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Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

Yalong Cao, Georg Oberdieck, Yukinobu Toda

Abstract

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.

Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

Abstract

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 -dimensional Donaldson-Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two surfaces and for the cotangent bundle of . Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a surface. This yields a conjectural formula for the number of isolated genus curves of minimal degree on a very general hyperkähler -fold of -type. The formula may be viewed as a -dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on 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 surfaces and generalized Kummer varieties.
Paper Structure (59 sections, 57 theorems, 447 equations, 3 tables)

This paper contains 59 sections, 57 theorems, 447 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Gopakumar-Vafa invariants
  3. GV/$\mathop{\rm DT}\nolimits_4$ correspondence
  4. Verification of conjectures I: $K3\times K3$
  5. Verification of conjectures II: $T^*\mathbb{P}^2$
  6. Verification of conjectures III: $K3^{[2]}$
  7. A Yau-Zaslow type formula on $K3^{[2]}$
  8. Appendix
  9. Gopakumar-Vafa invariants
  10. Gromov-Witten invariants
  11. Relations
  12. Definition of GV invariants
  13. Ideal geometry
  14. Ideal geometry: Primitive case
  15. Justification: GV in genus $1$
  16. ...and 44 more sections

Key Result

Theorem 6

(Proposition prop on GW for prod) Let $\gamma,\gamma'\in H^{4}(X)$, $\alpha\in H^6(X)$ and let be their Künneth decompositions. Then we have If $\beta$ is primitive, we have where with Eisenstein series: In particular, Conjecture intro conj on integrality holds for $X=S\times T$.

Theorems & Definitions (126)

  • Definition 1
  • Definition 2
  • Definition 3
  • Conjecture 4
  • Conjecture 5
  • Theorem 6
  • Theorem 7: Corollary \ref{['verify conjs']}
  • Theorem 8
  • Proposition 9
  • Proposition 10
  • ...and 116 more