Koszul duality between Betti and Cohomology numbers in Calabi-Yau case

TL;DR

This paper explores a duality between Betti and cohomology numbers for Calabi-Yau varieties, revealing formulas that connect algebraic and geometric invariants through a specific resolution technique.

Contribution

It establishes a Koszul duality between Betti and cohomology numbers in the Calabi-Yau setting using the box-product resolution of the diagonal.

Findings

Formulas relating Betti and cohomology numbers similar to classical formulas.

Application of the box-product resolution of the diagonal to establish duality.

Extension of known dualities to the Calabi-Yau case.

Abstract

Let $X$ be a smooth projective Calabi-Yau variety and $L$ a Koszul line bundle on $X$. We show that for Betti numbers of a maximal Cohen-Macaulay module over the homogeneous coordinate ring $A$ of $X$ there are formulas similar to the formulas for cohomology number. This similarity is realized via the box-product resolution of the diagonal $\Delta_X \subset X \times X$.