Curves on K3 surfaces and modular forms
By Davesh Maulik, R. Pandharipande, Richard P. Thomas. 2010
164 citations recorded by the source corpus.
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 proved. As a consequence, we prove the Katz–Klemm–Vafa conjecture evaluating λg 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 fibred 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 Pixton.
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/40c4d503dcc0d4a59d3310e0dfcecca4b5aae967
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In Geometry and Topology
Citations in the corpus
- Curve Counting on K3 × E, The Igusa Cusp Form χ10, and Descendent Integration (2016, background): “escendent theory of K3 surfaces in primitive classes is defined by (26) Fg τk1(γl1)···τkr(γlr) = X∞ h=0 D τk1(γl1)···τkr(γlr) ES g,βh qh−1 for g≥0. The descendent potential (26) is a quasimodular form [25]. The ring QMod = Q[E2(q),E4(q),E6(q)] of holomorphic quasimodular forms (of level 1) is the Q-a”
- Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, background): “The theories of local curves without any insertions are correspondent by [24,29].”
- Good degeneration of Quot-schemes and coherent systems (2011, unknown): “By going through the argument parallel to the proof of degeneration formula for Hilbert schemes of ideal sheaves, Maulik, Pandharipande and Thomas proved in [MPT10] the degeneration formula of PT stable pair invariants.”
- Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics (2019, methodology): “The Kawai-Yoshioka formula has played a central role in the Gromov-Witten and the stable pairs theory of K3 surfaces [40, 45, 46].”
- Localizing virtual cycles by cosections (2010, methodology): “In studying curve counting [28], Maulik-Pandharipande-Thomas use localization of the virtual cycle by a cosection to define the reduced Gromov-Witten and Pandharipande-Thomas invariants of a class of Calabi-Yau three-folds; in the appendix of the same paper, they provide an in-depth discussion on th”
- Refined curve counting on complex surfaces (2012, background): “In fact, while the Euler numbers of these spaces are known, the only known calculation of them (due to Maulik, Pandharipande and Thomas) is extremely indirect, and involves at least two uses of the Gromov–Witten/Pairs correspondence [51].”
- Torus localization and wall crossing for cosection localized virtual cycles (2015, background): “The cosection localized virtual fundamental class turned out to be quite useful ([2, 5, 7, 9, 13, 14, 16, 17, 19, 20, 23, 27, 29, 30, 31]).”