Noncommutative enhancements of contractions

TL;DR

This paper introduces a noncommutative sheaf-based enhancement of contraction loci in algebraic geometry, providing new tools for studying derived autoequivalences and unifying existing approaches.

Contribution

It develops a general framework for enhancing contraction loci with noncommutative sheaves, extending and unifying previous methods, and constructs a new autoequivalence of the derived category.

Findings

Constructs a global invariant using noncommutative sheaves D.

Provides conditions under which a new endofunctor is an autoequivalence.

Shows automatic autoequivalence conditions for codimension ≥ 3 loci.

Abstract

Given a contraction of a variety X to a base Y, we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions.