Maps, Sheaves and K3 Surfaces

Pandharipande, Rahul

doi:10.1093/oso/9780198784913.003.0005

Oxford Scholarship Online 2017

DOI: 10.1093/oso/9780198784913.003.0005

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Maps, Sheaves and K3 Surfaces

Rahul Pandharipande¹

Abstract: Abstract. The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3 surfaces. New results and conjectures (with D. Maulik) about descendent integration on K3 surfaces are announced. The recent proof of the Yau-Zaslow conjecture is surveyed .

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2021

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Cited by 3 publications

References 56 publications

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Sheaves of maximal intersection and multiplicities of stable log maps

Choi

Garrel

Katz

et al. 2021

Sel. Math. New Ser.

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A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.

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Sheaves of maximal intersection and multiplicities of stable log maps

Choi

Garrel

Katz

et al. 2021

Sel. Math. New Ser.

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Curves on K 3 surfaces and modular forms

Maulik

Pandharipande

Thomas

2010

Journal of Topology

129

230

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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.

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Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

Schürg

Toën

Vezzosi

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We show how a quasi-smooth derived enhancement of a Deligne-Mumford stack X naturally endows X with a functorial perfect obstruction theory in the sense of Behrend-Fantechi. This result is then applied to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps, for X = S an algebraic K3-surface, g ∈ N, and β = 0 in H 2 (S, Z) a curve class, we construct a derived stack RM red g,n (S; β) whose truncation is the usual stack M g,n (S; β) of pointed stable maps from curves of genus g to S hitting the class β, and such that the inclusion M g (S; β) ֒→ RM red g (S; β) induces on M g (S; β) a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov-Maulik-Pandharipande-Thomas [O-P2, M-P, M-P-T]. The approach we present here uses derived algebraic geometry and yields not only a full rigorous proof of the existence of a reduced obstruction theory -not relying on any result on semiregularity maps -but also a new global geometric interpretation. We give two further applications to moduli of complexes. For a K3-surface S we show that the stack of simple perfect complexes on S is smooth. This result was proved with different methods by Inaba ([In]) for the corresponding coarse moduli space. Finally, we construct a map from the derived stack of stable embeddings of curves (into a smooth complex projective variety X) to the derived stack of simple perfect complexes on X with vanishing negative Ext's, and show how this map induces a morphism of the corresponding obstruction theories when X is a Calabi-Yau threefold. An important ingredient of our construction is a perfect determinant map from the derived stack of perfect complexes to the derived stack of line bundles whose tangent morphism is, pointwise, Illusie's trace map for perfect complexes. We expect that this determinant map might be useful in other contexts as well.

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Maps, Sheaves and K3 Surfaces

Cited by 3 publications

References 56 publications

Sheaves of maximal intersection and multiplicities of stable log maps

Sheaves of maximal intersection and multiplicities of stable log maps

Curves on K 3 surfaces and modular forms

Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

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