Gromov-Witten theory of schemes in mixed characteristic

TL;DR

This paper extends Gromov-Witten theory to smooth projective schemes over Dedekind domains, establishing invariance across characteristics and deriving a genus zero reconstruction theorem using WDVV equations.

Contribution

It introduces Gromov-Witten classes and invariants in mixed characteristic and proves their deformation invariance and fundamental axioms.

Findings

Invariants are the same for fibers in different characteristics.

Genus zero invariants satisfy WDVV equations.

Reconstruction theorem for genus zero invariants in arbitrary characteristic.

Abstract

We define Gromov-Witten classes and invariants of smooth projective schemes of finite presentation over a Dedekind domain. We prove that they are deformation invariants and verify the fundamental axioms. For a smooth projective scheme 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.