Sheaves of maximal intersection and multiplicities of stable log maps

TL;DR

This paper advances the understanding of log Gromov-Witten invariants on surfaces by explicitly computing contributions from various curve components, introducing a moduli space of logarithmic sheaves, and comparing deformation theories.

Contribution

It introduces a moduli space of logarithmic sheaves for non-rigid curves and analyzes their contributions to log Gromov-Witten invariants, extending previous work on rigid curves.

Findings

Computed contributions of non-rigid irreducible curves in log Calabi-Yau surfaces.

Explicitly described moduli space components for unions of rigid curves.

Compared logarithmic deformation theory with relative stable maps.

Abstract

A great number of theoretical results are known about log Gromov-Witten invariants, but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov-Witten invariants. The first such calculation by Gross-Pandharipande-Siebert deals with multiple covers over rigid curves in the log Calabi-Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi-Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of "logarithmic" 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.