ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

4. Algebraic Geometry

1991 MS Classification: 14, 52, 81 Batyrev, Victor, Universtität Tübingen, Germany Mirror symmetry and toric geometry

A toric variety of dimension n over C is a normal algebraic variety V containing (C \ {0})^{n} as a Zariski open subset in such a way that the algebraic group action of (C \ {0})^{n} on itself extends to a regular action on V. A projective toric variety V is called a Gorenstein toric Fano variety if V has at worst Gorenstein singularities and the anticanonical sheaf on V is ample. A convex n-dimensional polyhedron Δ ⊂ R^{n} is called a reflexive polyhedron if the following conditions are satisfied: 1) all vertices of Δ belong to Z^{n}; 2) the point (0, …, 0) belongs to the interior of Δ; 3) all vertices of the polar polyhedron

Δ^{∗} := {(x_{1}, …, x_{n}) ∈ R^{n} : ∑_{i=1}^{n} x_{i}y_{i} ≥ −1 ∀(y_{1}, …, y_{n}) ∈ Δ}

belong to Z^{n}. There exists a 1-to-1 correspondence between the isomorphism classes of n-dimensional Gorenstein toric Fano varieties V and ndimensional reflexive polyhedra Δ ⊂ R^{n} considered up to a linear transformation from GL(n, Z). If Δ is a reflexive polyhedron, then the polar polyhedron Δ^{∗} is again reflexive. The duality between reflexive polyhedra Δ ↔ Δ^{∗} induces a duality V ↔ V^{∗} between the corresponding Gorenstein toric Fano varieties. It turns out that Calabi-Yau hypersurfaces X in V are mirror symmetric to Calabi-Yau hypersurfaces X^{∗} in V^{∗}. Recently this observation became a powerful tool in the investigation of the mirror symmetry phenomenon discovered by physicists. Reference: V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geometry 3 (1994)

ICM 1998 Berlin · Plenary and Invited Lectures