Decomposition of degenerate Gromov–Witten invariants
By D. Abramovich, Qile Chen, Mark Gross, Bernd S Siebert. Compositio Mathematica, 2017
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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.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/07a16d591fc3c39e1f071f5872941e67fb14081f
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In Computational Theory and Mathematics
In Geometry and Topology
Citations in the corpus
- Logarithmic Gromov–Witten theory with expansions (2019, background): “In the logarithmic theory, a parallel degeneration formula has not materialized, though there is encouraging recent work in this direction [4, 5, 43].”