Asymptotic vanishing of syzygies of algebraic varieties

TL;DR

This paper proves a conjecture on the asymptotic vanishing of syzygies for algebraic varieties, clarifying the behavior of minimal free resolutions as line bundle positivity increases.

Contribution

It confirms Ein--Lazarsfeld's conjecture on asymptotic syzygy vanishing and resolves the key case involving products of three projective spaces.

Findings

Proves asymptotic vanishing of syzygies for algebraic varieties.

Complements existing nonvanishing results to describe the full asymptotic picture.

Resolves the case of products of three projective spaces.

Abstract

The purpose of this paper is to prove Ein--Lazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein--Lazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.