Abramovich, Chen, Gross, Siebert — Decomposition Of Degenerate Gromov–Witten Invariants (2020)

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family \(X \longrightarrow B\) with singular fibre over \(b_0\in B\) yields a family \(\mathscr { M} (X/B,\beta ) \longrightarrow B\) of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over \(b_0\) in terms of rigid tropical maps to the tropicalization of \(X/B\) . This generalizes one aspect of known results in the case that the fibre \(X_{ b_0}\) is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.