On a logarithmic version of the derived McKay correspondence

TL;DR

This paper extends the derived McKay correspondence to log algebraic stacks, showing invariance of certain categories under log blow-up, thus broadening the scope of the original geometric correspondence.

Contribution

It introduces a logarithmic version of the derived McKay correspondence for log stacks with locally free structures, connecting infinite root stacks and valuativizations.

Findings

Category of coherent parabolic sheaves is invariant under log blow-up.

Establishes a new logarithmic framework for the derived McKay correspondence.

Generalizes Kawamata's result to log algebraic stacks.

Abstract

We globalize the derived version of the McKay correspondence of Bridgeland-King-Reid, proven by Kawamata in the case of abelian quotient singularities, to certain log algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible log blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under log blow-up, up to Morita equivalence.