Gromov-Witten theory, degenerations, and the tautological ring

TL;DR

Maulik and Ranganathan develop a comprehensive logarithmic degeneration framework to relate Gromov–Witten cycles to the tautological ring. They prove Chow-tautologicality for GW cycles of broad ambient geometries (complete intersections in products of projective spaces and many toric varieties with breakable Newton polytopes) and cohomological tautologicality for general fibers in normal crossings degenerations, via a reconstruction toolkit that iterates the log degeneration formula. Central innovations include a reconstruction program (exotic/non-exotic trading, rigidification), a universal P^1-bundle approach that reduces questions to base data on Picard-like stacks, and a robust log–absolute correspondence that expresses log cycles in terms of absolute strata data. The results provide strong evidence for Pandharipande–Levine–Pandharipande conjectures, illuminate how degenerations constrain GW theory, and suggest broad future directions for toric, Grassmannian, and K3/abelian settings.

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.

Figures (14)

• Figure 1: Products of dilated simplices are examples of breakable Newton polytopes and correspond to degenerations of hypersurfaces in products of projective space. Unimodular triangulations of $3\cdot\Delta_3$ and $2\cdot\Delta_2\times2\cdot\Delta_1$ are shown.
• Figure 2: An example of a polytope that is breakable but cannot be cut into just two pieces. This gives rise to an orbifold snc degeneration and can be used to produce four dimensional polytopes with analogous properties.
• Figure 3: A depiction of a degeneration with simple normal crossings special fiber, and the snc pair arising in the special fiber.
• Figure 4: A rigid tropical curve in a dual complex $\Delta$ on the left, given by the three interior edges. A non-rigid curve on the right that arises as a deformation. The dual complex is the triangle and the tropical curve is the union of the edges in the interior.
• Figure 5: The asymptotic star of a $1$-complex.

Theorems & Definitions (113)

• Definition 1.1.1: Disjoint contact orders
• Lemma 1.1.2
• proof
• Theorem 1.1.3
• Theorem 1.1.4
• Theorem 1.2.1
• Corollary 1.2.2
• Remark 1.2.3
• Definition 1.3.1: Broken toric bundle
• Theorem 1.3.2