Decomposition of degenerate Gromov–Witten invariants. (19th October 2020)

• Record Type: Journal Article
• Authors: Abramovich, Dan; Chen, Qile; Gross, Mark; Siebert, Bernd
• Is Part Of: Compositio mathematica. Volume 156:Number 10(2020)
• Page Start: 2020
• Page End: 2075
• Publication Date: 2020-10-19
• Subjects: logarithmic Gromov–Witten invariant, -- moduli stack, -- logarithmic stable map, -- degeneration, -- decomposition, -- tropical curve, -- tropical map, -- rigid tropical curve, -- Artin fan; 14N35, -- 14D23; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X20007393
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : 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.