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Decomposition of degenerate Gromov-Witten invariants. (English) Zbl 07283066
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.
MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14D23 Stacks and moduli problems
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