Curves on K3 surfaces and modular forms

Abstract

We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa conjecture evaluating ${\lambda }_g$ integrals (in all genera) in terms of explicit modular forms. Indeed, all K3 invariants in primitive classes are shown to be governed by modular forms. The method of proof is by degeneration to elliptically fibered rational surfaces. New formulas relating reduced virtual classes on K3 surfaces to standard virtual classes after degeneration are needed for both maps and sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds. Our approach uses a result of Kiem and Li to produce reduced classes. In Appendix A, we answer a number of questions about the relationship between the Kiem-Li approach, traditional virtual cycles, and symmetric obstruction theories. The interplay between the boundary geometry of the moduli spaces of curves, K3 surfaces, and modular forms is explored in Appendix B by A. Pixton.

An incorrect example in Appendix A, pointed out to us by Dominic Joyce, has been replaced by a reference to a new paper arXiv:1204.3958 containing a corrected example

Cited by in corpus (41)

• Reduced classes and curve counting on surfaces I: theory
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• Holomorphic anomaly equations and the Igusa cusp form conjecture
• Tropical refined curve counting from higher genera and lambda classes
• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
• A short proof of the Göttsche conjecture
• The Katz-Klemm-Vafa conjecture for K3 surfaces
• The quantum tropical vertex
• Semiregularity and obstructions of complete intersections
• Gromov-Witten/Pairs descendent correspondence for toric 3-folds
• Refined curve counting on complex surfaces
• Reduced classes and curve counting on surfaces II: calculations
• Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics
• Gromov-Witten invariants of the Hilbert schemes of points of a K3 surface
• Gromov-Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations
• Descendents on local curves: Rationality
• Ramanujan Identities and Quasi-Modularity in Gromov-Witten Theory
• Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting
• A proof of N.Takahashi's conjecture for $({\mathbb{P}}^2,E)$ and a refined sheaves/Gromov-Witten correspondence
• Donaldson-Thomas theory of $[{\mathbb{C}}^2/{\mathbb{Z}}_{n+1}]\times {\mathbb{P}}^1$
• Refined floor diagrams from higher genera and lambda classes
• Relative Donaldson-Thomas theory for Calabi-Yau 4-folds
• Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes
• Stable maps to Looijenga pairs: orbifold examples
• Almost closed 1-forms
• Sheaves on surfaces and virtual invariants
• Notes on the proof of the KKV conjecture
• Double nested Hilbert schemes and the local stable pairs theory of curves
• Stable pair invariants of surfaces and Seiberg-Witten invariants
• Quasimaps to moduli spaces of sheaves on a $K3$ surface
• Determinantal Calabi-Yau varieties in Grassmannians and the Givental $I$-functions
• Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex
• Curves on K3 surfaces in divisibility two
• Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
• Cosection localization and the Quot scheme ${\mathrm{Quot}}_S^l(\mathcal{E})$
• Relative orbifold Pandharipande-Thomas theory and the degeneration formula
• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
• Behrend's function is not constant on ${\mathrm{Hilb}}^n({\mathbb{A}}^3)$
• Gromov-Witten/Hilbert versus AdS3/CFT2 Correspondence
• The orbifold DT/PT vertex correspondence
• Simple obstructions and cone reduction