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Decomposition of degenerate Gromov–Witten invariants
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 10 / October 2020
- Published online by Cambridge University Press:
- 19 November 2020, pp. 2020-2075
- Print publication:
- October 2020
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- Article
- Export citation
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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.
