Homogeneity of cohomology classes associated with Koszul matrix factorizations

TL;DR

This paper proves the dimension property for cohomological field theories linked to homogeneous polynomials with isolated singularities, using Fourier-Mukai functors and Hochschild homology techniques.

Contribution

It establishes the dimension property for these theories within an algebraic framework, extending previous results to quasihomogeneous polynomials.

Findings

Dimension property proven for cohomological classes on moduli spaces

Homogeneity results applied to Koszul matrix factorizations

Method involves Fourier-Mukai functors and Hochschild homology

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.