Essential dimension of Hermitian spaces

N. Semenov, K. Zainoulline^{∗}

Abstract

We compute the essential dimension of Hermitian forms in the sense of O. Izhboldin. Apart from this we investigate the Chow motives of twisted incidence varieties and prove their incompressibility in dimensions 2^{r} − 1.

Let W be an n-dimensional vector space over a field L which is a quadratic extension of some subfield F with char F ≠ 2. Given a nonsingular Hermitian form h on W we can associate with it two smooth projective F -varieties:

• the (2n − 3)-dimensional variety V_{h} of h-isotropic L-lines in W and

• the projective quadric Q_{h} of dimension 2n − 2 defined by the quadratic form q_{h} associated with h, i.e., q_{h}(v) = h(v, v), v ∈ W.

The variety V_{h} is a projective homogeneous variety under the action of the unitary group associated with h. It is also a twisted form of the incidence variety, i.e., of the variety of flags consisting of a dimension one and codimension one linear subspaces in an n-dimensional vector space.

We will use the following classical result of Milnor-Husemoller (see [Le79]):

A quadratic form q on an F -vector space V is the underlying form of a Hermitian form over a quadratic field extension L = F (\sqrt{a}) iff dim V = 2n, q_{L} is hyperbolic, and det q = (−a)^{n} mod F^{2}.

∗Supported by DFG GI706/1-1 and INTAS 05-1000008-8118.

Keywords: Chow motives, Hermitian form, incidence variety, Milnor hypersurface, incompressibility.

1. Essential dimension. Following O. Izhboldin we define the essential dimension of a Hermitian form h as dim_{es}(h) = dim V_{h} − i(q_{h}) + 2, where i(q_{h}) stands for the first Witt index of q_{h} (cf. [KM03]).

The following theorem provides a Hermitian version of the main result of [KM03].

Theorem. Let Y be a complete F -variety with all closed points of even degree. Suppose that Y has a closed point of odd degree over F (V_{h}). Then dim_{es}(h) ≤ dim Y and if dim_{es}(h) = dim Y, then V_{h} is isotropic over F (Y).

Proof. In [Kr07] D. Krashen constructed a projective bundle of rank 1

Bl(Q_{h}) → V_{h},  (1)

where Bl(Q_{h}) is the blow-up of the quadric Q_{h} along some subvariety. In particular, the function field of Q_{h} is a purely transcendental extension of the function field of V_{h} of degree 1, and therefore our theorem follows from [KM03, Theorem 3.1].

2. Incompressibility. A smooth projective F -variety X is called incompressible if any rational map X −−→ X is dominant. The basic example of such varieties are anisotropic quadrics of dimensions 2^{r} − 1.

Theorem. Assume that V_{h} is anisotropic and dim V_{h} = 2^{r} − 1 for some r > 0. Then the variety V_{h} is incompressible.

Proof. We provide two independent proofs.

1. One can notice that V_{h} is a twisted form of the Milnor hypersurface H defined by the line bundle O(1) ⊗ O(1) on ℙ^{n−1} × ℙ^{n−1}, since over a separable closure of F they are given by the same equation ∑_{i=0}^{n−1} x_{i}y_{i} = 0 (see [LM, 2.5.3]). The incompressibility of V_{h} follows now from the Rost degree formula (see [Me03, §7]) and from an explicit computation of the characteristic numbers of H provided in [Me02, Proposition 7.2].

2. As an alternative proof one can notice that the incompressibility of V_{h} follows from the equality dim_{es}(h) = dim V_{h}. The latter can be deduced from the Hoffmann conjecture (proven in [Ka03]) if V_{h} is anisotropic and dim V_{h} = 2^{r} − 1. Indeed, if dim V_{h} = 2^{r} − 1, then dim q_{h} = 2^{r} + 2. Therefore i(q_{h}) = 1 or 2. But by the Milnor-Husemoller theorem i(q_{h}) must be even. Therefore dim_{es}(h) = dim V_{h}.

3. Chow motives. Using formula (1) D. Krashen proved the following formula relating the Chow motives of Q_{h} and V_{h}:

M(Q_{h}) ⊕ ⨁_{i=1}^{n−2} M(ℙ_{L}^{n−1}){i} ≃ M(V_{h}) ⊕ M(V_{h}){1}.  (2)

Consider the subcategory of the category of Chow motives with ℤ_{(2)}- or ℤ/2-coefficients generated by M(V_{h}). Since V_{h} is a projective homogeneous variety, the Krull-Schmidt theorem and the cancellation theorem hold in this subcategory (see [CM06, Corollary 35]). Computing the Poincaré polynomials of M(Q_{h}), M(V_{h}), and M(Spec L) over L we obtain the following explicit formulae:

P (Q_{h,L}, t) = \frac{(1−t^{n})(1+t^{n−1})}{1−t}, P (V_{h,L}, t) = \frac{(1−t^{n})(1−t^{n−1})}{(1−t)^{2}}, P (Spec L, t) = 2. (3)

Analyzing (2) and (3) we obtain that

Theorem. There exists a motive N_{h} such that

M(Q_{h}) ≃ \begin{cases}N_{h} ⊕ N_{h}\{1\}, & if n is even; \\ N_{h} ⊕ M(Spec L)\{n − 1\} ⊕ N_{h}\{1\}, & if n is odd; \end{cases} (4)

and

M(V_{h}) ≃ \begin{cases}N_{h} ⊕ ⨁_{i=0}^{(n−4)/2} M(ℙ_{L}^{n−1})\{2i + 1\}, & if n is even; \\ N_{h} ⊕ ⨁_{i=0}^{(n−3)/2} M(ℙ_{L}^{n−2})\{2i + 1\}, & if n is odd. \end{cases}  (5)

Observe that by the projective bundle theorem M(ℙ_{L}^{m}) ≃ ⨁_{i=0}^{m} M(Spec L){i}.

4. Higher forms of Rost motives. In [Vi00, Theorem 5.1] A. Vishik showed that given a quadratic form q over F divisible by an m-fold Pfister form φ, that is q = q′ ⊗ φ for some form q′, there exists a direct summand N of the motive M(Q_{q}) of the projective quadric Q_{q} associated with q such that

M(Q_{q}) ≃ \begin{cases}N ⊗ M(ℙ_{F}^{2^{m}−1}), & if dim q′ is even; \\ (N ⊗ M(ℙ_{F}^{2^{m}−1})) ⊕ M(Q_{φ})\{\frac{dim q}{2} − 2^{m−1}\}, & if dim q′ is odd. \end{cases}

In view of the Milnor-Husemoller theorem mentioned in the beginning, formula (4) provides a shortened proof of Vishik’s result for m = 1.

References

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