Logarithmic Gromov-Witten theory with expansions

Abstract

We construct 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 show that there exist proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$, equipped with virtual classes. Each point in such a moduli space parameterizes a map from a nodal curve to an expanded degeneration of $X$ that is dimensionally transverse to the strata. In the context of maps to a simple normal crossings degeneration, the virtual fundamental class is known to decompose as a sum over tropical maps. We use the expanded formalism to prove the degeneration formula -- we reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.