Homogeneity of cohomology classes associated with Koszul matrix factorizations

Abstract

In this work we prove the so called dimension property for the cohomological field theory associated with a homogeneous polynomial W with an isolated singularity, in the algebraic framework of arXiv:1105.2903. This amounts to showing that some cohomology classes on the Deligne-Mumford moduli spaces of stable curves, constructed using Fourier-Mukai type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in arXiv:math/0011032 for certain characteristic classes of Koszul matrix factorizations of 0. To reduce to this result we use the theory of Fourier-Mukai type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge-Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.

v1: 42 pages; v2: 47 pages, added a result on quasihomogeneous polynomials

References in corpus (6)

• Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence
• A Geometric Approach to Orlov's Theorem
• Matrix factorizations for nonaffine LG-models
• Coherent analogues of matrix factorizations and relative singularity categories
• Topological K-theory of complex noncommutative spaces
• Infinitesimal Derived Torelli Theorem for K3 surfaces