X by n open subsets U_{α} such that (E, ϕ) trivializes over each U_{α}. We show that all products p_{i_{1}}(E, ϕ) ∪ ⋯ ∪ p_{i_{n}}(E, ϕ) of n Pontryagin classes of (E, ϕ) vanish.

For each U_{α} write Z_{α} = X ∖ U_{α}. The restriction of the Pontryagin class p_{i_{α}}(E, ϕ) to U_{α} vanishes because it is a Pontryagin class of a symplectic bundle trivial on U_{α}. So p_{i_{α}}(E, ϕ) is the image of a class in A_{Z_{α}}(X) under extension of supports. It follows that p_{i_{1}}(E, ϕ) ∪ ⋯ ∪ p_{i_{n}}(E, ϕ) is the image of a class in A_{Z_{1}∩···∩Z_{n}}(X) = A_{∅}X = 0 under extension of supports. □

9. Asymptotic cohomology of quaternionic flag varieties

The direct system of trivial symplectic bundles h(𝒪) ↣ h(𝒪^{⊕2}) ↣ h(𝒪^{⊕3}) ↣ ⋯ over the base generates a direct system of quaternionic projective spaces HP^{0} ⟶^{i_{0}} HP^{1} ⟶^{i_{1}} HP^{2} → ⋯ and an inverse system of cohomology rings ⋯ → A(HP^{2}) → A(HP^{1}) → A(HP^{0}). Each HP^{n} has a rank 2 universal subbundle (𝒰_{n}, ϕ_{n}), and under the inclusion maps we have isomorphisms i_{n−1}^{∗}(𝒰_{n}, ϕ_{n}) ≅ (𝒰_{n−1}, ϕ_{n−1}). Theorem 8.1 and the functoriality of Pontryagin classes gives us the following theorem.

Theorem 9.1. Let A be a ring cohomology theory with a symplectic Thom structure. Then each map i_{n−1}^{A}: A(HP^{n}) → A(HP^{n−1}) in the inverse system of cohomology rings is surjective, and we have an isomorphism \underleftarrow{lim} A(HP^{n}) ≅ A(k)[[y]] with the indeterminate y corresponding to the element of \underleftarrow{lim} A(HP^{n}) given by the system of elements (p(𝒰_{n}, ϕ_{n}))_{n∈ℕ}.

More generally, let r ≥ 1. For any n ≥ r we will write HFlag(1^{r}; n) = HFlag(1, …, 1; n) with the 1 repeated r times. The system of trivial symplectic bundles also generates a direct system of flag bundles

HFlag(1^{r}; r) ↣ HFlag(1^{r}; r + 1) ↣ HFlag(1^{r}; r + 2) ↣ ⋯

and an inverse system of cohomology rings

⋯ → A(HFlag(1^{r}; r + 2)) → A(HFlag(1^{r}; r + 1)) → A(HFlag(1^{r}; r)).

Each HFlag(1^{r}; n) has r rank 2 universal symplectic subbundles (𝒰_{n}^{(1)}, ϕ_{n}^{(1)}), …, (𝒰_{n}^{(r)}, ϕ_{n}^{(r)}) plus a rank 2n − 2r universal symplectic subbundle (𝒱_{r,n}, ψ_{r,n}) and a decomposition as an orthogonal direct sum

h(𝒪^{⊕n}) ≅ (𝒰_{n}^{(1)}, ϕ_{n}^{(1)}) ⊥ ⋯ ⊥ (𝒰_{n}^{(r)}, ϕ_{n}^{(r)}) ⊥ (𝒱_{r,n}, ψ_{r,n})  (9.1)

For the inclusion maps j_{n}: HFlag(1^{r}; n) → HFlag(1^{r}; n + 1) we have natural isomorphisms j_{n}^{∗}(𝒰_{n+1}^{(i)}, ϕ_{n+1}^{(i)}) ≅ (𝒰_{n}^{(i)}, ϕ_{n}^{(i)}) for i = 1, …, r. The rest of this section will be devoted to proving the following theorem.

Theorem 9.2. Let A be a ring cohomology theory with a symplectic Thom structure. Let r ≥ 1 be an integer. Then the maps j_{n}^{A}: A(HFlag(1^{r}; n+1)) → A(HFlag(1^{r}; n)) are surjective, and we have

\underleftarrow{lim}_{n→∞} A(HFlag(1^{r}; n)) ≅ A(k)[[y_{1}, …, y_{r}]]  (9.2)

with the indeterminate y_{i} corresponding to the element of \underleftarrow{lim} A(HFlag(1^{r}; n)) given by the

system (p(𝒰_{n}^{(i)}, ϕ_{n}^{(i)}))_{n≥r}.

The Pontryagin polynomial of a symplectic bundle (E, ϕ) of rank 2r is

P_{E,ϕ}(t) = t^{r} − p_{1}(E, ϕ)t^{r−1} + p_{2}(E, ϕ)t^{r−2} − ⋯ + (−1)^{r}p_{r}(E, ϕ),

while the total Pontryagin class is

p_{t}(E, ϕ) = 1 + p_{1}(E, ϕ)t + ⋯ + p_{r}(E, ϕ)t^{r}.

We will need the following lemma, which is a weak version of the Cartan sum formula.

Lemma 9.3. Suppose the symplectic bundle (E, ϕ) is an orthogonal direct summand of the symplectic bundle (F, ψ) on X. Then the Pontryagin polynomial P_{E,ϕ}(t) divides the Pontryagin polynomial P_{F,ψ}(t) in A(X)[t].

Proof. There is an embedding i: HP_{X}(E, ϕ) ⊂ HP_{X}(F, ψ) such that the tautological rank 2 symplectic subbundle (𝒰_{1}, ψ|_{U_{1}}) on HP_{X}(F, ψ) restricts to the tautological rank 2 symplectic subbundle (𝒰_{2}, ϕ|_{U_{2}}) of HP_{X}(E, ϕ). Hence i^{A}: A(HP_{X}(F, ψ)) → A(HP_{X}(E, ϕ)) sends ζ_{1} = p(𝒰_{1}, ψ|_{U_{1}}) ↦ ζ_{2} = p(𝒰_{2}, ψ|_{U_{2}}). So we have P_{F,ψ}(ζ_{2}) = 0. The division of P_{F,ψ}(t) by the monic polynomial P_{E,ϕ}(t) yields a remainder R(t) ∈ A(HP_{X}(E, ϕ))[t] of degree at most \frac{1}{2} rk E − 1 such that R(ζ_{2}) = 0. So the remainder vanishes. □

Proof of Theorem 9.2. To prove the theorem we will calculate the rings A(HFlag(1^{r}; n)). Each of the relative quaternionic flag bundles is an iterated quaternionic projective bundle, so its cohomology ring is of the form

A(HFlag(1^{r}; n)) = A(k)[y_{1}, …, y_{r}]/I_{r,n}  (9.3)

The maps A(HFlag(1^{r}; n + 1) → A(HFlag(1^{r}; n)) are surjective because they are maps of A(k)-algebras which send a set of generators onto a set of generators.

The construction of HFlag(1^{r}; n) as an iterated flag bundle gives us

I_{r,n} = (P_{1}(y_{1}), P_{2}(y_{1}, y_{2}), …, P_{r}(y_{1}, …, y_{r}))  (9.4)

where each P_{i}(y_{1}, …, y_{i}) is the Pontryagin polynomial in the last variable y_{i} of the symplectic bundle

(G_{n,i}, γ_{n,i}) = (U_{n}^{(i)}, ϕ_{n}^{(i)}) ⊥ ⋯ ⊥ (U_{n}^{(r)}, ϕ_{n}^{(r)}) ⊥ (𝒱_{r,n}, ψ_{r,n}).  (9.5)

This bundle is an orthogonal direct summand of the h(𝒪^{⊕n}) of (9.1), so by Lemma 9.3 the polynomial P_{i}(y_{1}, …, y_{i}) divides P_{h(𝒪^{⊕n})}(y_{i}) = y_{i}^{n} in A(k)[y_{1}, …, y_{i}]/(P_{1}, …, P_{i−1}). From this we deduce

(y_{1}^{n}, y_{2}^{n}, …, y_{r}^{n}) ⊂ I_{r,n}.  (9.6)

Each (G_{n,i}, γ_{n,i}) is an orthogonal direct summand of (G_{n,i−1}, γ_{n,i}), so P_{G_{n,i},γ_{n,i}}(t) divides P_{G_{n,i−1},γ_{n,i−1}}(t). Since the difference in the degrees is 1, we have

P_{G_{n,i−1},γ_{n,i−1}}(t) = (t − z_{i})P_{G_{n,i},γ_{n,i}}(t), z_{i} = p_{1}(G_{n,i−1}, γ_{n,i−1}) − p_{1}(G_{n,i}, γ_{n,i}) (9.7)

We deduce p_{t}(G_{n,i−1}, γ_{n,i−1}) = (1 + z_{i}t)p_{t}(G_{n,i}, γ_{n,i}). Since p_{t}(G_{n,1}, γ_{n,1}) = p_{t}(h(𝒪^{⊕n})) = 1 we get

p_{t}(G_{n,i}, γ_{n,i}) = \frac{1}{∏_{m=1}^{i−1}(1 + z_{m}t)}

in A(k)/(P_{1}, …, P_{i−1})[t].

Write coeff(t^{i}, f(t)) for the coefficient of t^{i} in the power series or polynomial f(t). The Pontryagin polynomial and total Pontryagin class are related by the formula

P_{E,ϕ}(y) = coeff (t^{r+1}, \frac{p_{−t}(E, ϕ)}{1 − yt})

where p_{−t}(E, ϕ) means that one substitutes −t for t in the series p_{t}(E, ϕ). Hence we have

P_{i}(y_{1}, …, y_{i}) = coeff (t^{n−i+1}, \frac{1}{∏_{m=1}^{i−1}(1 − z_{m}t) · (1 − y_{i}t)}) .

Let h_{i}(u_{1}, …, u_{s}) be the i^{th} complete symmetric polynomial, the sum of all the monomials in u_{1}, …, u_{s} of degree i. Set h_{0} = 1. Their generating function is

H(t) = ∑_{i=0}^{∞} h_{i}(u_{1}, …, u_{s})t^{i} = \frac{1}{∏_{m=1}^{s}(1 − u_{m}t)}.

Thus we have

P_{i}(y_{1}, …, y_{i}) = h_{n−i+1}(z_{1}, …, z_{i−1}, y_{i}).  (9.8)

We claim that the z_{i} lie in the ideal (y_{1}, …, y_{r}) ⊂ A(k)[y_{1}, …, y_{r}]. That is because by the universal property of HFlag(1^{r}; n), fixing a decomposition h(𝒪^{⊕n}) = h(𝒪)^{⊥r} ⊥ h(𝒪^{⊕n−r}) gives a section s: k → HFlag(1^{r}; n) of the structural map. Since we have s^{∗}(𝒰_{n}^{(i)}, ϕ_{n}^{(i)}) = h(𝒪), we have s^{A}y_{i} = p(h(𝒪)) = 0. So we have (y_{1}, …, y_{r}) = ker s^{A}. However, the s^{∗}(G_{n,i}, γ_{n,i}) are also trivial, so from (9.7) we have s^{A}z_{i} = 0, proving the claim.

It now follows from (9.8) that we have P_{i}(y_{1}, …, y_{i}) ∈ (y_{1}, …, y_{i})^{n−i+1}. This gives us

I_{n,r} ⊂ (y_{1}, …, y_{n})^{n−r+1}  (9.9)

The inclusions (9.6) and (9.9) together give \underleftarrow{lim} A(HFlag(1^{r}; n)) = A(k)[[y_{1}, …, y_{r}]]. □

10. The splitting principle and the sum formula

The quaternionic flag varieties we studied in the previous section are iterated quaternionic projective bundles over quaternionic Grassmannians

HGr(r, n) ← HFlag(1, r − 1; n) ← ⋯ ← HFlag(1^{r−2}, 2; n) ← HFlag(1^{r}; n).  (10.1)

The quaternionic projective bundle theorem 8.2 implies that A(HFlag(1^{r}; n)) is a free module over A(HGr(r, n)) for any ring cohomology theory A with a symplectic Thom structure. More precisely, when one pulls back the universal rank 2r symplectic bundle (𝒰_{n}, ϕ_{n}) on HGr(r, n) to HFlag(1^{r}; n), it splits into the orthogonal direct sum of the r rank 2 universal symplectic subbundles

(𝒰_{n}^{(1)}, ϕ_{n}^{(1)}) ⊥ ⋯ ⊥ (𝒰_{n}^{(r)}, ϕ_{n}^{(r)})  (10.2)

on HFlag(1^{r}; n). For 1 ≤ i ≤ r −1 the bundle (𝒰_{n}^{(i)}, ϕ_{n}^{(i)}) is the pullback to HFlag(1^{r}; n) of the tautological rank 2 symplectic subbundles of the Quaternionic Projective Bundle Theorem for the i-th projective bundle of (10.1), which is an HP^{r−i}-bundle. Writing \overline{y}_{i} = p(𝒰_{n}^{(i)}, ϕ_{n}^{(i)}) we see the following.

Proposition 10.1. Let A be a ring cohomology theorem with a symplectic Thom structure. The projection q: HFlag(1^{r}; n) → HGr(r, n) induces a monomorphism q^{A}: A(HGr(r, n)) → A(HFlag(1^{r}; n)) under which A(HFlag(1^{r}; n)) is a free A(HGr(r, n))-module of rank r! with basis

𝔹_{r} = \{\overline{y}_{1}^{a_{1}} \overline{y}_{2}^{a_{2}} ⋯ \overline{y}_{r−1}^{a_{r−1}} | 0 ≤ a_{i} ≤ r − i for all i\}.  (10.3)

Let (E, ϕ) be a symplectic bundle of rank 2r on a scheme X, and let q: HFlag_{X}(E, ϕ) → X be the associated complete quaternionic flag bundle. The pullback of E splits as the orthogonal direct sum of the r universal symplectic subbundles of rank 2

q^{∗}(E, ϕ) ≅ (𝒰_{1}, ψ_{1}) ⊥ (𝒰_{2}, ψ_{2}) ⊥ ⋯ ⊥ (𝒰_{r}, ψ_{r}).  (10.4)

Write u_{i} = p(𝒰_{i}, ψ_{i}) ∈ A(HFlag_{X}(E, ϕ)). We will call the u_{i} the Pontryagin roots of (E, ϕ).

Theorem 10.2 (Symplectic splitting principle). The map q^{A}: A(X) → A(HFlag_{X}(E, ϕ)) is injective and makes A(HFlag_{X}(E, ϕ)) into a free two-sided A(X)-module of rank r! with basis the 𝔹_{r} of (10.3). Moreover, the Pontryagin classes p_{i}(E, ϕ) ∈ A(X) coincide after pullback by q^{A} with the elementary symmetric polynomials e_{i}(u_{1}, …, u_{r}) in the Pontryagin roots.

The proof uses two lemmas.

Lemma 10.3. Let R be a ring, and let a_{1}, …, a_{n} ∈ R be central elements such that a_{i} − a_{j} is not a zero divisor for all i ≠ j. If the polynomials t − a_{1}, …, t − a_{n} all divide h ∈ R[t], then ∏_{i=1}^{n}(t − a_{i}) also divides h.

Proof. By induction we may assume that h = g ∏_{i=1}^{n−1}(t−a_{i}). We then have g(a_{n}) ∏(a_{n}−a_{i}) = 0. Since the a_{n} − a_{i} are not zero divisors, we have g(a_{n}) = 0, and so t − a_{n} divides g. □

Lemma 10.4. Let (E, ϕ) be a symplectic bundle on an affine scheme X = Spec R. Suppose that E can be generated by n global sections. Then we can embed (E, ϕ) as a symplectic subbundle of the trivial symplectic bundle of rank 2n.

This lemma is well known (see for example [8]). Hypotheses like \frac{1}{2} ∈ R are not necessary for alternating forms.

Proof of Theorem 10.2. The first sentence of the theorem is simply a relative version of Proposition 10.1. It remains only to prove the second sentence.

We first treat the special case of the tautological rank 2r symplectic subbundle (𝒰_{r,n}, ϕ_{r,n}) on HGr(r, n). Let p_{i} ∈ \underleftarrow{lim} A(HGr(r, n)) be the element corresponding to the inverse system of Pontryagin classes (p_{i}(𝒰_{r,n}, ϕ_{r,n}))_{n≥r}. Recall the y_{i} ∈ \underleftarrow{lim} A(HFlag(1^{r}; n)) of Theorem 9.2

given by the system (p(𝒰_{n}^{(i)}, ϕ_{n}^{(i)}))_{n≥r}. Since (𝒰_{n}^{(i)}, ϕ_{n}^{(i)}) is an orthogonal direct summand of the pullback to HFlag(1^{r}; n) of the tautological bundle (𝒰_{r,n}, ϕ_{r,n}) of HGr(r, n), the Pontryagin polynomial t − p(𝒰_{n}^{(i)}, ϕ_{n}^{(i)}) divides the Pontryagin polynomial

P_{𝒰_{r,n},ϕ_{r,n}}(t) = t^{r} − p_{1}(𝒰_{r,n}, ϕ_{r,n})t^{r−1} + ⋯ + (−1)^{r}p_{r}(𝒰_{r,n}, ϕ_{r,n}).

The quotient polynomial of degree r−1 at level n restricts to the quotient at level n−1 because the quotients and remainders for division by monic polynomials with central coefficients are unique. So the quotient polynomials also form an inverse system, and t − y_{i} divides P (t) = t^{r} − p_{1}t^{r−1} + ⋯ + (−1)^{r}p_{r} in the inverse limit A(k)[[y_{1}, …, y_{r}]]. Lemma 10.3 applies, and we get ∏_{i=1}^{r}(t − y_{i}) = P (t). Hence the p_{i} are the elementary symmetric polynomials in the y_{i}. It follows that the p_{i}(𝒰_{r,n}, ϕ_{r,n}) are the elementary symmetric polynomials in the u_{i} = p(𝒰_{n}^{(i)}, ϕ_{n}^{(i)}).

We next treat the case where X is affine. By Lemma 10.4 (E, ϕ) can be embedded as an orthogonal direct summand of some trivial symplectic bundle. This is classified by a map f: X → HGr(r, n). Since f^{A} and the map for the corresponding quaternionic flag bundles pull back the Pontryagin classes and roots of (𝒰_{r,n}, ϕ_{r,n}) to those of (E, ϕ), the Pontryagin classes of (E, ϕ) are also the elementary symmetric polynomials in the Pontryagin roots.

Finally suppose X is general. Recall from §2 that we are assuming that our schemes are quasi-compact with an ample family of line bundles. Therefore there is an affine bundle g: Y → X with Y an affine scheme. The Pontryagin classes of g^{∗}(E, ϕ) are the elementary symmetric polynomials in its Pontryagin roots. Then g^{A} and the induced map on the cohomology of the quaternionic flag bundles are isomorphisms and send the Pontryagin classes and roots of (E, ϕ) to those of g^{∗}(E, ϕ). So the Pontryagin classes of (E, ϕ) are the elementary symmetric polynomials in its Pontryagin roots. □

Theorem 10.5 (Cartan sum formula). Suppose (F, ψ) ≅ (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}) is an orthogonal direct sum of symplectic bundles over a scheme X. Then for all i we have

p_{t}(F, ψ) = p_{t}(E_{1}, ϕ_{1})p_{t}(E_{2}, ϕ_{2}),  (10.5)
p_{i}(F, ψ) = p_{i}(E_{1}, ϕ_{1}) + ∑_{j=1}^{i−1} p_{i−j}(E_{1}, ϕ_{1})p_{j}(E_{2}, ϕ_{2}) + p_{i}(E_{2}, ϕ_{2}).  (10.6)

The first Pontryagin class is additive, and the top Pontryagin class is multiplicative.

Proof. Consider the fiber bundles

p: HFlag_{X}(E_{1}, ϕ_{1}) ×_{X} HFlag_{X}(E_{2}, ϕ_{2}) → X,  q: HFlag_{X}(F, ψ) → X

We have orthogonal direct sum decompositions

p^{∗}(E_{1}, ϕ_{1}) ≅ ⊥_{i=1}^{r}(𝒰_{i}, ϕ_{i}), p^{∗}(E_{2}, ϕ_{2}) ≅ ⊥_{j=1}^{s}(𝒰′_{j}, ϕ′_{j}), q^{∗}(F, ψ) ≅ ⊥_{ℓ=1}^{r+s}(𝒱_{ℓ}, ψ_{ℓ}), (10.7)

By Theorem 10.2 over HFlag(E_{1}, ϕ_{1}) ×_{X} HFlag(E_{2}, ϕ_{2}) we have

p_{t}(E_{1}, ϕ_{1})p_{t}(E_{2}, ϕ_{2}) = ∏_{i=1}^{r}(1 + p(𝒰_{i}, ϕ_{i})t) ∏_{j=1}^{s}(1 + p(𝒰′_{j}, ϕ′_{j})t)  (10.8)

while over HFlag(F, ψ) we have p_{t}(F, ψ) = ∏_{ℓ=1}^{r+s}(1+p(𝒱_{ℓ}, ψ_{ℓ})t). We also have the orthogonal direct sum p^{∗}(F, ψ) = ⊥_{i=1}^{r} p^{∗}(𝒰_{i}, ϕ_{i}) ⊥ ⊥_{j=1}^{s} p^{∗}(𝒰′_{j}, ϕ′_{j}). This decomposition is classified by a unique map

f: HFlag(E_{1}, ϕ_{1}) ×_{X} HFlag(E_{2}, ϕ_{2}) → HFlag_{X}(F, ψ)

such that

f^{∗}(𝒱_{ℓ}, ψ_{ℓ}) ≅ \begin{cases}p^{∗}(𝒰_{ℓ}, ϕ_{ℓ}) & for ℓ = 1, …, r, \\ p^{∗}(𝒰′_{ℓ−r}, ϕ′_{ℓ−r}) & for ℓ = r + 1, …, r + s. \end{cases}

It follows that when we pull p_{t}(F, ψ) back along f, we get

p_{t}(F, ψ) = ∏_{i=1}^{r}(1 + p(𝒰_{i}, ϕ_{i})t) ∏_{j=1}^{s}(1 + p(𝒰′_{j}, ϕ′_{j})t) = p_{t}(E_{1}, ϕ_{1})p_{t}(E_{2}, ϕ_{2}).

Equating the terms of degree i in this equality of series gives formula (10.6). □

The Cartan sum formula, Proposition 8.5, and nilpotence combine to show that the total Pontryagin class may be defined for Grothendieck-Witt classes of symplectic bundles, giving maps

p_{t}: GW^{−}(X) → A(X)[t]^{×},  (10.9)

functorial in X, and sending sums to products. Note that since symplectic bundles have only finitely many nonzero Pontryagin classes, and they are all nilpotent, the same holds for virtual differences of bundles as well. Hence the image is in A(X)[t]^{×}. Actually the morphism sends

p_{t}: G̃W^{−}(X) = \frac{GW^{−}(X)}{ℤ[h(𝒪)]} → A(X)[t]^{×},

and in particular, the first Pontryagin class is an additive map p_{1} : G̃W^{−}(X) → A(X).

11. Cohomology of quaternionic Grassmannians

We recall some facts about symmetric polynomials. They may mostly be found in Macdonald’s book [10, Chap. 1, §§1–3].

Let Λ_{r} ⊂ ℤ[y_{1}, …, y_{r}] be the ring of symmetric polynomials in r variables. Let e_{i} denote the i-th elementary symmetric polynomial, and h_{i} the i-th complete symmetric polynomial, the sum of all the monomials of degree i. Set e_{0} = h_{0} = 1. We have Λ_{r} = ℤ[e_{1}, …, e_{r}]. The generating functions are

E(t) = ∑_{i≥0} e_{i}t^{i} = ∏_{j=1}^{r}(1 + y_{i}t),  H(t) = ∑_{i≥0} h_{i}t^{i} = ∏_{j=1}^{r}(1 − y_{i}t)^{−1}.  (11.1)

So we have

E(t)H(−t) = 1,  h_{m} + ∑_{i=1}^{r}(−1)^{r}e_{i}h_{m−i} = 0.  (11.2)

So we also have Λ_{r} = ℤ[h_{1}, …, h_{r}]. The h_{i} with i > r are nonzero but are dependent on h_{1}, …, h_{r}.

Let λ = (λ_{1}, λ_{2}, …, λ_{r}) be a partition of length l(λ) ≤ r. Write δ = (r − 1, r − 2, …, 1, 0) and a_{λ+δ} = det(y_{i}^{λ_{j}+r−j})_{1≤i,j≤r}. Then a_{λ+δ} is a skew-symmetric polynomial and therefore divisible by the Vandermonde determinant a_{δ}. The quotient s_{λ} = a_{λ+δ}/a_{δ} is the Schur polynomial for λ. It is symmetric of degree |λ| = ∑ λ_{i}. One has s_{(1^{i})} = e_{i} and s_{(i)} = h_{i}. The a_{λ+δ} with l(λ) ≤ r form a ℤ-basis of the skew-symmetric polynomials in r variables, so the s_{λ} with l(λ) ≤ r form a ℤ-basis of Λ_{r}. Denote by λ′ the partition dual to λ. One has formulas

s_{λ} = det(e_{λ′_{i}−i+j})_{1≤i,j≤m} = det(h_{λ_{i}−i+j})_{1≤i,j,≤r},  (11.3)

for m ≥ l(λ′) and r ≥ l(λ).

The ring of symmetric functions Λ in an infinite number of variables is the inverse limit of the Λ_{r} in the category of graded rings. It is a polynomial ring in infinitely many indeterminates

Λ = ℤ[e_{1}, e_{2}, …] = ℤ[h_{1}, h_{2}, …]

It has an involution ω with ω(e_{i}) = h_{i} and ω(h_{i}) = e_{i} for all i and ω(s_{λ}) = s_{λ′} for all partitions λ. The quotient map Λ ↠ Λ_{r} is the quotient by the ideal (e_{r+1}, e_{r+2}, …). The ring Λ has as ℤ-basis the Schur functions s_{λ} with λ ranging over all partitions. If, however, l(λ) = λ′_{1} > r, then the first row of the first determinant of (11.3) is (e_{λ′_{1}}, e_{λ′_{1}} + 1, …, e_{λ′_{1}} + m − 1). All these entries are sent to 0 in Λ_{r}. Set

Π_{r,n−r} = {partitions λ with length l(λ) = λ′_{1} ≤ r and with λ_{1} ≤ n − r}

The set Π_{r,n−r} has \binom{n}{r} members. For the ideal I_{n−r} = (h_{n−r+1}, h_{n−r+2}, …, h_{n}) ⊂ Λ_{r} the quotient map Λ_{r} ↠ Λ_{r}/I_{n−r} sends s_{λ} ↦ 0 for all λ ∉ Π_{r,n−r}. The quotient Λ_{r}/I_{n−r} is free over ℤ with basis {s_{λ} | λ ∈ Π_{r,n−r}}.

Now suppose (E, ϕ) is a symplectic bundle of rank 2r over S. Let q: HFlag_{S}(E, ϕ) → S be the associated complete quaternionic flag bundle, let

q^{∗}(E, ϕ) ≅ (𝒰_{1}, ψ_{1}) ⊥ (𝒰_{2}, ψ_{2}) ⊥ ⋯ ⊥ (𝒰_{r}, ψ_{r}).

be the splitting of the pullback of E as the orthogonal direct sum of the r universal symplectic subbundles of rank 2 on HFlag_{S}(E, ϕ), and let y_{i} = p(𝒰_{i}, ψ_{i}) ∈ A(HFlag_{S}(E, ϕ)) be the Pontryagin roots of (E, ϕ). For any partition λ of length l(λ) ≤ r write

s_{λ}(E, ϕ) = s_{λ}(y_{1}, …, y_{r}) = det(p_{λ′_{i}−i+j}(E, ϕ))_{1≤i,j≤m} ∈ A(S)  (11.4)

with m ≥ l(λ′).

We now complete the calculation of the cohomology of HFlag(1^{r}; n) begun in §9.

Theorem 11.1. On HFlag(1^{r}; n) let y_{1}, …, y_{r} be the first Pontryagin classes of the r tautological rank 2 subbundles. Then A(HFlag(1^{r}; n)) = A(k)[y_{1}, …, y_{r}]/I_{r,n} with

I_{r,n} = (h_{n}(y_{1}), h_{n−1}(y_{1}, y_{2}), …, h_{n−r+1}(y_{1}, …, y_{r})),  (11.5)

and if we now write h_{i} for the complete symmetric polynomial in all r variables y_{1}, …, y_{r}, then we also have

A(HFlag(1^{r}; n)) = A(k)[y_{1}, …, y_{r}]/(h_{n}, h_{n−1}, …, h_{n−r+1}).  (11.6)

Proof. Using the Cartan sum formula, we see that the classes z_{i} of (9.7) are actually z_{i} = y_{i}. Hence (9.8) can be rewritten as P_{i}(y_{1}, …, y_{i}) = h_{n−i+1}(y_{1}, …, y_{i}), and we get (11.5).

Since the complete symmetric polynomials are the sums of all monomials of a given degree, we have

h_{m}(y_{1}, …, y_{i−1}, y_{i}) = h_{m}(y_{1}, …, y_{i−1}) + y_{i}h_{m−1}(y_{1}, …, y_{i−1}, y_{i}).

This formula can be used to show that one can add one variable to each of the generators of I_{r,n} which have less than r variables without changing the ideal. One repeats this until all the generators have r variables. This gives (11.6). □

Theorem 11.2. Let (𝒰, ϕ) be the tautological symplectic bundle of rank 2r on HGr(r, n), and let (𝒰^{⊥}, ψ) be its orthogonal complement. Then for any ring cohomology theory A with a symplectic Thom structure and any S the map

A(S)[e_{1}, …, e_{r}]/(h_{n−r+1}, …, h_{n}) ⟶^{≅} A(HGr(r, n) × S)  (11.7)

sending e_{i} ↦ p_{i}(𝒰, ϕ) for all i is an isomorphism of rings, the map

A(S)^{⊕\binom{n}{r}} ⟶^{(s_{λ}(𝒰,ϕ))_{λ∈Π_{r,n−r}}} A(HGr(r, n) × S)  (11.8)

is an isomorphism of two-sided A(S)-modules, and for all partitions λ we have

s_{λ}(𝒰, ϕ) = (−1)^{|λ|}s_{λ′}(𝒰^{⊥}, ψ).  (11.9)

Proof. The proof is much like the case of ordinary Grassmannians. It is enough to consider S = k. Write A_{r,n} = A(k)[e_{1}, …, e_{r}]/(h_{n−r+1}, …, h_{n}). Since A(HGr(r, n)) → A(HFlag(1^{r}; n)) is injective, the complete symmetric polynomials h_{i} with i > n − r vanish in A(HGr(r, n)) as well. We thus get the map γ: A_{r,n} → A(HGr(r, n)) sending e_{i} ↦ p_{i}(𝒰, ϕ). By Proposition 10.1

𝔹_{r} = {y_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{r−1}^{a_{r−1}} | 0 ≤ a_{i} ≤ r − i for all i}.

is a basis of A(HFlag(1^{r}; n)) as a two-sided A(HGr(r, n))-module, and it is also a basis of ℤ[y_{1}, …, y_{r}] as a free module over the ring of symmetric polynomials ℤ[e_{1}, …, e_{r}] and therefore also a basis of A(k)[y_{1}, …, y_{r}]/(h_{n−r+1}, …, h_{n}) as a two-sided free A_{r,n}-module. By Theorem 11.1 the map of free modules A(k)[y_{1}, …, y_{r}]/(h_{n−r+1}, …, h_{n}) → A(HFlag(1^{r}; n)) is an isomorphism. So γ: A_{r,n} → A(HGr(r, n)) is also an isomorphism.

Over the quaternionic Grassmannian HGr(r, n) the orthogonal direct sum (𝒰, ϕ) ⊥ (𝒰^{⊥}, ψ) is the trivial symplectic bundle h(𝒪)^{⊕n} and has vanishing Pontryagin classes. So the Cartan sum formula (10.5) gives p_{t}(𝒰, ϕ)p_{t}(𝒰^{⊥}, ψ) = 1. For the generating series of (11.1) we have γ(E(t)) = p_{t}(𝒰, ϕ). So from the identity E(t)H(−t) = 1 of (11.2), we see we have γ(H(−t)) = p_{t}(𝒰^{⊥}, ψ) and thus (−1)^{i}γ(h_{i}) = p_{i}(𝒰^{⊥}, ψ). The formula s_{λ}(𝒰, ϕ) = (−1)^{|λ|}s_{λ′}(𝒰^{⊥}, ψ) now follows from (11.3). The sign change h_{i} ↦ (−1)^{i}h_{i} comes from the involution of Λ_{r} sending f ↦ (−1)^{deg(f)}f for all homogeneous symmetric polynomials. Hence the sign (−1)^{|λ|} in front of s_{λ′}(𝒰^{⊥}, ψ). □

The usual Mayer-Vietoris argument now gives the following generalization of Theorem 11.2.

Theorem 11.3. Let (E, ϕ) be a symplectic bundle of rank 2n over S. Let (𝒰, ϕ|_{𝒰}) be the tautological subbundle of rank 2r on HGr_{S}(r, E, ϕ), and let (𝒰^{⊥}, ϕ|_{U^{⊥}}) be its orthogonal complement. Then for any ring cohomology theory with a symplectic Thom structure

A(S)^{⊕\binom{n}{r}} ⟶_{≅}^{(s_{λ}(𝒰,ϕ|_{𝒰} ))_{λ∈Π_{r,n−r}}} A(HGr_{S}(r, E, ϕ)),

A(S)^{⊕\binom{n}{r}} ⟶_{≅}^{(s_{λ}(𝒰^{⊥},ϕ|_{𝒰^{⊥}} ))_{λ∈Π_{n−r,r}}} A(HGr_{S}(r, E, ϕ))

are isomorphisms of two-sided A(S)-modules.

Let (𝒰_{r,n}, ϕ_{r,n}) ↪ h(𝒪)^{⊕n} be the universal tautological symplectic bundle of rank 2r on HGr(r, n). Let η_{r,n}: 𝒰_{r,n} ⊂ h(𝒪)^{⊕n} be the inclusion. Consider the inclusions of symplectic subbundles

\xymatrix{𝒰_{r,n} \ar@{^{(}->}[r]^{^{( \begin{matrix}η_{r,n} \\ 0 \end{matrix} )}} & h(𝒪)^{⊕n} ⊕ h(𝒪) = h(𝒪)^{⊕n+1}, \\ h(𝒪) ⊕ 𝒰_{r,n} \ar@{^{(}->}[r]^{_{( \begin{matrix}1 & 0 \\ 0 & η_{r,n} \end{matrix} )}} & h(𝒪) ⊕ h(𝒪)^{⊕n} = h(𝒪)^{⊕n+1}.}  (11.10)

They are classified by maps α_{r,n}: HGr(r, n) → HGr(r, n + 1) and β_{r,n}: HGr(r, n) → HGr(r + 1, n + 1) respectively. Let γ_{r,n} = β_{r,n+1}α_{r,n} = α_{r+1,n+1}β_{r,n}: HGr(r, n) → HGr(r + 1, n + 2). We have direct systems of quaternionic Grassmannians

HGr(r, r) ⟶^{α_{r,r}} HGr(r, r + 1) ⟶^{α_{r,r+1}} HGr(r, r + 2) → · → HGr(r, n) ⟶^{α_{r,n}} ⋯

HGr(0, 0) ⟶^{γ_{0,0}} HGr(1, 2) ⟶^{γ_{1,2}} HGr(2, 4) → ⋯ → HGr(n, 2n) ⟶^{γ_{n,2n}} ⋯

(11.11)

Theorem 11.4. For any ring cohomology theory A with a symplectic Thom structure and for any S the maps

(α_{r,n} × 1_{S})^{A}: A(HGr(r, n + 1) × S) → A(HGr(r, n) × S)
(β_{r,n} × 1_{S})^{A}: A(HGr(r + 1, n + 1) × S) → A(HGr(r, n) × S)

are split surjective, and we have isomorphisms

A(S)[[p_{1}, …, p_{r}]] ⟶^{≅} \underleftarrow{lim}_{n→∞} A(HGr(r, n) × S)  (11.12)
A(S)[[p_{1}, p_{2}, p_{3}, …]] ⟶^{≅} \underleftarrow{lim}_{n→∞} A(HGr(n, 2n) × S)  (11.13)

with each variable p_{i} sent to the inverse system of i^{th} Pontryagin classes (p_{i}(𝒰_{r,n}))_{n≥r} or (p_{i}(𝒰_{n,2n}))_{n∈ℕ}.

The theorem follows from the explicit generators and relations for the A(HGr(r, n) × S) given in Theorem 11.2 in the same way as for ordinary Grassmannians.

12. Recovering Thom classes from Pontryagin classes

In this section we show that a symplectic Thom structure is determined by its system of Pontryagin classes. This section is joint work with Alexander Nenashev.

Definition 12.1. A Pontryagin structure on a ring cohomology theory B on a category of schemes is a rule assigning to every rank 2 symplectic bundle (E, ϕ) over a scheme S in the category a central element p(E, ϕ) ∈ B(S) with the following properties:

(1) For (E_{1}, ϕ_{1}) ≅ (E_{2}, ϕ_{2}) we have p(E_{1}, ϕ_{1}) = p(E_{2}, ϕ_{2}).

(2) For a morphism f: Y → S we have f^{B}(p(E, ϕ)) = p(f^{∗}(E, ϕ)).

(3) For the tautological rank 2 symplectic subbundle (𝒰, ϕ|_{𝒰}) on HP^{1} the maps

(1, p(𝒰, ϕ|_{𝒰})): B(S) ⊕ B(S) → B(HP^{1} ×S)

are isomorphisms.

(4) For a rank 2 symplectic space (V, ϕ) viewed as a trivial symplectic bundle over k we have p(V, ϕ) = 0 in B(k).

The Pontryagin classes associated to a symplectic Thom structure by formula (7.1) form a Pontryagin structure because of the functoriality of the Thom classes, the Quaternionic Projective Bundle Theorem 8.1, and Proposition 7.7.

Theorem 12.2. Every Pontryagin structure on a ring cohomology theory is the system of Pontryagin classes of a unique symplectic Thom structure whose classes are given by formula (12.4) below.

The rest of this section is devoted to proving this theorem. Our strategy goes as follows. Let (E, ϕ) be a rank 2 symplectic bundle over S. We will study the HP^{1} bundle associated to the rank 4 symplectic bundle (F, ψ) with

F = 𝒪_{S} ⊕ E ⊕ 𝒪_{S},  ψ = (\begin{matrix}0 & 0 & 1 \\ 0 & ϕ & 0 \\ −1 & 0 & 0 \end{matrix}) .  (12.1)

The bundle (F, ψ) is of the form studied in Theorem 4.1, so the HP^{1} bundle has the properties of that proposition. Write

\begin{matrix}P = HP_{S}^{1}(F, ψ), \\ N^{−} = HP_{S}^{1}(F, ψ) ∩ Gr_{S}(2, E ⊕ 𝒪_{S}), \end{matrix}  \begin{matrix}N^{+} = HP_{S}^{1}(F, ψ) ∩ Gr_{S}(2, 𝒪_{S} ⊕ E), \\ S = N^{+} ∩ N^{−} = HP_{S}^{0}(E, ϕ). \end{matrix}

Write (U, ψ|_{U}) for the universal rank 2 symplectic subbundle over P. In the closed embedding HP_{S}^{0}(E, ϕ) ⊂ HP_{S}^{1}(F, ψ) of Theorem 4.1, the small quaternionic projective bundle is now an HP^{0} bundle over S and is therefore S itself. The restriction of (U, ψ|_{U}) to this S = HP^{0}(E, ϕ) is the tautological rank 2 symplectic subbundle of the rank 2 symplectic bundle (E, ϕ). So it is (E, ϕ) itself. It now follows from Theorem 4.1 that the two summands of the normal bundle N_{1} and N_{2} satisfy

N^{+} ≅ E^{∨} ≅ E  N^{−} ≅ E^{∨} ≅ E  (12.2)

We have the following diagram.

[figure: E = N^{−}; f; π; P; s; h; U; S; z; g; N^{+}; i; i; P; z_{U}]  (12.3)

The loci S, N^{+}, N^{−}, P and the vector bundles U → P and E → S are as above, the maps i, f, g, and the map S → N^{−} are the inclusions, and h, π, and the map N^{−} → S are the projections to S. We have hf = π. The section s: P → U corresponds to the vector bundle map s_{+} : 𝒪_{P} → U of Theorem 4.1(d) whose zero locus is N^{+}. In particular, s is transversal to the zero section z_{U} of U. The locus N^{−} can be identified with the vector bundle E by the isomorphism of (12.2), and z and π correspond to the zero section and structural map of the vector bundle. The intersections z_{U}(P) ∩ s(P) = N^{+} and N^{+} ∩ N^{−} = S are transversal.

We now prove a series of lemmas. The first one applies to a slightly more general situation.

Lemma 12.3. Let B be a ring cohomology theory with a Pontryagin structure. Let (G, ω) be a rank 4 symplectic bundle over a scheme S, and let (𝒰, ω|_{𝒰}) be its tautological rank 2 symplectic subbundle. Then the map (1, p(𝒰, ω|_{𝒰})): B(S) ⊕ B(S) → B(HP_{S}^{1}(G, ω)) is an isomorphism.

Proof. The lemma is true for a trivial (G, ω), which has HP_{S}^{1}(G, ω) = HP^{1} ×S, by the axioms of a Pontryagin structure. The general case follows by a Mayer-Vietoris argument. □

Lemma 12.4. Let B be a ring cohomology theory with a Pontryagin structure. In the situation of (12.1)–(12.3) let \overline{e}^{B}: B_{X}(P) → B(P) be the supports extension map. Then there exists a unique element θ_{E,ϕ} ∈ B_{X}(P) satisfying \overline{e}^{B}(θ_{E,ϕ}) = p(U, ψ|_{U}). Moreover, θ_{E,ϕ} is B(S)-central, and the map θ: B(S) → B_{X}(P) sending α ↦ h^{B}(α) ∪ θ_{E,ϕ} is an isomorphism.

Proof. We are in the situation of Theorem 5.1. This gives us two pieces of information. First, write Y = P ∖ N^{+}, let j: Y ↪ P be the inclusion, and let q = hj: Y → S be the projection. Theorem 5.2(a) then says that q^{B}: B(S) → B(Y) is an isomorphism.

Second, by Theorem 5.1 itself there are maps

Y ⟵^{g_{1}} Y_{1} ⟵_{≅}^{g_{2}} Y_{2} ⟶^{q} HGr_{S}(0, E, ϕ) = S

with g_{1} an 𝔸^{1}-bundle, g_{2} an 𝔸^{0}-bundle, and q an 𝔸^{5}-bundle. Moreover, g_{2}^{∗}g_{1}^{∗}(U|_{Y}) has two tautological sections e, f over Y_{2} which together give an isometry with the trivial rank 2

symplectic bundle (𝒪_{S}^{⊕2}, ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )). Therefore g_{2}^{B}g_{1}^{B}j^{B}(p(U, ψ|_{U})) = p(g_{2}^{∗}g_{1}^{∗}j^{∗}(U, ψ|_{U})) is the

pullback by Y_{2} → k of the Pontryagin class of the trivial symplectic bundle over k, which vanishes by the axioms of a Pontryagin structure. So we have g_{2}^{B}g_{1}^{B}j^{B}(p(U, ψ|_{U})) = 0. Moreover, since g_{1} and g_{2} are affine bundles, g_{2}^{B}g_{1}^{B} is an isomorphism. So we have j^{B}(p(U, ψ|_{U})) = 0.

Now consider the diagram of two-sided B(S)-modules.

\xymatrix{0 \ar[r] & B(S) \ar[r]^{(0,1)} \ar@{.>}[d]^{≅}_{θ} & B(S) ⊕ B(S) \ar[r]^{(1,0)} \ar[d]_{(1,p(U,ψ|_{U}))∘h^{B}}^{≅} & B(S) \ar[r] \ar[d]_{q^{B}}^{≅} & 0 \\ ⋯ \ar[r]_{∂=0} & B_{N^{+}}(P) \ar[r]_{_{\overline{e}_{B}}} & B(P) \ar[r]_{j^{B}} & B(Y) \ar[r]_{∂=0} & ⋯}

The bottom row is the localization exact sequence. The middle vertical arrow is an isomorphism by Lemma 12.3, and q^{B} is an isomorphism by the discussion above. Since we have q^{B} = j^{B}h^{B}, that also implies that j^{B} is surjective, and therefore ∂ vanishes. The righthand square commutes because we have j^{B}(p(U, ψ|_{U})) = 0. It now follows that the diagram can be completed by a unique isomorphism θ making the lefthand square commute. Setting θ_{E,ϕ} = θ(1_{S}), we get an element with all the desired properties because θ is an isomorphism of two-sided B(S)-modules. □

Theorem 12.5. Let B be a ring cohomology theory with a Pontryagin structure. In the situation of (12.1)–(12.3) let θ_{E,ϕ} ∈ B_{X}(P) be the unique element satisfying \overline{e}^{B}(θ_{E,ϕ}) = p(U, ψ|_{U}) of Lemma 12.4. Then the assignment

th(E, ϕ) = −f^{B}(θ_{E,ϕ}) ∈ B_{S}(E)  (12.4)

defines a symplectic Thom structure on B whose Pontryagin classes are the given Pontryagin structure.

Proof. We verify the axioms of a symplectic Thom structure.

First, the th(E, ϕ) are supposed to be B(S)-central. By Lemma 12.3 the element θ_{E,ϕ} ∈ B_{N^{+}}(P) is B(S)-central, and since f: E = N^{−} → P is compatible with the maps π and h to S, the map f^{B}: B_{N^{+}}(P) → B_{S}(E) is a map of two-sided B(S)-modules. So f^{B}(θ_{E,ϕ}) = − th(E, ϕ) is B(S)-central.

The functoriality conditions on the th(E, ϕ) follow from the functoriality of the constructions of (12.1)–(12.3).

The maps ∪ th(E, ϕ): B(S) → B_{S}(E) are isomorphisms because they are compositions

B(S) ⟶_{≅}^{−θ} B_{N^{+}}(P) ⟶_{≅}^{f^{B}} B_{S}(E)

of maps which are isomorphisms because of Lemma 12.4 and Proposition 4.3.

Hence the th(E, ϕ) define a symplectic Thom structure.

Finally, the Pontryagin classes −z^{B}e^{B}(th(E, ϕ)) defined by the symplectic Thom structure are the original ones because using the commutative diagram

\xymatrix{B_{X}(P) \ar[r]^{^{\overline{e}^{B}}} \ar[d]^{f^{B}} & B(P) \ar[d]^{f^{B}} \\ B_{S}(E) \ar[r]^{e^{B}} & B(E) \ar[r]^{z^{B}} & B(S)}

we see that we have

−z^{B}e^{B}(th(E, ϕ)) = z^{B}e^{B}f^{B}(θ_{E,ϕ}) = z^{B}f^{B}\overline{e}^{B}(θ_{E,ϕ}) = z^{B}f^{B}(p(U, ψ)) = p(E, ϕ)

using functoriality of the Pontryagin classes. □

Theorem 12.6. Let B be a ring cohomology theory with a Pontryagin structure. Suppose that the system of Pontryagin classes are those of some symplectic Thom structure with classes th(E, ϕ). In the situation of (12.1)–(12.3) let θ_{E,ϕ} ∈ B_{N^{+}}(P) be the unique element satisfying \overline{e}^{B}(θ_{E,ϕ}) = p(U, ψ|_{U}) of Lemma 12.4. Then we have th(E, ϕ) = −f^{B}(θ_{E,ϕ}) ∈ B_{S}(E).

Thus the symplectic Thom structure inducing the Pontryagin structure is unique.

Proof. In Proposition 7.6 we saw that p(U, ψ|_{U}) is the image of − th(U, ψ|_{U}) under the composition

B_{P} (U) ⟶^{s^{B}} B_{X}(P) ⟶^{\overline{e}^{B}} B(P).

Since θ_{E,ϕ} is the unique element with \overline{e}^{B}(θ_{E,ϕ}) = p(U, ψ|_{U}), we have θ_{E,ϕ} = −s^{B}(th(U, ψ|_{U})). To complete the proof of the lemma, we need to prove f^{B}s^{B}(th(U, ψ|_{U})) = th(E, ϕ).

Now consider the diagram.

\xymatrix{E \ar[d]^{π} \ar[r]^{i_{1}} & E ⊕ E \ar[d]^{p_{2}} \ar[r]^{f_{1}} & U \ar[d] \\ S \ar[r]^{z} & E \ar[u]^{Δ} \ar[r]^{f} & P \ar[u]^{s}}

According to Theorem 4.1(e), the pullbacks to N^{−} of the two bundles U → P and E → S are isomorphic. Pulling further back along the isomorphism E ≅ W gives an isomorphism j_{2}^{∗}f^{∗}U ≅ π^{∗}E = E ⊕ E of bundles over E. According to Lemma 4.2, the section s of U whose zero locus is X pulls back to the tautological diagonal section Δ of π^{∗}E = E ⊕ E. This gives the righthand square. The lefthand square is clear. The functoriality of the Thom classes gives th(E, ϕ) = i_{1}^{B}f_{1}^{B}(th(U, ψ)) ∈ B_{S}(E). The first projection p_{1} : E ⊕ E → E is an 𝔸^{2}-bundle with p_{1}^{−1}(S) = 0⊕E, and it has sections i_{1} and Δ. It follows that the maps i_{1}^{B} and Δ^{B}: B_{0⊕E}(E ⊕ E) = B_{E}(π^{∗}E) → B_{S}(E) are equal. Substituting, and using also sf = f_{1}Δ, we get the desired formula

th(E, ϕ) = i_{1}^{B}f_{1}^{B}(th(U, ψ)) = Δ^{B}f_{1}^{B}(th(U, ψ)) = f^{B}s^{B}(th(U, ψ)) = −f^{B}(θ_{E,ϕ}).  □

Proof of Theorem 12.2. Theorem 12.5 showed that every Pontryagin structure consists of the Pontryagin classes associated to the symplectic Thom structure given by formula (12.4), and Theorem 12.6 showed that this was the only symplectic Thom structure inducing the given Pontryagin structure. □

13. Thom classes of higher rank bundles

In this section we define Thom classes for higher rank symplectic bundles and prove some of their properties. The construction proceeds as in §12 only in higher rank and using top Pontryagin classes. But now that Theorem 12.2 has been proven establishing the equivalence of symplectic Thom structures and Pontryagin structures, we may return to using a ring cohomology theory A with a symplectic Thom structure. So we now have at our disposal our results on quaternionic Grassmannians.

Let (E, ϕ) be a symplectic bundle of rank 2r over S, and let (F, ψ) be the rank 2r + 2 symplectic bundle of (4.2) or (12.1). Set

\begin{matrix}P = HGr_{S}(r, F, ψ), \\ S = N^{+} ∩ N^{−} = HGr_{S}(r, E, ϕ), \end{matrix} N^{+} = HGr_{S}(r, F, ψ) ∩ Gr_{S}(2r, 𝒪_{S} ⊕ E), N^{−} = HGr_{S}(r, F, ψ) ∩ Gr_{S}(2r, E ⊕ 𝒪_{S}).  (13.1)

Write (U, ψ|_{U}) for the universal rank 2r symplectic subbundle over P. The restriction of (U, ψ|_{U}) to S = HGr_{S}(r, E, ϕ) is again (E, ϕ). We have the same isomorphisms N^{+} ≅ E and N^{−} ≅ E as in (12.2) and the same commutative diagram (12.3). The main difference is that the relative dimensions over S of the various loci and bundles are now

dim_{S} S = 0, dim_{S} E = dim_{S} N^{+} = dim_{S} N^{−} = 2r, dim_{S} P = 4r, dim_{S} U = 6r.

Proposition 13.1. Let A be a ring cohomology theory with a symplectic Thom structure. In the situation above let \overline{e}^{A}: A_{N^{+}}(P) → A(P) be the supports extension map. Then there exists a unique element θ_{E,ϕ} ∈ A_{N^{+}}(P) satisfying \overline{e}^{A}(θ_{E,ϕ}) = p_{r}(U, ψ|_{U}). Moreover, θ_{E,ϕ} is A(S)-central, and the map θ: A(S) → A_{N^{+}}(P) sending α ↦ h^{A}(α) ∪ θ_{E,ϕ} is an isomorphism.

Proof. We claim we have an isomorphism of two-sided A(S)-modules

(1, p_{1}(U, ψ|_{U}), …, p_{r}(U, ψ|_{U})): A(S)^{⊕r+1} ⟶^{≅} A(P).

This is because Theorem 11.3 establishes that A(P) is a free two-sided A(S)-module with basis s_{λ}(𝒰, ψ|_{U}) for λ ∈ Π_{r,1} = {(1^{i}) | 0 ≤ i ≤ r}. But those particular Schur polynomials are 1, e_{1}, …, e_{r}, so the characteristic classes are 1, p_{1}, …, p_{r}.

Similarly, if we write (𝒰_{r−1}, ϕ|_{𝒰_{r−1}}) for the tautological rank 2r − 2 symplectic subbundle on HGr_{S}(r − 1, E, ϕ), we have an isomorphism of two-sided A(S)-modules

(1, p_{1}(𝒰_{r−1}, ϕ|_{𝒰_{r−1}}), …, p_{r−1}(𝒰_{r−1}, ϕ|_{𝒰_{r−1}})): A(S)^{⊕r} ⟶^{≅} A(HGr_{S}(r − 1, E, ϕ)).

Let Y = P ∖ N^{+} and let j: Y ↪ P be the inclusion. By Theorem 5.2(b) there is a map σ: A(HGr_{S}(r − 1, E, ϕ)) → Y ⊂ P classifying the rank 2r symplectic subbundle 𝒪 ⊕ 𝒰_{r−1} ⊕ 𝒪 of the pullback of F, and the pullback map

σ^{A}: A(Y) ⟶^{≅} A(HGr_{S}(r − 1, E, ϕ))

is an isomorphism. Composing with the restriction gives a map σ^{A}j^{A}: A(P) → A(Y) ≅ A(HGr_{S}(r − 1, E, ϕ)) sending 1 ↦ 1 and (in simplified notation)

p_{i}(U) ↦ p_{i}(𝒪 ⊕ 𝒰_{r−1} ⊕ 𝒪) = \begin{cases}p_{i}(𝒰_{r−1}) & for i = 1, …, r − 1, \\ 0 & for i = r, \end{cases}

the equalities coming from the Cartan sum formula. Therefore in the localization sequence

⋯ ⟶^{∂} A_{N^{+}}(P) ⟶^{\overline{e}^{A}} A(P) ⟶^{j^{A}} A(Y) ⟶^{∂} ⋯

the map j^{A} is split surjective, ∂ vanishes, and \overline{e}^{A} is split injective with image the free direct summand A(S) · p_{r}(U, ψ|_{U}). So there exists indeed a unique element θ_{E,ϕ} ∈ A_{X}(P) satisfying \overline{e}^{A}(θ_{E,ϕ}) = p_{r}(U, ψ|_{U}). Moreover, θ_{E,ϕ} is A(P)-central like the Pontryagin class and therefore also A(S)-central, and the map θ: A(S) → A_{X}(P) sending α ↦ h^{A}(α) ∪ θ_{E,ϕ} is indeed an isomorphism. □

Theorem 13.2. Let A be a ring cohomology theory with a symplectic Thom structure. For a symplectic bundle (E, ϕ) of rank 2r on S, let P, N^{+}, N^{−}, f, etc., be as in (13.1) and (12.3), and let θ_{E,ϕ} ∈ A_{N^{+}}(P) be the unique element satisfying \overline{e}^{A}(θ_{E,ϕ}) = p(U, ψ|_{U}) of Proposition 13.1. Then the assignment

th(E, ϕ) = (−1)^{r}f^{A}(θ_{E,ϕ}) ∈ A_{S}(E)  (13.2)

gives a system of classes with the following properties:

(1) Each th(E, ϕ) ∈ A_{S}(E) is A(S)-central.

(2) For an isomorphism γ: (E, ϕ) ≅ (E_{1}, ϕ_{1}) we have th(E, ϕ) = γ^{A} th(E_{1}, ϕ_{1}).

(3) For u: T → S, writing u_{E}: u^{∗}E → E for the pullback, we have u_{E}^{A}(th(E, ϕ)) = th(u^{∗}(E, ϕ)) ∈ A_{T} (u^{∗}E).

(4) The maps ∪ th(E, ϕ): A(S) → A_{S}(E) are isomorphisms.

(5) We have th((E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2})) = q_{1}^{A} th(E_{1}, ϕ_{1}) ∪ q_{2}^{A} th(E_{2}, ϕ_{2}), where q_{1}, q_{2} are the

projections from E_{1} ⊕ E_{2} onto its factors. Moreover, for e^{A}: A_{S}(E) → A(E) the extension of supports map, and z^{A}: A(E) → A(S) the restriction to the zero section, we have

p_{r}(E, ϕ) = (−1)^{r}z^{A}e^{A}(th(E, ϕ)),  (13.3)

while for (E, ϕ) of rank 2 the class th(E, ϕ) just defined is the same as the class in the symplectic Thom structure.

The classes th(E, ϕ) are called the symplectic Thom classes.

First part of the proof of Theorem 13.2. The proof of (1)–(4) and (13.3) is identical to the proof of Theorem 12.5. The coincidence of the two symplectic Thom classes for rank 2 bundles was proven in Theorem 12.6. We will prove (5) at the end of this section. □

We may now generalize the direct image maps of (7.2). When i: X → Y is a closed embedding of smooth varieties of even codimension 2r whose normal bundle N = N_{X/Y} is equipped with a specified symplectic form ϕ, we say that i: X → Y has a symplectic normal bundle (N, ϕ). We can compose the isomorphism of Theorem 13.2(4) with the deformation to the normal bundle isomorphisms when they exist and extension supports maps gives direct image maps.

i_{A,♭}: A(X) ⟶_{≅}^{∪ th(N,ϕ)} A_{X}(N) ⟶_{≅}^{d_{X/Y}} A_{X}(Y), i_{A,♮}: A(X) ⟶^{i_{A,♭}} A_{X}(Y) → A(Y)

The analogues of Propositions 7.3, 7.4, 7.5 and 7.6 hold for these more general direct image maps except that for a symplectic bundle (E, ϕ) of rank 2r equations (7.4) and (7.5) should read

p_{r}(E, ϕ) = (−1)^{r}\overline{e}^{A}s^{A} th(E, ϕ),  (13.4)
i_{A,♮}i^{A}(b) = \overline{e}^{A}i_{A,♭}i^{A}(b) = (−1)^{r}b ∪ p_{r}(E, ϕ).  (13.5)

Now let (E, ϕ) be symplectic bundle of rank 2n − 2 on S, and let (F, ψ) = (E, ϕ) ⊥ h(𝒪_{S}) be as in (4.2). Let N^{+} = HGr_{S}(r, F, ψ) ∩ Gr(2r, 𝒪_{S} ⊕ E) and Y = HGr_{S}(r, F, ψ) ∖ N^{+}. Let u: N^{+} → HGr_{S}(r, E, ϕ) be the bundle map of Theorem 4.1, and let i: N^{+} ↪ HGr_{S}(r, F, ψ) be the inclusion. Let σ: HGr_{S}(r − 1, E, ϕ) → Y ↪ HGr_{S}(r, F, ψ) be the map of Theorem 5.2. Let τ = e^{A}q^{A}i_{A,♭}.

\xymatrix{⋯ \ar[r]^{∂=0} & A_{N^{+}}(HGr_{S}(r, F, ψ)) \ar[r]^{e^{A}} & A(HGr_{S}(r, F, ψ)) \ar[r] \ar@{-->}[dr]_{σ^{A}} & A(Y) \ar[r]^{∂=0} \ar[d]_{≅} & ⋯ \\  & A(HGr_{S}(r, E, ϕ)) \ar[u]_{≅}^{i_{A,♭}q^{A}} \ar@{-->}[ur]^{τ} &  & A(HGr_{S}(r − 1, E, ϕ))} (13.6)

Write 𝒰_{r} for the tautological bundles on HGr_{S}(r, E, ϕ), write 𝒰_{r−1} for the one on HGr_{S}(r − 1, E, ϕ), and 𝒱_{r} for the one on HGr_{S}(r, F, ψ). Write s_{λ}(𝒰_{r}) for the Schur polynomials in the Pontryagin classes as in (11.4).

Theorem 13.3. The maps τ and σ act on the A(S)-bases of the cohomology of the quaternionic Grassmannian bundles by

τ: s_{λ}(𝒰_{r}) ↦ (−1)^{r}p_{r}(𝒱_{r})s_{λ}(𝒱_{r}) = (−1)^{r}s_{λ+(1^{r})}(𝒱_{r})  for λ ∈ Π_{r,n−1−r}
σ: s_{λ}(𝒱_{r}) ↦ \begin{cases}s_{λ}(𝒰_{r−1}) & for λ ∈ Π_{r−1,n−r}, \\ 0 & for λ ∈ Π_{r,n−r} ∖ Π_{r−1,n−r}. \end{cases}

Proof. For a partition λ of length l(λ) ≤ r, one has equality l(λ) = r if and only if the dual partition is of the form λ′ = (r, λ′_{2}, …, λ′_{m}). When that occurs formula (11.3) gives

s_{λ} = |\begin{matrix}e_{r} & 0 & ⋯ & 0 \\ e_{λ′_{2}−1} & e_{λ′_{2}} & ⋯ & e_{λ′_{2}+m−2} \\ ⋮ & ⋮ & ... & ⋮ \\ e_{λ′_{m}−m+1} & e_{λ′_{m}−m+2} & ⋯ & e_{λ′_{m}} \end{matrix}| = e_{r}s_{λ−(1^{r})}.

in Λ_{r}. Therefore, keeping in mind that the pullbacks of the tautological subbundles along the projection q: N^{+} → HGr_{S}(r, E, ϕ) and the inclusion i: N^{+} → HGr_{S}(r, F, ψ) are isometric, we have

τ(s_{λ}(𝒰_{r})) = e^{A}i_{A,♭}q^{A}(s_{λ}(𝒰_{r})) = e^{A}i_{A,♭}i^{A}(s_{λ}(𝒱_{r})) = (−1)^{r}p_{r}(𝒱_{r})s_{λ}(𝒱_{r}) = (−1)^{r}s_{λ+(1^{r})}(𝒱_{r}),

using (13.5).

The map σ classifies the rank 2r symplectic bundle 𝒪 ⊕ 𝒰_{r−1} ⊕ 𝒪 on HGr_{S}(r−1, E, ϕ), so we have

σ^{A}(s_{λ}(𝒱_{r})) = s_{λ}(𝒪 ⊕ 𝒰_{r−1} ⊕ 𝒪) = s_{λ}(𝒰_{r−1})

because the Pontryagin classes of 𝒰_{r−1} and 𝒪⊕𝒰_{r−1}⊕𝒪 are equal by the Cartan sum formula. For the λ ∈ Π_{r−1,n−r} this is one of the elements of the basis of A(HGr_{S}(r − 1, E, ϕ) as a twosided A(S)-module. For λ ∈ Π_{r,n−r} ∖ Π_{r−1,n−r}, i.e. those with l(λ) = r, we have s_{λ}(𝒰_{r−1}) = p_{r}(𝒰_{r−1})s_{λ−(1^{r})}(𝒰_{r−1}) = 0, since the tautological symplectic subbundle on HGr_{S}(r − 1, E, ϕ) is of rank 2r − 2 and has p_{r}(𝒰_{r−1}) = 0. □

End of the proof of Theorem 13.2. We now prove property (5) of the theorem. Suppose rk E_{1} = 2r_{1} and rk E_{2} = 2r_{2}. Let

P_{r_{1}+r_{2}} = HGr_{S}(r_{1} + r_{2}, (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}) ⊥ h(𝒪_{S})),
P_{r_{1}} = HGr_{S}(r_{1}, (E_{1}, ϕ_{1}) ⊥ h(𝒪_{S})),
P_{r_{2}} = HGr_{S}(r_{2}, (E_{2}, ϕ_{2}) ⊥ h(𝒪_{S})),
F_{r_{1},r_{2}} = HFlag_{S}(r_{1}, r_{2}; (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}) ⊥ h(𝒪_{S})),
G_{r_{1}} = HGr_{S}(r_{1}, (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}) ⊥ h(𝒪_{S})),
G_{r_{2}} = HGr_{S}(r_{2}, (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}) ⊥ h(𝒪_{S}))

We have maps

\xymatrix{P_{r_{1}} \ar[r]^{t_{1}} & F_{r_{1},r_{2}} \ar[d]^{ρ} \ar[r]^{ρ_{1}} \ar[dr]^{ρ_{2}} & G_{r_{1}} \\ P_{r_{2}} \ar@{-->}[ur]^{t_{2}} & P_{r_{1}+r_{2}} & G_{r_{2}}}

defined as follows. Over F_{r_{1},r_{2}} there are orthogonal tautological symplectic subbundles U_{1}, U_{2} of ranks 2r_{1} and 2r_{2} respectively. The maps ρ_{i} are classified by the U_{i} and ρ by U_{1} ⊕ U_{2}. All three projections are quaternionic Grassmannian bundles.

Over P_{r_{1}} there is a tautological rank 2r_{1} subbundle 𝒱_{1} ⊂ E_{1}⊕h(𝒪). The map t_{1} is classified by the pair of orthogonal symplectic subbundles 𝒱_{1} ⊥ E_{2} ⊂ E_{1} ⊕ E_{2} ⊕ h(𝒪). The map t_{2} is defined analogously, reversing the roles of E_{1} and E_{2}.

In P_{r_{1}}, P_{r_{2}} and P_{r_{1}+r_{2}} there are the loci and maps of (12.3)

[figure: E_{1} = N_{1}^{−}; f_{1}; π_{1}; P_{r_{1}}; s_{1}; h_{1}; U_{1}; S_{1}; z_{1}; g_{1}; N_{1}^{+}; i_{1}; i_{1}; P_{r_{1}}; z_{U_{1}}; E_{2} = N_{2}^{−}; f_{2}; π_{2}; P_{r_{2}}; s_{2}; h_{2}; U_{2}; S_{2}; z_{2}; g_{2}; N_{2}^{+}; i_{2}; i_{2}; P_{r_{2}}; z_{U_{2}}]

[figure: E_{1} ⊕ E_{2} = N^{−}; f; π; P_{r_{1}+r_{2}}; s; h; U; S; z; g; N^{+}; i; i; P_{r_{1}+r_{2}}; z_{U}]

Writing μ_{1} : 𝒪_{E_{1}} → π_{1}^{∗}E_{1} and μ_{2} : 𝒪_{E_{2}} → π_{2}^{∗}E_{2} for the tautological sections. The compositions

γ_{1} : E_{1} = N_{1}^{−} ⟶^{f_{1}} P_{r_{1}} ⟶^{t_{1}} F_{r_{1},r_{2}}  γ_{2} : E_{2} = N_{2}^{−} ⟶^{f_{2}} P_{r_{2}} ⟶^{t_{2}} F_{r_{1},r_{2}}

are the maps classified by the orthogonal pairs of subbundles which are the images of

π_{1}^{∗}E_{1} ⟶^{( \begin{matrix}0 \\ 1 \\ 0 \\ μ_{1}^{∨}ϕ_{1} \end{matrix} )} 𝒪_{E_{1}} ⊕ π_{1}^{∗}E_{1} ⊕ π_{1}^{∗}E_{2} ⊕ 𝒪_{E_{1}} \underleftarrow{( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} )} π_{1}^{∗}E_{2}

π_{2}^{∗}E_{1} \underrightarrow{( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} )} 𝒪_{E_{2}} ⊕ π_{2}^{∗}E_{1} ⊕ π_{2}^{∗}E_{2} ⊕ 𝒪_{E_{2}} ⟵^{( \begin{matrix}0 \\ 0 \\ 1 \\ μ_{2}^{∨}ϕ_{2} \end{matrix} )} π_{2}^{∗}E_{2}

The restriction of ρ: F_{r_{1},r_{2}} → P_{r_{1}+r_{2}} to the inverse image of E_{1} ⊕ E_{2} = N^{−} has a section γ: N^{−} → ρ^{−1}(N^{−}) ⊂ F_{r_{1},r_{2}} classified by the orthogonal pair of subbundles

π^{∗}E_{1} ⟶^{( \begin{matrix}0 \\ 1 \\ 0 \\ μ_{1}^{∨}ϕ_{1} \end{matrix} )} 𝒪_{E_{1}⊕E_{2}} ⊕ π^{∗}E_{1} ⊕ π^{∗}E_{2} ⊕ 𝒪_{E_{1}⊕E_{2}} ⟵^{( \begin{matrix}0 \\ 0 \\ 1 \\ μ_{2}^{∨}ϕ_{2} \end{matrix} )} π^{∗}E_{2}

The restriction of γ: E_{1} ⊕ E_{2} → F_{r_{1},r_{2}} to each factor E_{i} thus coincides with γ_{i}: E_{i} → F_{r_{1},r_{2}}.

In G_{r_{1}} there is the locus \overline{N}_{1}^{+} = G_{r_{1}} ∩ Gr_{S}(2r_{1}, 𝒪_{S} ⊕ E_{1} ⊕ E_{2}) and there is an analogous locus \overline{N}_{2}^{+} ⊂ G_{r_{2}}. We have ρ_{1}^{−1}(\overline{N}_{1}^{+}) ∩ ρ_{2}^{−1}(\overline{N}_{2}^{+}) = ρ^{−1}(N^{+}).

Call U_{1} all the tautological bundles of rank 2r_{1}, call U_{2} all the tautological bundles of rank 2r_{2}, and call U all the tautological bundles of rank 2r_{1} + 2r_{2}.

By Proposition 13.1 and Theorem 13.3 the extension of supports maps A\overline{N}{+ \atop i}(G_{r_{i}}) → A(G_{r_{i}}) and A_{N_{i}^{+}}(P_{r_{i}}) → A(P_{r_{i}}) and A_{N^{+}}(P_{r_{1}+r_{2}}) → A(P_{r_{1}+r_{2}}) are all split injective. Since the ρ_{i}: F_{r_{1},r_{2}} → G_{r_{i}} and ρ: F_{r_{1},r_{2}} are quaternionic Grassmannian bundles, we may pull back along the ρ_{i} and ρ and see that the extension of supports maps A_{ρ_{i}^{−1}(\overline{N}_{i}^{+})}(F_{r_{1},r_{2}}) → A(F_{r_{1},r_{2}}) and A_{ρ^{−1}(N^{+})}(F_{r_{1},r_{2}}) are also split injective.

In the notation of (13.6) let

\overline{θ}_{E_{1},ϕ_{1}} = (−1)^{r}i_{A,♭}q^{A}(1_{\overline{N}_{1}^{+}∩\overline{N}_{1}^{−}}) ∈ A\overline{N}{+ \atop 1}(G_{r_{1}}).

Then by Theorem 13.3 and the functoriality of the Pontryagin classes

\overline{θ}_{E_{1},ϕ_{1}} ∈ A\overline{N}{+ \atop 1}(G_{r_{1}}), ρ_{1}^{A}(\overline{θ}_{E_{1},ϕ_{1}}) ∈ A_{ρ_{1}^{−1}(\overline{N}_{1}^{+})}(F_{r_{1},r_{2}}), t_{1}^{A}ρ_{1}^{A}(\overline{θ}_{E_{1},ϕ_{1}}) ∈ A_{N_{1}^{+}}(P_{r_{1}}),

are the unique classes in their respective cohomology groups whose images under the extension of supports maps are the Pontryagin classes p_{r_{1}}(U_{1}, ϕ_{1}). Therefore t_{1}^{A}ρ_{1}^{A}(\overline{θ}_{E_{1},ϕ_{1}}) is the θ_{E_{1},ϕ_{1}} of Theorem 12.6, and th(E_{1}, ϕ_{1}) = γ_{1}^{A}ρ_{1}^{A}(\overline{θ}_{E_{1},ϕ_{1}}) ∈ A_{S}(E_{1}), while q_{1}^{A} th(E_{1}, ϕ_{1}) = γ^{A}ρ_{1}^{A}(\overline{θ}_{E_{1},ϕ_{1}}) ∈ A_{E_{2}}(E_{1} ⊕ E_{2}).

There are analogous classes

\overline{θ}_{E_{2},ϕ_{2}} ∈ A\overline{N}{+ \atop 2}(G_{r_{2}}),  ρ_{2}^{A}(\overline{θ}_{E_{2},ϕ_{2}}) ∈ A_{ρ_{2}^{−1}(\overline{N}_{2}^{+})}(F_{r_{1},r_{2}}),

such that ρ_{2}^{A}(\overline{θ}_{E_{2},ϕ_{2}}) is the unique class whose extension of supports is p_{r_{2}}(U_{2}, ϕ_{2}) ∈ A(F_{r_{1},r_{2}}) and γ^{A}ρ_{2}^{A}(\overline{θ}_{E_{2},ϕ_{2}}) = q_{2}^{A} th(E_{2}, ϕ_{2}).

Similarly there are classes

θ_{E_{1}⊕E_{2},ϕ_{1}⊕ϕ_{2}} ∈ A_{N^{+}}(P_{r_{1}+r_{2}}),  ρ^{A}(θ_{E_{1}⊕E_{2},ϕ_{1}⊕ϕ_{2}}) ∈ A_{ρ^{−1}(N^{+})}(F_{r_{1},r_{2}}),

which are the unique classes whose images under the extension of supports maps are the Pontryagin classes p_{r_{1}+r_{2}}(U). We have

th(E_{1} ⊕ E_{2}, ϕ_{1} ⊕ ϕ_{2}) = f^{A}θ_{E_{1}⊕E_{2},ϕ_{1}⊕ϕ_{2}} = γ^{A}ρ^{A}θ_{E_{1}⊕E_{2},ϕ_{1}⊕ϕ_{2}}.

Now

ρ_{1}^{A}(\overline{θ}_{E_{1},ϕ_{1}})ρ_{2}^{A}(\overline{θ}_{E_{2},ϕ_{2}}) ∈ A_{ρ_{1}^{−1}(\overline{N}_{1}^{+})∩ρ_{2}^{−1}(\overline{N}_{2}^{+})}(F_{r_{1},r_{2}}) = A_{ρ^{−1}(N^{+})}(F_{r_{1},r_{2}})

is a class whose image under the extension of supports map is p_{r_{1}}(U_{1}, ϕ_{1})p_{r_{2}}(U_{2}, ϕ_{2}) = p_{r_{1}+r_{2}}(U, ϕ). (Top Pontryagin classes multiply.) Since ρ^{A}(θ_{E_{1}⊕E_{2},ϕ_{1}⊕ϕ_{2}}) was the unique class with that property we have

ρ_{1}^{A}(\overline{θ}_{E_{1},ϕ_{1}})ρ_{2}^{A}(\overline{θ}_{E_{2},ϕ_{2}}) = ρ^{A}(θ_{E_{1}⊕E_{2},ϕ_{1}⊕ϕ_{2}}).

Applying γ^{A} now gives

q_{1}^{A}(th(E_{1}, ϕ_{1})) q_{2}^{A}(th(E_{2}, ϕ_{2})) = th(E_{1} ⊕ E_{2}, ϕ_{1} ⊕ ϕ_{2}).  □

14. Symplectic orientations

A ring cohomology theory can be symplectically oriented by any of five structures satisfying different axioms. We have already seen two of them: a symplectic Thom structure (Definition 7.1) and a Pontryagin structure (Definition 12.1). Here are the other three (cf. [14, Definitions 3.26, 3.32, 3.1]).

Definition 14.1. A Pontryagin classes theory on a ring cohomology theory A on a category of schemes is a system of assignments to every symplectic bundle (E, ϕ) over every scheme S in the category of elements p_{i}(E, ϕ) ∈ A(S) for all i ≥ 1 satisfying

(1) For (E_{1}, ϕ_{1}) ≅ (E_{2}, ϕ_{2}) we have p_{i}(E_{1}, ϕ_{1}) = p_{i}(E_{2}, ϕ_{2}) for all i.

(2) For a morphism f: Y → S we have f^{A}(p_{i}(E, ϕ)) = p_{i}(f^{∗}(E, ϕ)) for all i.

(3) For the tautological rank 2 symplectic subbundle (𝒰, ϕ|_{𝒰}) on HP^{1} the maps

(1, p_{1}(𝒰, ϕ|_{𝒰})): A(S) ⊕ A(S) → A(HP^{1} ×S)

are isomorphisms for all S.

(4) For a rank 2 symplectic space (V, ϕ) viewed as a trivial symplectic bundle over k we have p_{1}(V, ϕ) = 0 in A(k).

(5) For an orthogonal direct sum of symplectic bundles (E, ϕ) ≅ (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}) we have p_{i}(E, ϕ) = p_{i}(E_{1}, ϕ_{1}) + ∑_{j=1}^{i−1} p_{i−j}(E_{1}, ϕ_{1})p_{j}(E_{2}, ϕ_{2}) + p_{i}(E_{2}, ϕ_{2}) for all i.

(6) For (E, ϕ) of rank 2r we have p_{i}(E, ϕ) = 0 for i > r.

One may also set p_{0}(E, ϕ) = 1 and even p_{i}(E, ϕ) = 0 for i < 0.

Definition 14.2. A symplectic Thom classes theory on a ring cohomology theory A on a category of schemes is a system of assignments to every symplectic bundle (E, ϕ) over every scheme X in the category of an element th(E, ϕ) ∈ A_{X}(E) satisfying conditions (1)–(5) of Theorem 13.2.

Definition 14.3. A symplectic orientation on a ring cohomology theory A on a category of schemes is a system of assignments to every symplectic bundle (E, ϕ) over every scheme X in the category and every closed subset Z ⊂ X with X ∖ Z in the category of an isomorphism th_{Z}^{E,ϕ}: A_{Z}(X) → A_{Z}(E) with the following properties.

(1) Let π: E → X be the structure map, and let π^{A}: A_{Z}(X) → A_{π^{−1}(Z)}(E) be the pullback. Then for all a ∈ A(X) and b ∈ A_{Z}(X) one has

th_{Z}^{E,ϕ}(a ∪ b) = th_{X}^{E,ϕ}(a) ∪ π^{A}b,  th_{Z}^{E,ϕ}(b ∪ a) = π^{A}b ∪ th_{X}^{E,ϕ}(a).

(2) For every isometry of symplectic bundles ϕ: (E, ϕ) → (F, ψ) the following diagram commutes

[figure: A_{Z}(X); ≅; ^{th{F,ψ \atop Z}}; 1; A_{Z}(F); ≅; ϕ^{A}; A_{Z}(X); ≅; _{th{E,ϕ \atop Z}}; A_{Z}(E).]

(3) For every morphism f: X′ → X with Z′ ⊂ X′ closed and f^{−1}(Z) ⊂ Z′, then for (E′, ϕ′) = f^{∗}(E, ϕ) and g: E′ → E the pullback of f along π: E → X, the following diagram commutes

\xymatrix{A_{Z}(X) \ar[r]_{≅}^{^{th{E,ϕ \atop Z}}} \ar[d]_{f^{A}} & A_{Z}(E) \ar[d]^{g^{A}} \\ A_{Z′}(X′) \ar[r]_{≅}^{_{th{E′,ϕ′ \atop Z′}}} & A_{Z′}(E′).}

(4) For every pair of sympectic bundles (E_{1}, ϕ_{1}) and (E_{2}, ϕ_{2}) over a scheme X, with structural maps p_{i}: E_{i} → X, the following diagram commutes

\xymatrix{A_{Z}(X) \ar[r]_{≅}^{^{th{E1,ϕ1 \atop Z}}} \ar[d]^{≅}_{_{th{E2,ϕ2 \atop Z}}} & A_{Z}(E_{1}) \ar[d]^{_{th{p∗ 1E2,p∗ 1ϕ2 \atop Z}}}_{≅} \\ A_{Z}(E_{2}) \ar[r]^{_{th{p∗ 2E1,p∗ 2ϕ1 \atop Z}}}_{≅} & A_{Z}(E_{1} ⊕ E_{2}).}

Property (1) of Definition 14.3 implies that the th_{Z}^{E,ϕ} are A(X)-bimodule maps, but at least formally it is somewhat stronger ([14, Lemma 3.33] did not seem obvious to the second author with the formally weaker property).

Theorem 14.4. Let A be a ring cohomology theory.

(a) There are inverse bijections

{symplectic Thom structures on A} ⟷ {Pontryagin structures on A}

given by the formulas (7.1) and (12.4).

(b) There are inverse bijections

{Pontryagin class structures on A} ⟷ {Pontryagin structures on A}

given by forgetfulness and the map which assigns to a Pontryagin structure the Pontryagin classes assigned by Definition 8.3 to the associated symplectic Thom structure.

(c) There are inverse bijections

{symplectic Thom class structures on A} ⟷ {symplectic Thom structures on A}

given by forgetfulness and Theorem 13.2.

(d) There are inverse bijections

{symplectic Thom class structures on A} ⟷ {symplectic orientations on A}

given by the formulas th_{Z}^{E,ϕ}(b) = π^{A}b ∪ th(E, ϕ) and th(E, ϕ) = th_{X}^{E,ϕ}(1_{X}).

Proof. (a) This is Theorem 12.2.

(b) The right-to-left map is well-defined because the Pontryagin classes associated to a symplectic Thom structure satisfy the functoriality axioms (1)(2) by simple arguments, axiom (3) concerning HP^{1} by Theorem 8.1, the triviality axiom (4) by Proposition 7.7, the Cartan sum formula (5) by Theorem 10.5, and the dimension axiom (6) holds by definition. The rightto-left-to-right roundtrip assigns to a given Pontryagin structure the Pontryagin structure associated to the associated Thom structure. This is the identity map because of Theorem 12.5. The left-to-right map is injective because of the splitting principle (Theorem 10.2) and the Cartan sum formula (cf. [14, Theorem 3.27]).

(c) The right-to-left map is well-defined by Theorem 13.2, and the right-to-left-to-right roundtrip is the identity by the last phrase in that theorem. The left-to-right map is injective because of the splitting principle and the multiplicativity formula (5) for the symplectic Thom classes (cf. [14, Lemma 3.34]).

(d) Left to the reader. □

15. More on the cohomology of the open stratum

This is a version of our original proof of Theorem 3.4(b). We use the following geometry, with (V, ϕ) a symplectic space of dimension 2n + 2 over a field and GrSp(2, V, ϕ) ⊂ Gr(2, V) the closed subvariety parametrizing totally isotropic subspaces.

Theorem 15.1. The open stratum X_{0} ⊂ HP^{n} is a dense open subvariety of an open subvariety Y_{0} ⊂ Gr(2, V) with complement Z_{0} = Y_{0} ∖ X_{0} = Y_{0} ∩ GrSp(2, V, ϕ), and there is an open subvariety Y′_{0} ⊂ Y_{0} containing Z_{0} for which there exists a diagram in which the arrows have the stated properties and the horizontal arrows commute with each other and with all upward arrows and with the two solid downward arrows.

\xymatrix{Z_{0} \ar[r]_{closed} \ar[d]^{restriction of the 𝔸^{2n}-bundle} \ar[rr]^{closed} & Y′_{0} \ar@{^{(}->}[r]_{open} \ar[d]^{𝔸^{2n}-bundle} & Y_{0} \ar@{.>}[d]^{different ; 𝔸^{2n}-bundle} & X_{0} = Y_{0} ∖ Z_{0} \ar@{^{(}->}[l]_{open} \\ ℙ^{2n−1} \ar[r]^{closed} \ar[u] \ar[rr]_{hyperplane} & ℙ^{2n} ∖ 0 \ar@{^{(}->}[r]^{open} \ar[u] & ℙ^{2n} \ar[u] & 𝔸^{2n} \ar@{^{(}->}[l]_{open} \ar[u]}  (15.1)

Proof. In (V, ϕ) we have a 1-dimensional subspace E = ⟨e⟩, a vector f with ϕ(e, f) = 1, and the ϕ-nondegenerate subspace F = ⟨e, f⟩^{⊥}. We have E^{⊥} = E ⊕ F. We set

X_{0} = {U ⊂ V | U ⊄ E^{⊥} and ϕ|_{U} is nondegenerate},
Y_{0} = {U ⊂ V | U ⊄ E^{⊥}},
Z_{0} = Y_{0} ∖ X_{0} = Y_{0} ∩ GrSp(2, V, ϕ),
Y′_{0} = {U ⊂ V | U ⊄ E^{⊥} and U ∩ E = {0}}.

Clearly Y_{0} is an open subvariety of the Grassmannian, and X_{0} and Y′_{0} are open in Y_{0}. Moreover, any U in Y_{0} ∖ Y′_{0} is of the form U = ⟨u, v⟩ with 0 ≠ u ∈ E and v ∉ E^{⊥}. We therefore have (up to nonzero multiples) u = e and v = f +v_{0} with v_{0} ∈ E^{⊥}. We then have ϕ(u, v) = 1, so ϕ|_{U} is nondegenerate. So we have Y_{0} ∖ Y′_{0} ⊂ X_{0} and Z_{0} ⊂ Y′_{0}.

For the vector bundle structures, we use Białynicki-Birula’s theory of cell decompositions induced by 𝔾_{m}-actions. Consider the one-parameter subgroup λ_{1} : 𝔾_{m} ↪ GL(V) given by

λ_{1}(t) · v = \begin{cases}v & for v ∈ E^{⊥} = ⟨e⟩ ⊕ F , \\ t^{−1}v & for v ∈ ⟨f⟩. \end{cases}

The fixed-point locus of the induced action on Gr(2, V) has two components

Γ_{0} = Gr(2, E^{⊥}),  Γ_{1} = {⟨v, f⟩ | v ∈ E^{⊥}} ≅ ℙ(E^{⊥}) = ℙ^{2n}

For each component there is a stratum Υ_{i} = {x ∈ Gr(2, V) | lim_{t→0} λ_{1}(t)x ∈ Γ_{i}}. The assignment x ↦ lim_{t→0} λ_{1}(t)x gives maps π_{1} : Υ_{i} → Γ_{i} which send any U ∈ Gr(2, V) to its associated graded space with respect to the filtration on U induced by 0 ⊂ E^{⊥} ⊂ V. The maps π_{1} : Υ_{i} → Γ_{i} are vector bundles identifiable with the normal bundles N_{Γ_{i}/Υ_{i}}. In this case we have

Υ_{0} = Gr(2, E^{⊥}) = Γ_{0},  Υ_{1} = Y_{0}.

The fiber of π_{1} : Y_{0} → ℙ^{2n} over ⟨v, f⟩ ∈ ℙ^{2n} consists of points corresponding to spaces ⟨v, u+f⟩ with u ∈ E^{⊥}. These points are parametrized by the classes \overline{u} ∈ E^{⊥}/⟨v⟩. So Y_{0} is isomorphic to the tautological rank 2n quotient bundle 𝒬 on ℙ^{2n}.

Now λ_{1} does not preserve the symplectic form ϕ because it does not act on e and f with opposite weights. So to study the part of Y_{0} in GrSp(2, V, ϕ) we replace it by λ_{2} : 𝔾_{m} ↪ Sp(2, V, ϕ) acting linearly on V with

λ_{2}(t) · v = \begin{cases}tv & for v ∈ ⟨e⟩, \\ v & for v ∈ F , \\ t^{−1}v & for v ∈ ⟨f⟩. \end{cases}

The induced action of λ_{2} on Gr(2, V) comes with four fixed-point loci

Γ′_{0} = {⟨e, v⟩ | v ∈ F }, Γ′_{1} = Gr(2, F), Γ′_{2} = {⟨e, f⟩}, Γ′_{3} = {⟨u, f⟩ | u ∈ F } = ℙ^{2n−1}

and vector bundles π_{2} : Υ′_{i} → Γ′_{i} much as before. This time, however, we have Y_{0} = Υ′_{2} ∪ Υ′_{3} and Y′_{0} = Υ′_{3}. We have Y_{0} ∖ Y′_{0} = Υ′_{2} = {⟨e, v + f⟩ | v ∈ F } ≅ 𝔸^{2n}, and we may see that it intersects Γ_{1} = ℙ^{2n} in the unique point ⟨e, f⟩ = Γ′_{2}. We will call this point 0 ∈ ℙ^{2n}.

Now the Białynicki-Birula vector bundle Y′_{0} = Υ′_{3} → Γ′_{3} = ℙ^{2n−1} is of rank 2n + 1. The base, the fixed point set ℙ^{2n−1} = {⟨u, f⟩ | u ∈ ℙ(F)}, is entirely within GrSp(2, V, ϕ). Since the one-parameter subgroup respects GrSp(2, V, ϕ), the locus Z_{0} is also a bundle over ℙ^{2n−1}, indeed a subbundle of Y′_{0} of rank 2n.

As a bundle Y′_{0} is isomorphic to N_{ℙ^{2n−1}/Y′_{0}}. At any point U\overline{u} = ⟨u, f⟩ ∈ ℙ^{2n−1}, the fiber of the normal bundle is canonically isomorphic to the quotient of the subspace of ϕ ∈ Hom_{k}(U\overline{u}, V/U\overline{u}) = T_{U\overline{u}} Gr(2, V) corresponding to first order deformations ⟨u + ϕ(u)ε, f + ϕ(f)ε⟩ which lie within Y′_{0} modulo those lying within ℙ^{2n−1}. These first-order deformations are thus the ⟨u+αeε, f +(v +βe)ε⟩ with \overline{v} ∈ F/⟨u⟩. Thus Y′_{0} is isomorphic to a vector bundle N → ℙ^{2n−1} which we can split as N = N_{0} ⊕ L with

N_{0} ⟶_{≅}^{(\overline{v},β)} 𝒬 ⊕ 𝒪,  L ⟶_{≅}^{α} 𝒪(1),

The subbundle of N corresponding to Z_{0} corresponds to the first-order deformations satisfying

ϕ(u + αeε, f + (v + βe)ε) = (α + ϕ(u, v))ε = 0.

So it is defined by the equation α = −ϕ(u, v). This bundle is thus the graph of a vector bundle map ψ: N_{0} → L which fiberwise is (\overline{v}, β) ↦ −ϕ(u, v) (with u a function of the base). Let N_{1} ⊂ N be the graph of ψ. We still have a direct sum N = N_{1} ⊕ L. We now have a quotient map N ↠ N/N_{1} ≅ L which makes Y′_{0} ≅ N into a rank 2n vector bundle over L, with Z_{0} ≅ N_{1} the subbundle lying over the zero section ℙ^{2n−1} of L. Finally the subbundle L consists of the points ⟨αe + u, f⟩ with \overline{u} ∈ ℙ^{2n−1}, which is ℙ^{2n} ∖ 0. □

Second proof of Theorem 3.4(b). By Theorem 3.2 it is enough to treat the open stratum X_{0}. By Theorem 15.1 X_{0} ⊂ Y_{0} is an open subvariety with complement Z_{0}, and there is a closed embedding of pairs (ℙ^{2n}, 𝔸^{2n}) → (Y_{0}, X_{0}). This yields a morphism of localization long exact sequences

\xymatrix{A_{Z_{0}}(Y_{0}) \ar[r] \ar[d] & A(Y_{0}) \ar[r] \ar[d]_{≅} & A(X_{0}) \ar[r]^{∂} \ar[d]_{i^{A}} & A_{Z_{0}}(Y_{0}) \ar[d] \ar[r] & A(Y_{0}) \ar[d]_{≅} \\ A_{ℙ^{2n−1}}(ℙ^{2n}) \ar[r] & A(ℙ^{2n}) \ar[r] & A(𝔸^{2n}) \ar[r]^{∂} & A_{ℙ^{2n−1}}(ℙ^{2n}) \ar[r] & A(ℙ^{2n})}  (15.2)

The map A(Y_{0}) → A(ℙ^{2n}) is an isomorphism because Y_{0} → ℙ^{2n} is a vector bundle. By excision A_{Z_{0}}(Y_{0}) → A_{ℙ^{2n−1}}(ℙ^{2n}) is isomorphic to A_{Z_{0}}(Y′_{0}) → A_{ℙ^{2n−1}}(ℙ^{2n} ∖ 0), which again is an isomorphism because of the vector bundles in Theorem 15.1 and strong homotopy invariance. The map labeled i^{A} is therefore also an isomorphism by the five lemma. Since i^{A}t^{A}: A(k) → A(𝔸^{2n}) is an isomorphism by homotopy invariance, it follows that t^{A} is an isomorphism. □

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Steklov Institute of Mathematics, Saint Petersburg

Laboratoire J.-A. Dieudonné (UMR 6621 du CNRS), Département de mathématiques, Université de Nice – Sophia Antipolis, 06108 Nice Cedex 02, France