Deformation Theory of Log Pseudo-holomorphic Curves and Logarithmic Ruan-Tian Perturbations

TL;DR

This paper develops a deformation theory for log pseudo-holomorphic curves in symplectic pairs, introducing a logarithmic Ruan-Tian perturbation framework to define Gromov-Witten invariants in this setting.

Contribution

It introduces a deformation theory for log curves and constructs a logarithmic analog of Ruan-Tian perturbations, enabling Gromov-Witten invariants for semi-positive pairs.

Findings

Transversality for simple maps in each stratum

Construction of Gromov-Witten type invariants in arbitrary genus

Framework for future log Gromov-Witten invariants using local perturbations

Abstract

In a previous paper [FT1], for any logarithmic symplectic pair (X,D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural set up for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove that the subspace of simple maps in each stratum is cut transversely. Such perturbations enable a geometric construction of Gromov-Witten type invariants for certain semi-positive pairs (X, D) in arbitrary genera. In future works, we will use local perturbations and a gluing theorem to construct log Gromov- Witten invariants of arbitrary such pair (X, D).