Logarithmic Gromov–Witten theory with expansions

Abstract

We construct a version of relative Gromov-Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we construct virtually smooth and proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$. Each point in such a moduli space parameterizes maps from nodal curves to expanded degenerations of $X$ that are dimensionally transverse to the strata. We use the expanded formalism to reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.