Decomposition of degenerate Gromov–Witten invariants
Dan Abramovich, Qile Chen, Mark Gross, Bernd Siebert
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.
Topics & Concepts
MathematicsLogarithmDegenerate energy levelsDecompositionPure mathematicsModuliModuli spaceMathematical analysisCombinatoricsPhysicsBiologyEcologyQuantum mechanicsAlgebraic Geometry and Number TheoryPolynomial and algebraic computationAdvanced Differential Equations and Dynamical Systems