Elliptic genera of toric varieties and applications to mirror symmetry

1999/04/22 by Lev Borisov, Lev A. Borisov, Anatoly Libgober · 5 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Botany #Calabi–Yau manifold #Elliptic curve #Genus #Geometry and complex manifolds #Gravitational singularity #Hypersurface #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Mirror symmetry #Pure mathematics #Toric variety #hep-th #math.AG #msc:14J32 #msc:14M25 #msc:81T40

paper · pdf · doi:10.1007/s002220000058

32 pages, LaTeX

arxiv created 1999/04/22 · openalex publication_date 2000/05/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

The paper contains a proof that elliptic genus of a Calabi-Yau manifold is a Jacobi form, finds in which dimensions the elliptic genus is determined by the Hodge numbers and shows that elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror coincide up to sign. The proof of the mirror property is based on the extension of elliptic genus to Calabi-Yau hypersurfaces in toric varieties with Gorenstein singularities.

Citations

• Complex vector bundles and Jacobi forms
• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Mirror Symmetry and Algebraic Geometry
• Topological Methods in Algebraic Geometry

Cited by

• Lattice points in polytope boundaries and formal geometric quantization of singular Calabi Yau hypersurfaces in toric varieties
• Tracking the symmetries of \mathbb Z₃-orbifold K3s within the Mathieu groups
• The Chiral de Rham Complex and Positivity of the Equivariant Signature of the Loop Space
• Elliptic classes of Schubert varieties via Bott–Samelson resolution
• Vertex operator algebra [wikipedia]