Gromov-Witten theory, degenerations, and the tautological ring

Abstract

Gromov-Witten (GW) theory produces Chow and cohomology classes on the moduli of curves, and there are several conjectures/speculations about their relation to the tautological ring. We develop new degeneration techniques to address these. In Chow, we show that GW cycles of complete intersections in products of projective spaces (and more generally a broad class of toric varieties) with restricted insertions are tautological. This gives significant evidence for a 2010 speculation of Pandharipande that GW cycles of varieties over the algebraic numbers are tautological. In particular, the 0-cycle for curves on the quintic threefold is proportional to a zero stratum in the moduli space of stable curves. In cohomology, we show that in normal crossings degenerations, GW classes of the general fiber lie in the span of absolute GW classes of the special fiber strata. This confirms a 2006 conjecture of Levine-Pandharipande for targets that degenerate into elementary pieces, including complete intersections in products of projective spaces and many toric varieties. Our proofs rely on several reconstruction theorems in logarithmic GW theory, which make the logarithmic degeneration formula an inductive tool to compute GW cycles via snc degenerations. We prove a folklore conjecture that logarithmic GW cycles of a pair are determined by absolute invariants of the strata. We prove a conjecture of Urundolil Kumaran and the second author that GW cycles of toric pairs are tautological, and analogous results for broken toric bundles. We also develop tools to study GW cycles with vanishing cohomology and strengthen the logarithmic degeneration formula to allow iteration.