Global matrix factorizations

TL;DR

This paper explores matrix factorizations and curved module categories for Landau-Ginzburg models, establishing their properties and computing Hochschild cohomology, with implications for their smoothness, properness, and Calabi-Yau conditions.

Contribution

It extends the theory of matrix factorizations for Landau-Ginzburg models by establishing model structures, identifying generators, and computing Hochschild cohomology.

Findings

Categories are smooth and proper when the singular locus of W is proper.

Hochschild cohomology matches derived endomorphisms of the diagonal curved module.

Categories are Calabi-Yau when the total space X is Calabi-Yau.

Abstract

We study matrix factorization and curved module categories for Landau-Ginzburg models (X,W) with X a smooth variety, extending parts of the work of Dyckerhoff. Following Positselski, we equip these categories with model category structures. Using results of Rouquier and Orlov, we identify compact generators. Via To\"en's derived Morita theory, we identify Hochschild cohomology with derived endomorphisms of the diagonal curved module; we compute the latter and get the expected result. Finally, we show that our categories are smooth, proper when the singular locus of W is proper, and Calabi-Yau when the total space X is Calabi-Yau.