Limits of stable maps in a semi-stable degeneration

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.

The earlier version of this paper (with a different title) was an outline within the symplectic category. The is the first detailed version in the Kahler category

References in corpus (6)

• Logarithmic Geometry and Moduli
• Gromov Witten invariants of exploded manifolds
• Decomposition of degenerate Gromov-Witten invariants
• The degeneration formula for stable log maps
• Basic Riemannian Geometry and Sobolev Estimates used in Symplectic Topology
• Tropical gluing formulae for Gromov-Witten invariants