Showing papers by "Rahul Pandharipande published in 2010"
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TL;DR: In this article, it was shown that ordered product factorizations in the tropical vertex group are equivalent to calculations of genus zero relative Gromov-Witten invariants of toric surfaces.
Abstract: Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove that ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus zero relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory
201 citations
TL;DR: In this article, the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ2 was determined and the operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system.
Abstract: We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ2. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.
159 citations
TL;DR: In this article, Pixton et al. studied the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes and proved a Gromov-Witten/Pairs correspondence for toric 3-folds.
Abstract: We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equiv- alences relating the reduced Gromov-Witten invariants of K3 sur- faces to characteristic numbers of stable pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa con- jecture evaluating λ 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 ra- tional 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.
149 citations
TL;DR: In this article, Pixton et al. studied the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes and proved the Katz-Klemm-Vafa conjecture in terms of explicit 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.
106 citations
TL;DR: The local Donaldson-Thomas theory of curves is solved by localization and degeneration methods in this article, which complete a triangle of equivalences relating Gromov-Witten theory, Donaldson−Thomas theory, and the quantum cohomology of the Hilbert scheme of points of the plane.
Abstract: The local Donaldson–Thomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating Gromov–Witten theory, Donaldson–Thomas theory, and the quantum cohomology of the Hilbert scheme of points of the plane.
86 citations
TL;DR: In this paper, the Yau-Zaslow conjecture for all curve classes on K3 surfaces was shown to hold for all curves in the moduli of the STU model.
Abstract: The Yau-Zaslow conjecture determines the reduced genus 0 Gromov-Witten invariants of K3 surfaces in terms of the Dedekind eta function. Classical intersections of curves in the moduli of K3 surfaces with Noether-Lefschetz divisors are related to 3-fold Gromov-Witten theory via the K3 invariants. Results by Borcherds and Kudla-Millson determine the classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise.
Via a detailed study of the STU model (determining special curves in the moduli of K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.
81 citations
TL;DR: In this article, Pandharipande et al. proved new results about the rays and symmetries of scattering diagrams of commutators, including previous conjectures by Gross-Siebert and Kontsevich.
Abstract: Elements of the tropical vertex group are formal families of symplec- tomorphisms of the 2-dimensional algebraic torus. Commutators in the group are related to Euler characteristics of the moduli spaces of quiver representations and the Gromov-Witten theory of toric surfaces. After a short survey of the subject (based on lectures of Pandharipande at the 2009 Geometry summer school in Lisbon), we prove new results about the rays and symmetries of scattering diagrams of commutators (including previous conjectures by Gross-Siebert and Kontsevich). Where possible, we present both the quiver and Gromov-Witten perspectives.
58 citations
TL;DR: In this article, basic properties of the quantum differential equation of the Hilbert scheme of points in the plane were discussed and an exact solution to the connection problem from the Donaldson-Thomas point q = 0 to the Gromov-Witten point Q = -1 was obtained.
Abstract: We discuss here basic properties of the quantum differential equation of the Hilbert scheme of points in the plane. Our emphasis is on intertwining operators (which shift equivariant parameters) and their applications. In particular, we obtain an exact solution to the connection problem from the Donaldson-Thomas point q = 0 to the Gromov-Witten point q = -1.
31 citations
TL;DR: In this article, the rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the scaling 2-torus) including relative conditions and odd degree insertions for higher genus curves.
Abstract: We study the stable pairs theory of local curves in 3-folds with descendent insertions. The rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the scaling 2-torus) including relative conditions and odd degree insertions for higher genus curves. The capped 1-leg descendent vertex (equivariant with respect to the 3-torus) is also proven to be rational. The results are obtained by combining geometric constraints with a detailed analysis of the poles of the descendent vertex.
23 citations
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TL;DR: The double point relation defines a natural theory of algebraic cobordism for bundles on varieties as discussed by the authors, which is an extension of scalars of standard algebraic Cobordism.
Abstract: The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.
20 citations
TL;DR: In this article, the k-th power of the cotangent line class on the moduli space of stable 1-pointed genus g curves is found for k >= 2g.
Abstract: Simple boundary expressions for the k-th power of the cotangent line class on the moduli space of stable 1-pointed genus g curves are found for k >= 2g. The method is by virtual localization on the moduli space of maps to the projective line. As a consequence, nontrivial tautological classes in the kernel of the push-forward map associated to the irreducible boundary divisor of the moduli space of stable g+1 curves are constructed. The geometry of genus g+1 curves then provides universal equations in genus g Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities conjectured recently by K. Liu and H. Xu.
01 Jan 2010
TL;DR: In this article, Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 5-folds, and the contributions of moving multiple covers of genus 0 curves to the genus 1 invariants are determined.
Abstract: Gromov–Witten theory is used to define an enumerative geometry of curves in Calabi–Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the genus 1 Gromov–Witten invariants. The resulting invariants, conjectured to be integral, are analogous to the previously defined BPS counts for Calabi–Yau 3 and 4-folds. We comment on the situation in higher dimensions where new issues arise. Two main examples are considered: the local Calabi–Yau $\mathbb{P}^2$ with normal bundle $\oplus_{i=1}^{3} \mathcal{O} (-1)$ and the compact Calabi–Yau hypersurface $X_7 \subset \mathbb{P}^6$. In the former case, a closed form for our integer invariants has been conjectured by G. Martin. In the latter case, we recover in low degrees the classical enumeration of elliptic curves by Ellingsrud and Stromme.
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TL;DR: In this article, the moduli spaces of curves, K3 surfaces, maps, and sheaves were discussed in the problem session of the AGNES conference in Amherst (April 2010).
Abstract: The article contains a few questions and speculations related to the moduli spaces of curves, K3 surfaces, maps, and sheaves presented in the problem session of the AGNES conference in Amherst (April 2010)
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TL;DR: In this article, the rationality of the descendent partition function for stable pairs on nonsingular toric 3-folds was proved for all log Calabi-Yau geometries of the form (X,K3) where X is a non-simplex toric three-fold.
Abstract: We prove the rationality of the descendent partition function for stable pairs on nonsingular toric 3-folds. The method uses a geometric reduction of the 2- and 3-leg descendent vertices to the 1-leg case. As a consequence, we prove the rationality of the relative stable pairs partition functions for all log Calabi-Yau geometries of the form (X,K3) where X is a nonsingular toric 3-fold.