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Gopakumar-Vafa invariants and Macdonald formula II

This paper establishes the cohomological Gopakumar-Vafa/Pandharipande-Thomas correspondence for the local plane and quadric in all effective curve classes and Euler characteristics by proving that the strict supports of stable pair vanishing cycle direct images are closures of transverse unions of smooth connected curves, thereby reducing the problem to an identity of semisimplified perverse direct images on the Chow variety via the family Macdonald formula.

Original authors: Lutian Zhao

Published 2026-09-16

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CC BY 4.0

Original authors: Lutian Zhao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a branch dedicated to counting and classifying shapes that exist in higher dimensions. These are not the simple triangles and circles of a child's drawing, but complex, multi-layered structures that can twist and fold in ways our three-dimensional intuition cannot easily grasp. One of the most persistent challenges in this field is understanding how these shapes behave when they are slightly altered or when they break apart into smaller pieces. Mathematicians have developed different "languages" or methods to describe these objects. Some methods focus on the shapes themselves, while others look at the bundles of information, or sheaves, that can be wrapped around them. For decades, a major goal has been to prove that these different languages are actually describing the same underlying reality, just using different vocabulary. If they are indeed equivalent, it means that a calculation done in one language can be translated perfectly into another, unlocking new ways to solve problems that were previously impossible.

This paper, authored by Lutian Zhao, tackles a specific and difficult version of this translation problem. The author focuses on two particular types of surfaces: the familiar flat plane and a shape that looks like a stretched-out square, known as a quadric. In the mathematical universe, these surfaces are often studied by attaching a special kind of three-dimensional space to them, creating a "local" environment where curves can live. The central question is how to count the ways these curves can sit inside this space, especially when the curves are allowed to be broken, disconnected, or have multiple layers. The paper aims to prove that two specific counting methods—one based on stable pairs (which are essentially a curve with a specific marking) and another based on counting sheaves (bundles of data)—yield identical results for every possible curve class and every possible Euler characteristic (a number that describes the shape's topological complexity).

The journey to this proof begins by looking at the "supports" of these mathematical objects. In simple terms, a support is the specific location or region where a mathematical object actually exists. The author first had to determine exactly where these objects could live. It turns out that the most complex objects, which might seem to exist in messy, non-reduced forms, are actually built from simpler, smooth, and connected curves that intersect each other cleanly, like roads crossing at a single point without forming a traffic jam. The paper rigorously proves that every possible configuration of these curves is essentially a collection of these smooth, transverse intersections. This is a crucial step because it narrows down the infinite possibilities to a manageable set of geometric scenarios.

Once the possible locations were mapped out, the author used a powerful technique called wall-crossing. Imagine a landscape where the rules for what counts as a stable object change as you move across a boundary, or a "wall." By carefully analyzing how the counts change as one crosses these walls, the author could relate the complex counting problem to a much simpler one. The paper demonstrates that for the two surfaces in question, the behavior of these objects is constrained to a specific, finite range of complexity. Outside of this range, the objects simply do not exist. Inside this range, the author shows that the counting problem can be reduced to a calculation on the "reduced" locus, which is the space of smooth, non-repeating curves.

The final breakthrough comes from applying a known formula, originally developed for symmetric products of curves, to this specific setting. The author proves that the complex data gathered from the stable pairs is exactly the same as the data gathered from the sheaf-counting method. This is not just a numerical coincidence; the paper establishes a deep structural identity between the two, showing that they are two sides of the same coin. The proof relies on a clever use of "point modifications," which are operations that add or remove a single point from a curve. By studying how these operations interact, the author derives a relationship that forces the counting numbers to align perfectly.

The result is a complete and rigorous confirmation of the correspondence between these two counting methods for the local plane and the quadric. The paper rules out the possibility that there are any "hidden" configurations or strange, non-reduced cycles that would break this correspondence. Instead, it confirms that the entire structure is built from the clean, transverse unions of smooth curves. This work provides a solid foundation for understanding how different mathematical perspectives on curve counting are unified, offering a clear and complete picture of these geometric objects in these specific environments. It stands as a definitive proof, leaving no ambiguity about the relationship between the stable pair invariants and the sheaf-theoretic invariants in these cases.

Technical Summary: Gopakumar-Vafa Invariants and Macdonald Formula II

Problem Statement
This paper addresses the cohomological Gopakumar-Vafa (GV) / Pandharipande-Thomas (PT) correspondence for local surfaces, specifically the local plane ( $X = \text{Tot}(K_{\mathbb{P}^2})$ ) and the local quadric ( $X = \text{Tot}(K_{\mathbb{F}_0})$ ). While previous work established this correspondence for reduced curves and specific curve classes, a complete proof for all effective curve classes and Euler characteristics required a rigorous understanding of the supports of the perverse sheaves arising from stable pair moduli spaces.

The core difficulty lies in the fact that the Macdonald formula, which relates the cohomology of symmetric products of curves to that of compactified Jacobians, applies naturally to reduced curves. However, stable pair moduli spaces contain contributions from non-reduced cycles (multiple covers, non-reduced divisors). A "support theorem" is necessary to demonstrate that the simple perverse constituents of the direct images of vanishing cycles are supported on closures of loci of transverse unions of smooth connected curves, thereby allowing the reduction to the reduced-curve case.

Methodology
The proof employs a synthesis of wall-crossing techniques, dimensional reduction, and Hecke correspondences on derived stacks of surface pairs.

Support Analysis via Wall-Crossing:
The author utilizes Toda's stability conditions and wall-crossing formulas to relate the PT direct images in different chambers. By applying an induction hypothesis on the curve degree $d = L \cdot \beta$ , the paper expresses the change of chamber as convolutions of rank-one factors (stable pairs) with semistable sheaf factors.
- Key Tool: The cohomological integrality theorem for smooth stacks (BDINP25) is applied to the semistable sheaf factors, showing their direct images belong to a specific category of complexes with controlled supports.
- Duality: Toda's duality exchanges Euler characteristics $\chi$ and $-\chi$ , reducing the range of $\chi$ that needs to be analyzed to a finite interval $[1-g_\beta, g_\beta-1]$ .
Dimensional Reduction and Slope Bounds:
To handle the finite range of $\chi$ , the paper introduces a lower bound on the Harder-Narasimhan (HN) slopes of the surface sheaves underlying the stable pairs.
- Dimensional Reduction: Using Kinjo's dimensional reduction, the vanishing cycle direct images on the shifted cotangent stack are identified with dualizing complexes on the base stack of surface pairs.
- Induction: The induction hypothesis is applied to the rank-one factors in the HN filtration, which have strictly smaller curve degrees.
Hecke Correspondences and Commutators:
The paper constructs point-modification operators (creation and removal of points) on bounded stacks of surface pairs over the Chow variety.
- Relative Quot Stacks: By considering a pencil of divisors and point modifications on a smooth divisor $D$ , the author defines operators $e_\chi$ and $f_\chi$ .
- Commutator Relation: A key step involves computing the commutator $f_\chi e_\chi - e_{\chi-1} f_{\chi-1}$ . Using a relative Quot construction and analyzing the difference of two compositions over a derived Hecke stack, the paper proves this commutator acts as a scalar multiplication by $(D \cdot \beta)$ on the dualizing complex.
- Vanishing Argument: This commutator relation, combined with the finiteness of the $\chi$ range, forces the relevant objects in the Verdier quotient (modulo the desired supports) to vanish, thereby proving the support theorem.
Application of the Macdonald Formula:
Once the support theorem is established, the problem reduces to the reduced locus. The paper applies the family Macdonald formula (MSV21) on the reduced locus of the Chow variety. This relates the cohomology of the relative Hilbert schemes to the intersection cohomology of the compactified Jacobians, yielding the correspondence as an identity of semisimplified perverse direct images.

Key Contributions and Results

Theorem 1.1 (Support Theorem): For $S = \mathbb{P}^2$ and $S = \mathbb{F}_0$ , every simple constituent of the perverse cohomology sheaf $p^i K_{\chi, \beta}$ (the direct image of the PT vanishing cycle sheaf) has strict support $\Sigma_\lambda$ , where $\Sigma_\lambda$ is the closure of the locus of transverse unions of smooth connected curves with component classes partitioning $\beta$ . Crucially, every such support meets the locus of reduced divisors $B^{\text{red}}_\beta$ .
Theorem 1.4 (GV/PT Correspondence): There is an isomorphism of formal series of perverse sheaves on the Chow variety:
$Z_{PT}(q, Q) \cong \text{Exp}_{\text{Chow}}\left( \sum_{\beta > 0} A_\beta(q) Q_\beta \right)$
where $Z_{PT}$ is the stable pair series and the right-hand side is constructed from the cohomological GV data $A_\beta(q)$ via symmetric powers and addition maps.
Extension to Non-Reduced Cycles: The paper proves that the correspondence holds for all effective curve classes, not just those containing smooth members, by showing that non-reduced contributions are determined by the reduced locus via intermediate extension.
Refined Product Formula: By taking hypercohomology and stalks, the result recovers the numerical GV/PT correspondence and its local version, as well as the refined product formula involving Lefschetz gradings.

Significance
The paper claims to complete the cohomological GV/PT correspondence for local surfaces in all effective curve classes and Euler characteristics. Its significance lies in:

Rigorous Support Control: It provides the necessary support theorem to extend the Macdonald formula from reduced curves to all one-cycles, resolving a gap in previous literature (specifically [Zha26]).
Unification of Methods: It successfully integrates wall-crossing, dimensional reduction, and Hecke algebra techniques (commutator relations) to prove a structural result about perverse sheaves on Chow varieties.
Generalization: It generalizes the correspondence from irreducible curves to arbitrary effective classes, including those with negative arithmetic genus and non-ample classes on the quadric, by carefully analyzing the behavior of semistable sheaves in ruling classes.

The work relies on the foundational results of Maulik, Shen, Toda, and others, but the specific contribution is the proof of the support theorem and the subsequent derivation of the full correspondence via the intermediate extension of the reduced-curve identity.

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