Limits of stable maps in a semi-stable degeneration

TL;DR

This paper investigates the limits of stable maps in semi-stable degenerations from an analytic perspective, establishing conditions, deformation theory, and a degeneration formula with explicit examples.

Contribution

It provides an analytic approach to stable map limits in degenerations, complementing existing algebraic methods, and derives a new degeneration formula with detailed deformation-obstruction analysis.

Findings

Stable maps satisfy specific combinatorial and analytical conditions in the central fiber.

A deformation-obstruction theory for the moduli spaces is developed.

An explicit example illustrating the degeneration formula is worked out.

Abstract

Given a semistable degeneration with a simple normal crossings central fiber, Abramovich-Chen-Gross-Siebert [3] proved a degeneration formula that relates the moduli spaces of stable maps in smooth fibers to certain moduli spaces of log-smooth maps in the central fiber. In this paper, we study the same problem from an analytic point of view. We prove that the limiting stable maps in the central fiber satisfy specific combinatorial and analytical conditions. Furthermore, we explain the deformation-obstruction theory of the moduli spaces arising from these conditions, derive a degeneration formula, and work out an explicit example. The earlier version [8] of this paper contains an outline of these ideas for the symplectic category.