Gromov-Witten theory of tame Deligne-Mumford stacks in mixed characteristic

TL;DR

This paper extends Gromov-Witten theory to tame Deligne-Mumford stacks over Dedekind domains, establishing invariance across characteristics and deriving a reconstruction theorem for genus zero invariants.

Contribution

It introduces Gromov-Witten classes and invariants for stacks over Dedekind domains and proves their deformation invariance and characteristic independence.

Findings

Invariants are deformation invariant across characteristics

Genus zero invariants satisfy WDVV equations

Reconstruction theorem for genus zero invariants

Abstract

We define Gromov-Witten classes and invariants of smooth proper tame Deligne-Mumford stacks of finite presentation over a Dedekind domain. We prove that they are deformation invariants and verify the fundamental axioms. For a smooth proper tame Deligne-Mumford stack over a Dedekind domain, we prove that the invariants of fibers in different characteristics are the same. We show that genus zero Gromov-Witten invariants define a potential which satisfies the WDVV equation and we deduce from this a reconstruction theorem for genus zero Gromov-Witten invariants in arbitrary characteristic.