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Kai Behrend - One of the best experts on this subject based on the ideXlab platform.
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Donaldson-Thomas type invariants via microlocal geometry
Annals of Mathematics, 2009Co-Authors: Kai BehrendAbstract:We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.
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Super-rigid Donaldson-Thomas Invariants
Mathematical Research Letters, 2007Co-Authors: Kai Behrend, Jim BryanAbstract:We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson-Thomas correspondence of \cite{MNOP} for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for the quintic threefold in degrees one and two.
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Super-rigid Donaldson-Thomas invariants
arXiv: Algebraic Geometry, 2006Co-Authors: Kai Behrend, Jim BryanAbstract:We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson-Thomas correspondence for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for the quintic threefold in degrees one and two.
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Donaldson-Thomas invariants via microlocal geometry
arXiv: Algebraic Geometry, 2005Co-Authors: Kai BehrendAbstract:We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.
Yuuji Tanaka - One of the best experts on this subject based on the ideXlab platform.
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On the Moduli Space of Donaldson-Thomas Instantons
2016Co-Authors: Yuuji TanakaAbstract:In alignment with a programme by Donaldson and Thomas, Thomas [48] constructed a deformation invariant for smooth projective Calabi-Yau threefolds, which is now called the Donaldson-Thomas invariant, from the moduli space of (semi-)stable sheaves by using algebraic geometry techniques. In the same paper [48], Thomas noted that certain perturbed Hermitian-Einstein equations might possibly produce an analytic theory of the invariant. This article sets up the equations on symplectic 6-manifolds, and gives the local model and structures of the moduli space coming from the equations. We then describe a Hitchin-Kobayashi style correspondence for the equations on compact Kahler threefolds, which turns out to be a special case of results by Alvarez-Consul and Garcia-Prada [1].
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A removal singularity theorem of the Donaldson–Thomas instanton on compact Kähler threefolds
Journal of Mathematical Analysis and Applications, 2014Co-Authors: Yuuji TanakaAbstract:Abstract We consider a perturbed Hermitian–Einstein equation, which we call the Donaldson–Thomas equation, on compact Kahler threefolds. In [12] , we analysed some analytic properties of solutions to the equation, in particular, we proved that a sequence of solutions to the Donaldson–Thomas equation has a subsequence which smoothly converges to a solution to the Donaldson–Thomas equation outside a closed subset of the Hausdorff dimension two. In this article, we prove that some of these singularities can be removed.
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a weak compactness theorem of the Donaldson thomas instantons on compact kahler threefolds
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Yuuji TanakaAbstract:In Tanaka [18], we introduced a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian–Einstein equation perturbed by Higgs fields, and called it a Donaldson–Thomas equation, to analytically approach the Donaldson–Thomas invariants. In this article, we consider the equation on compact Kahler threefolds, and study some of the analytic properties of solutions to them, using analytic methods in higher-dimensional Yang–Mills theory developed by Nakajima (1987) [14], Nakajima (1988) [15] and Tian (2000) [20] with some additional arguments concerning an extra nonlinear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson–Thomas equation of a unitary vector bundle over a compact Kahler threefold has a converging subsequence outside a closed subset whose real two-dimensional Hausdorff measure is finite, provided that the L2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-compactness theorem of solutions to the equations on compact Kahler threefolds.
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a weak compactness theorem of the Donaldson thomas instantons on compact k ahler threefolds
arXiv: Differential Geometry, 2008Co-Authors: Yuuji TanakaAbstract:In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact K\"ahler threefolds, and study some of analytic properties of solutions to them, using analytic methods in higher-dimensional Yang-Mills theory developed by Nakajima and Tian with some additional arguments concerning an extra non-linear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson-Thomas equation of a unitary vector bundle over a compact K\"ahler threefold has a converging subsequence outside a closed subset whose real 2-dimensional Hausdorff measure is finite, provided that the L^2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-compactness theorem of solutions to the equations on compact K\"ahler threefolds.
Jim Bryan - One of the best experts on this subject based on the ideXlab platform.
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the Donaldson thomas theory of k 3 e via the topological vertex
The Abel Symposium, 2017Co-Authors: Jim BryanAbstract:We give a general overview of the Donaldson-Thomas invariants of elliptic fibrations and their relation to Jacobi forms. We then focus on the specific case of where the fibration is S × E, the product of a K3 surface and an elliptic curve. Oberdieck and Pandharipande conjectured (Oberdieck and Pandharipande, K3 Surfaces and Their Moduli, Progress in Mathematics, vol. 315 (Birkhauser/Springer, Cham, 2016), pp. 245–278, arXiv:math/1411.1514) that the partition function of the Gromov-Witten/Donaldson-Thomas invariants of S × E is given by minus the reciprocal of the Igusa cusp form of weight 10. For a fixed primitive curve class in S of square 2h − 2, their conjecture predicts that the corresponding partition functions are given by meromorphic Jacobi forms of weight − 10 and index h − 1. We calculate the Donaldson-Thomas partition function for primitive classes of square − 2 and of square 0, proving strong evidence for their conjecture. Our computation uses reduced Donaldson-Thomas invariants which are defined as the (Behrend function weighted) Euler characteristics of the quotient of the Hilbert scheme of curves in S × E by the action of E. Our technique is a mixture of motivic and toric methods (developed with Kool in (Bryan and Kool, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex (2016), arXiv:math/1608.07369)) which allows us to express the partition functions in terms of the topological vertex and subsequently in terms of Jacobi forms. We compute both versions of the invariants: unweighted and Behrend function weighted Euler characteristics. Our Behrend function weighted computation requires us to assume Conjecture 18 in (Bryan and Kool, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex (2016), arXiv:math/1608.07369).
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generating functions for colored 3d young diagrams and the Donaldson thomas invariants of orbifolds
Duke Mathematical Journal, 2010Co-Authors: Benjamin Young, Jim BryanAbstract:We derive two multivariate generating functions for three-dimensional (3D) Young diagrams (also called plane partitions). The variables correspond to a coloring of the boxes according to a finite Abelian subgroup G of SO (3). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C 3/G]. We need only the vertex operator methods of Okounkov, Reshetikhin, and Vafa for the easy case G = Z n; to handle the considerably more difficult case G = Z 2 × Z 2, we also use a refinement of the author's recent q-enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C 3/G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the hard Lefschetz condition.
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Super-rigid Donaldson-Thomas Invariants
Mathematical Research Letters, 2007Co-Authors: Kai Behrend, Jim BryanAbstract:We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson-Thomas correspondence of \cite{MNOP} for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for the quintic threefold in degrees one and two.
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Super-rigid Donaldson-Thomas invariants
arXiv: Algebraic Geometry, 2006Co-Authors: Kai Behrend, Jim BryanAbstract:We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson-Thomas correspondence for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for the quintic threefold in degrees one and two.
Zhenbo Qin - One of the best experts on this subject based on the ideXlab platform.
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Donaldson-Thomas invariants of certain Calabi-Yau 3-folds
arXiv: Algebraic Geometry, 2010Co-Authors: Zhenbo QinAbstract:We compute the Donaldson-Thomas invariants for two types of Calabi-Yau 3-folds. These invariants are associated to the moduli spaces of rank-2 Gieseker semistable sheaves. None of the sheaves are locally free, and their double duals are locally free stable sheaves investigated earlier by Donaldson and Thomas, Li and Qin respectively. We show that these Gieseker moduli spaces are isomorphic to some Quot-schemes. We prove a formula for Behrend's functions when torus actions present with positive dimensional fixed point sets, and use it to obtain the generating series of the relevant Donaldson-Thomas invariants in terms of the McMahon function. Our results might shed some light on the wall-crossing phenomena of Donaldson-Thomas invariants.
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THE GROMOV–WITTEN AND Donaldson–THOMAS CORRESPONDENCE FOR TRIVIAL ELLIPTIC FIBRATIONS
International Journal of Mathematics, 2007Co-Authors: Dan Edidin, Zhenbo QinAbstract:We study the Gromov–Witten and Donaldson–Thomas correspondence conjectured in [16, 17] for trivial elliptic fibrations. In particular, we verify the Gromov–Witten and Donaldson–Thomas correspondence for primary fields when the threefold is E × S where E is a smooth elliptic curve and S is a smooth surface with numerically trivial canonical class.
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The Gromov-Witten and Donaldson-Thomas correspondence for trivial elliptic fibrations
arXiv: Algebraic Geometry, 2006Co-Authors: Dan Edidin, Zhenbo QinAbstract:We study the Gromov-Witten and Donaldson-Thomas correspondence conjectured in [MNOP1, MNOP2], for trivial elliptic fibrations. In particular, we verify the Gromov-Witten and Donaldson-Thomas correspondence for primary fields when the threefold is $E \times S$ where $E$ is a smooth elliptic curve and $S$ is a smooth surface with numerically trivial canonical class.
Rahul Pandharipande - One of the best experts on this subject based on the ideXlab platform.
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The local Donaldson–Thomas theory of curves
Geometry & Topology, 2010Co-Authors: Andrei Okounkov, Rahul PandharipandeAbstract:Let X be a nonsingular projective variety of dimension 3 over C . Gromov–Witten theory is defined by integration over the moduli space of stable maps to X , and Donaldson–Thomas theory is defined by integration over the moduli space of ideal sheaves of X (see Donaldson–Thomas [5], Maulik et al [24; 25] and Thomas [34]). If X is quasi-projective, the Gromov–Witten and Donaldson–Thomas theories may not be well-defined. However, if X is the total space of a rank 2 bundle over a nonsingular projective curve, N ! C; local Gromov–Witten and Donaldson–Thomas theories are defined via equivariant residues (see Bryan–Pandharipande [4] and Maulik et al [24]). The Gromov–Witten and Donaldson–Thomas theories of X relative to a nonsingular surface S X are defined via moduli spaces of maps and sheaves with boundary conditions along S . See Eliashberg–Givental–Hofer [6], Ionel–Parker [12], Li–Ruan [16], Li [17; 18] and Maulik et al [25] for various treatments of the subject.
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the local Donaldson thomas theory of curves
Geometry & Topology, 2010Co-Authors: Andrei Okounkov, Rahul PandharipandeAbstract:Let X be a nonsingular projective variety of dimension 3 over C . Gromov–Witten theory is defined by integration over the moduli space of stable maps to X , and Donaldson–Thomas theory is defined by integration over the moduli space of ideal sheaves of X (see Donaldson–Thomas [5], Maulik et al [24; 25] and Thomas [34]). If X is quasi-projective, the Gromov–Witten and Donaldson–Thomas theories may not be well-defined. However, if X is the total space of a rank 2 bundle over a nonsingular projective curve, N ! C; local Gromov–Witten and Donaldson–Thomas theories are defined via equivariant residues (see Bryan–Pandharipande [4] and Maulik et al [24]). The Gromov–Witten and Donaldson–Thomas theories of X relative to a nonsingular surface S X are defined via moduli spaces of maps and sheaves with boundary conditions along S . See Eliashberg–Givental–Hofer [6], Ionel–Parker [12], Li–Ruan [16], Li [17; 18] and Maulik et al [25] for various treatments of the subject.
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gromov witten theory and Donaldson thomas theory i
Compositio Mathematica, 2006Co-Authors: D Maulik, Andrei Okounkov, Nikita Nekrasov, Rahul PandharipandeAbstract:We conjecture an equivalence between the Gromov–Witten theory of 3-folds and the holomorphic Chern–Simons theory of Donaldson and Thomas. For Calabi–Yau 3-folds, the equivalence is defined by the change of variables is the Euler characteristic parameter of Donaldson–Thomas theory. The conjecture is proven for local Calabi–Yau toric surfaces.
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Gromov-Witten theory and Donaldson-Thomas theory, I
Compositio Mathematica, 2006Co-Authors: D Maulik, Andrei Okounkov, Nikita Nekrasov, Rahul PandharipandeAbstract:We conjecture an equivalence between the Gromov–Witten theory of 3-folds and the holomorphic Chern–Simons theory of Donaldson and Thomas. For Calabi–Yau 3-folds, the equivalence is defined by the change of variables $e^{iu}=-q$ , where $u$ is the genus parameter of Gromov–Witten theory and $q$ is the Euler characteristic parameter of Donaldson–Thomas theory. The conjecture is proven for local Calabi–Yau toric surfaces.
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The local Donaldson-Thomas theory of curves
arXiv: Algebraic Geometry, 2005Co-Authors: Andrei Okounkov, Rahul PandharipandeAbstract: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. The quantum differential equation of the Hilbert scheme of points of the plane has a natural interpretation in the local Donaldson-Thomas theory of curves. The solution determines the 1-legged equivariant vertex.
