an:07283066
Zbl 07283066
Abramovich, Dan; Chen, Qile; Gross, Mark; Siebert, Bernd
Decomposition of degenerate Gromov-Witten invariants
EN
Compos. Math. 156, No. 10, 2020-2075 (2020).
00455941
2020
j
14N35 14D23
logarithmic Gromov-Witten invariant; moduli stack; logarithmic stable map; degeneration; decomposition; tropical curve; tropical map; rigid tropical curve; Artin fan
Summary: 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.