Logarithmic enumerative geometry for curves and sheaves

TL;DR

This work develops a comprehensive logarithmic enhancement of Gromov–Witten and Donaldson–Thomas theory for simple normal crossings pairs $(X|D)$, introducing a logarithmic Nakajima basis to encode boundary data and a family of boundary insertions that couple across multiple boundary components. It establishes a robust degeneration package: a strong, vertex-by-vertex degeneration formula that expresses invariants of a general fiber in terms of invariants attached to the special fiber’s strata, including both cycle-theoretic and numerical versions and generating-function consequences. The framework proves compatibility between the logarithmic GW/DT correspondence and degenerations, implying that conjectures verified on all strata propagate to the general fiber, and provides practical computational tools via exotic insertions and the Nakajima basis. Through detailed construction of moduli spaces (expansions, log maps, Hilbert schemes) and tropical/combinatorial models (cone complexes, Artin fans, subdivisions), the paper unifies tropical, logarithmic, and degeneration techniques to advance a complete logarithmic curve/sheaf theory with concrete examples and applications.

Abstract

We propose a logarithmic enhancement of the Gromov-Witten/Donaldson-Thomas correspondence, with descendants, and study its behavior under simple normal crossings degenerations. The formulation of the logarithmic correspondence requires a matching of tangency conditions with relative insertions. This is achieved via a version of the Nakajima basis for the cohomology of the Hilbert schemes of points on a logarithmic surface. Next, we establish a strong form of the degeneration formula in logarithmic DT theory - the numerical DT invariants of the general fiber of a degeneration are determined by the numerical DT invariants attached to strata of the special fiber. The GW version of this result, which we prove in all target dimensions, strengthens currently known formulas. A key role is played by a certain exotic class of insertions, introduced here, that impose non-local incidence conditions coupled across multiple boundary strata of the target geometry. Finally, we prove compatibility of the new logarithmic GW/DT correspondence with degenerations. In particular, the logarithmic conjecture for all strata of the special fiber of a degeneration implies the traditional GW/DT conjecture on the general fiber. Compatibility is a strong constraint, and can be used to calculate logarithmic DT invariants. Several examples are included to illustrate the nature and utility of the formula.

Figures (8)

• Figure 1: A rigid Chow $1$-complex corresponding to a component in the moduli space $\mathsf{PT}_{2h,\chi}({\mathcal{P}}^2_0\times\mathbb{P}^1)$. The component is virtually birational to $\mathsf{PT}_{2h,\chi}(\mathbb{P}^2\times\mathbb{P}^1)$. The vertices corresponding to $p_i^{\mathsf{trop}}$ are components of the expansion that contain the special fibers of $p_i(t)$.
• Figure 2: A rigid Chow $1$-complex corresponding for conics in the degenerate target ${\mathcal{P}}^2_0\times\mathbb{P}^1$.
• Figure 3: A combinatorial type exhibiting the non-flatness of the evaluation map from the PT moduli space.
• Figure 4: The picture on the left is the dual Newton subdivision and the picture on the right is the dual Chow $1$-complex. The picture has been decorated also with the choice of partition along the legs.
• Figure 5: A schematic of the slices of of $\Sigma_{\mathcal{Y}}$ over different points of the base ${\mathbb R}_{\geq 0}$.

Theorems & Definitions (112)

• Conjecture A
• Theorem B
• Theorem C
• Definition 1.1.1: Cone spaces, smoothness, morphisms, and face maps
• Definition 1.2.1
• Definition 1.6.1
• Definition 1.7.1
• Proposition 1.7.2
• proof
• Proposition 1.7.3