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Georg Oberdieck

Publications and source records attributed to Georg Oberdieck.

At least 19 recordsLinked to original sources
arXiv2026

Gromov-Witten theory of abelian varieties in families and modular forms

This is the first paper in a series on the Gromov-Witten theory of the universal abelian variety over the moduli space of principally polarized abelian varieties of dimension $h$. We conjecture that the generating series of Gromov-Witten classes, when summed over the degree against the principal polarization, is a cycle-valued quasimodular form for $\mathrm{SL}_2(\mathbb{Z})$ and satisfy a holomorphic anomaly equation. These conjectures generalize the quasimodularity of the Gromov-Witten theory of elliptic curves to higher dimension and raise interesting questions regarding enumerative mirror symmetry for abelian varieties. In genus $1$ it specializes to a conjecture of Greer and Lian which was proven by Iribar Lopez after tautological projection. We also discuss a special family of abelian varieties with a conjectural relation to Siegel quasimodular forms of higher genus. The main result of the paper is a proof of the conjectures in genus $2$ after tautological projection. For that we introduce quotient Gromov-Witten invariants which are indexed by the characteristic polynomial of the curve class and are shown to determine all descendent Gromov-Witten invariants satisfying a degree conditions. We then give an explicit formula for all genus $2$ quotient invariants after tautological projection as the Shimura lift of the product of two Eisenstein series. The formula is based on a curious modular identity derived in a joint appendix with Brandon Williams.

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arXiv2026

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

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arXiv2026

The multiple cover formula for $K3$ and abelian surfaces

All reduced descendent Gromov-Witten invariants of $K3$ and abelian surfaces in primitive curve classes can be calculated by the methods of \cite{BOPY,MPT}. To handle the imprimitive curve classes, a multiple cover formula was conjectured in \cite{ObPand} for $K3$ surfaces and in \cite{O_NLGW} for abelian surfaces. We prove here that both descendent multiple cover formulas are implied by the conjectural families GW/PT correspondence for semipositive relative 3-folds with primary insertions. The implication is proven by showing that the multiple cover formula for $S$ can be recast as a property of an appropriate localization vertex for the relative 3-fold Gromov-Witten theory of $(S\times \mathbb{P}^1/S_0 \cup S_\infty)$. The families GW/PT correspondence then transfers the multiple cover formula from the Gromov-Witten side to the stable pairs side where the formula is proven geometrically by studying cosections and applying universality properties. Along the way, we prove a DT/PT correspondence for the reduced theories of $(S\times \mathbb{P}^1/S_0 \cup S_\infty)$ using the wallcrossing techniques of Kuhn-Liu-Thimm \cite{KLT2,KLT}.

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arXiv2025

On the descendent Gromov-Witten theory of a K3 surface

We study the reduced descendent Gromov-Witten theory of K3 surfaces in primitive curve classes. We present a conjectural closed formula for the stationary theory, which generalizes the Bryan-Leung formula. We also prove a new recursion that allows to remove descendent insertions of $1$ in many instances. Together this yields an efficient way to compute a large class of invariants (modulo the conjecture on the stationary part). As a corollary we conjecture a surprising polynomial structure which underlies the Gromov-Witten invariants of the K3 surface.

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arXiv2025

Quasi-modular forms for the orthogonal group and Gromov-Witten theory of Enriques surfaces

We develop the theory of almost-holomorphic and quasimodular forms for orthogonal groups of a lattice of signature $(2,n)$ through orthogonal lowering and raising operators. The interactions with the regularized theta lift of Borcherds is a central theme. Our main results are: (i) the constant-term morphism, which sends an almost-holomorphic modular form to its associated quasimodular form, is an isomorphism, (ii) description of spaces of quasimodular forms in terms of vector-valued modular forms, (iii) the lowering and raising operators satisfy equivariance properties with the theta lift, (iv) a weight-depth inequality which is a necessary and sufficient criterion for the theta lift of an almost-holomorphic modular form to be almost-holomorphic, (v) an explicit formula for the series expansion of the lift of any almost-holomorphic modular form, (vi) the Fourier-Jacobi coefficients of an orthogonal quasimodular form are quasi-Jacobi forms. As a geometric application, we conjecture that the Gromov-Witten potentials of Enriques and bielliptic surfaces are orthogonal quasimodular forms and satisfy holomorphic anomaly equations with respect to the lowering operators on quasimodular forms. We show that parallel statements for an arbitrary K3 or abelian-surface fibration do not hold.

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arXiv2024

Curve counting on the Enriques surface and the Klemm-Mariño formula

We determine the Gromov-Witten invariants of the local Enriques surfaces for all genera and curve classes and prove the Klemm-Mariño formula. In particular, we show that the generating series of genus $1$ invariants of the Enriques surface is the Fourier expansion of a certain power of Borcherds automorphic form on the moduli space of Enriques surfaces. We also determine all Vafa-Witten invariants of the Enriques surface. The proof uses the correspondence between Gromov-Witten and Pandharipande-Thomas theory. On the Gromov-Witten side we prove the relative Gromov-Witten potentials of an elliptic Enriques surfaces are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. On the sheaf side, we relate the Pandharipande-Thomas invariants of the Enriques-Calabi-Yau threefold in fiber classes to the $2$-dimensional Donaldson-Thomas invariants by a version of Toda's formula for local K3 surfaces. Altogether, we obtain sufficient modular constraints to determine all invariants from basic geometric computations.

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arXiv2024

Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

We conjecture that the generating series of Gromov-Witten invariants of the Hilbert schemes of $n$ points on a K3 surface are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture in genus $0$ and for at most $3$ markings - for all Hilbert schemes and for arbitrary curve classes. In particular, for fixed $n$, the reduced quantum cohomologies of all hyperkähler varieties of $K3^{[n]}$-type are determined up to finitely many coefficients. As an application we show that the generating series of $2$-point Gromov-Witten classes are vector-valued Jacobi forms of weight $-10$, and that the fiberwise Donaldson-Thomas partition functions of an order two CHL Calabi-Yau threefold are Jacobi forms.

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arXiv2024

Towards refined curve counting on the Enriques surface II: Motivic refinements

We study the motivic Pandharipande-Thomas invariants of the Enriques Calabi-Yau threefolds in fiber curve classes by basic computations and analysis of a wallcrossing formula of Toda. Motivated by our results we conjecture a formula for the perverse Hodge numbers of the compactified Jacobian fibration of linar systems on Enriques surfaces in terms of its Betti numbers. This leads to an asymptotic for said Hodge numbers and raises questions about the behaviour of the extremal Hodge numbers.

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arXiv2024

Towards refined curve counting on the Enriques surface I: K-theoretic refinements

We conjecture an explicit formula for the $K$-theoretically refined Vafa-Witten invariants of the Enriques surface. By a wall-crossing argument the conjecture is equivalent to a new conjectural formula for the K-theoretically refined Pandharipande-Thomas invariants of the local Enriques surface. Evidence for the conjecture is given in several cases. We also comment on the case of K3 surfaces previously studied by Thomas.

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arXiv2023

Quantum cohomology of the Hilbert scheme of points on an elliptic surface

We determine the quantum multiplication with divisor classes on the Hilbert scheme of points on an elliptic surface $S \to Σ$ for all curve classes which are contracted by the induced fibration $S^{[n]} \to Σ^{[n]}$. The formula is expressed in terms of explicit operators on Fock space. The structure constants are meromorphic quasi-Jacobi forms of index $0$. Combining with work of Hu-Li-Qin, this determines the quantum multiplication with divisors on the Hilbert scheme of elliptic surfaces with $p_g(S)>0$. We also determine the equivariant quantum multiplication with divisor classes for the Hilbert scheme of points on the product $E \times \mathbb{C}$. The proof of our formula is based on Nesterov's Hilb/PT wall-crossing, a newly established GW/PT correspondence for the product of an elliptic surface times a curve, and new computations in the Gromov-Witten theory of an elliptic curve.

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arXiv2023

Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes

Let $S$ be a K3 surface. We study the reduced Donaldson-Thomas theory of the cap $(S \times \mathbb{P}^1) / S_{\infty}$ by a second cosection argument. We obtain four main results: (i) A multiple cover formula for the rank 1 Donaldson-Thomas theory of $\mathrm{K3} \times E$, leading to a complete solution of this theory. (ii) Evaluation of the wall-crossing term in Nesterov's quasi-map wallcrossing between the punctual Hilbert schemes and Donaldson-Thomas theory of $\mathrm{K3} \times \text{Curve}$. (iii) A multiple cover formula for the genus $0$ Gromov-Witten theory of punctual Hilbert schemes. (iv) Explicit evaluations of virtual Euler numbers of Quot schemes of stable sheaves on K3 surfaces.

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arXiv2023

Pandharipande-Thomas theory of elliptic threefolds, quasi-Jacobi forms and holomorphic anomaly equations

Let $π: X \to B$ be an elliptically fibered threefold satisfying $c_3(T_X \otimes ω_X)=0$. We conjecture that the $π$-relative generating series of Pandharipande-Thomas invariants of $X$ are quasi-Jacobi forms and satisfy two holomorphic anomaly equations. For elliptic Calabi-Yau threefolds our conjectures specialize to the Huang-Katz-Klemm conjecture. The proposed formulas constitute the first case of holomorphic anomaly equations in Pandharipande-Thomas theory. We prove our conjectures for the equivariant Pandharipande-Thomas theory of $\mathbb{C}^2 \times E$ when specialized to the anti-diagonal action. For $K3 \times \mathbb{C}$ we state reduced versions of our conjectures. As a corollary we find an explicit conjectural formula for the stationary theory generalizing the Katz-Klemm-Vafa formula for K3 surfaces. Further evidence is available for $\mathbb{P}^2 \times E$ based on earlier work of the second author. To deal with elliptic threefolds with $c_3(T_X \otimes ω_X) \neq 0$ we show that the moduli space of $π$-stable pairs is represented by a proper algebraic space. We conjecture that the associated $π$-stable pair invariants form quasi-Jacobi forms.

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arXiv2022

Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on $K3$ surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the $K3$ surface. As an application we establish the higher rank Segre-Verlinde correspondence for $K3$ surfaces as conjectured by Göttsche and Kool.

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arXiv2022

Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds

As an analogy to Gopakumar-Vafa conjecture on Calabi-Yau 3-folds, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. When $X$ is holomorphic symplectic, Gromov-Witten invariants vanish and one can consider the corresponding reduced theory. In a companion work, we propose a definition of Gopakumar-Vafa type invariants for such a reduced theory. In this paper, we give them a sheaf theoretic interpretation via moduli spaces of stable pairs.

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arXiv2022

Gromov-Witten theory and Noether-Lefschetz theory for holomorphic-symplectic varieties

We use Noether-Lefschetz theory to study the reduced Gromov--Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov-Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov-Witten techniques we also determine the generating series of Noether-Lefschetz numbers of a general pencil of Debarre-Voisin varieties. This reproves and extends a result of Debarre, Han, O'Grady and Voisin on HLS divisors on the moduli space of Debarre-Voisin fourfolds.

math.AG↗
arXiv2022

Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds, Klemm and Pandharipande for Calabi-Yau 4-folds, Pandharipande and Zinger for Calabi-Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced $4$-dimensional Donaldson-Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two $K3$ surfaces and for the cotangent bundle of $\mathbb{P}^2$. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a $K3$ surface. This yields a conjectural formula for the number of isolated genus $2$ curves of minimal degree on a very general hyperkähler $4$-fold of $K3^{[2]}$-type. The formula may be viewed as a $4$-dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on $K3$ surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on $K3$ surfaces and generalized Kummer varieties.

math.AG↗
arXiv2021

Marked relative invariants and GW/PT correspondences

We introduce marked relative Pandharipande-Thomas (PT) invariants for a pair $(X,D)$ of a smooth projective threefold and a smooth divisor. These invariants are defined by integration over the moduli space of $r$-marked stable pairs on $(X,D)$, and appear naturally when degenerating diagonal insertions via the Li-Wu degeneration formula. We propose a Gromov-Witten (GW) / PT correspondence for marked relative invariants. We show compatibility of the conjecture with the degeneration formula and a splitting formula for relative diagonals. The results provide new tools to prove GW/PT correspondences for varieties with vanishing cohomology. As an application we prove the GW/PT correspondence for: (i) all Fano complete intersections, and (ii) the reduced theories of $(S \times C, S \times \{ z_1, \ldots, z_N \})$ where $S$ is a K3 surface and $C$ is a curve, for all curve classes which have divisibility at most $2$ over the K3 surface. In the appendix we introduce a notion of higher-descendent invariants which can be seen as an analogue of the nodal Gromov-Witten invariants defined by Argüz, Bousseau, Pandharipande and Zvonkine in \cite{ABPZ}. We show that the higher-descendent invariants reduce to marked relative invariants with diagonal insertions.

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