Deformations of algebras defined by tilting bundles

TL;DR

This paper constructs noncommutative algebras derived from deformations of schemes with tilting bundles, extending to equivariant cases and applying to rational surface singularities and symplectic reflection algebras.

Contribution

It proves tilting bundles lift to deformations and produces graded deformations of endomorphism algebras, linking geometric deformations to algebraic structures.

Findings

Lifting tilting bundles to infinitesimal deformations

Construction of graded deformations in equivariant settings

Applications to rational surface singularities and symplectic reflection algebras

Abstract

In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformations of that scheme and then expanding this result to $\mathbb{C}^*$-equivariant deformations over schemes with a good $\mathbb{C}^*$-action. In both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra, and this is a graded deformation in the $\mathbb{C}^*$-equivariant case. We apply our results to rational surface singularities, generalising the deformed preprojective algebras, and also to symplectic situations where the deformations produced are related to symplectic reflection algebras.