What is a stable log map?

TL;DR

This paper establishes an equivalence between algebraic and symplectic notions of stable log maps for smooth projective varieties with divisors, and refines the symplectic convergence concept to define a symplectic analogue of algebraic log maps.

Contribution

It proves an equivalence between algebraic and symplectic stable log maps and introduces a refined symplectic convergence notion for log Gromov-Witten theory.

Findings

Proved equivalence between algebraic and symplectic stable log maps.

Defined the symplectic analogue of fine saturated algebraic log maps.

Refined the notion of log Gromov convergence in symplectic topology.

Abstract

Let $X$ be a smooth projective variety over $\mathbb{C}$ with a simple normal crossings divisor $D\subset X$. We compare the notions of stable log maps to $(X,D)$ in algebraic geometry and symplectic topology. In particular, we prove an equivalence between fine (basic) algebraic log maps and symplectic log maps, and we define the symplectic analogue of fine saturated algebraic log maps by refining the notion of log Gromov convergence.