$S_n$-Equivariant Sheaves and Kozsul Cohomology

TL;DR

This paper offers a new perspective on Kozsul cohomology, connecting it to sheaves and Bridgeland-King-Reid equivalence, and demonstrates polynomial growth of certain dimensions in high degrees.

Contribution

It introduces a novel interpretation of Kozsul cohomology using $S_n$-equivariant sheaves and explores its implications in higher dimensions.

Findings

Kozsul cohomology interpretation aligns with Hilbert scheme in low dimensions

Proves polynomial behavior of $K_{p,q}(B,L)$ in large degree cases

Provides new tools for understanding cohomology in algebraic geometry

Abstract

We give a new interpretation of Kozsul cohomology, which is equivalent under the Bridgeland-King-Reid equivalence to Voisin's Hilbert scheme interpretation in dimensions 1 and 2, but is different in higher dimensions. As an application, we prove that the dimension $K_{p,q}(B,L)$ is a polynomial in $d$ for $L=dA+P$ with $A$ ample and $d$ large enough.