Papers by Marco Gualtieri
Proceedings of the London Mathematical Society
A stable generalized complex structure is one that is generically symplectic but degenerates alon... more A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a formalism which allows us to view such structures as symplectic forms with singularities of logarithmic or elliptic type. This allows us to define two period maps: one for deformations in which the background 3-form flux is fixed, and one for which the flux is allowed to vary. As a result, we prove the unobstructedness of each of these deformation problems. We use the same approach to establish local classification theorems for the degeneracy locus as well as for analogues of Lagrangian submanifolds called Lagrangian branes.
A Celebration of the Mathematical Legacy of Raoul Bott
We describe how generalized complex geometry, which interpolates between complex and symplectic g... more We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists. T-duality relates topologically distinct torus bundles, and prescribes a method for transporting geometrical structures between them. We describe how this relation may be understood as a Courant algebroid isomorphism between the spaces in question. This then allows us to transport Dirac structures, generalized Riemannian metrics, generalized complex and generalized Kähler structures, extending the Buscher rules well-known to physicists. Finally, we re-interpret T-duality as a Courant reduction, and explain that T-duality between generalized complex manifolds may be viewed as a generalized complex submanifold (D-brane) of the product, in a way that establishes a direct analogy with the Fourier-Mukai transform.
The Stokes groupoids
Journal für die reine und angewandte Mathematik (Crelles Journal)
We construct and describe a family of groupoids over complex curves which serve as the universal ... more We construct and describe a family of groupoids over complex curves which serve as the universal domains of definition for solutions to linear ordinary differential equations with singularities. As a consequence, we obtain a direct, functorial method for resumming formal solutions to such equations.
Deformation of Dirac Structures via $L_\infty $ Algebras
International Mathematics Research Notices
The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra th... more The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra that depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty $ algebra instead. We develop a simplified method for describing this $L_\infty $ algebra and use it to prove that the $L_\infty $ algebras corresponding to different transversals are canonically $L_\infty $–isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures on Lie groups, and Lie bialgebras. Finally, we apply our result to a classical problem in the deformation theory of complex manifolds; we provide explicit formulas for the Kodaira–Spencer deformation complex of a fixed small deformation of a complex manifold, in terms of the deformation complex of the original manifold.
Bulletin of the London Mathematical Society
A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisso... more A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper we show that the isomorphism class of the holomorphic Poisson structure occurring in this local model is independent of the choice of gauge equivalence, and is hence the unique local invariant of generalized complex manifolds. We use this result to prove that the complex locus of a generalized complex manifold naturally inherits the structure of a complex analytic space.
Mathematische Annalen
We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic a... more We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentum map, we obtain a complete classification of such manifolds in terms of decorated log affine polytopes, hence extending the classification of symplectic toric manifolds achieved by Atiyah, Guillemin-Sternberg, Kostant, and Delzant.
Journal of Geometry and Physics
We study type one generalized complex and generalized Calabi-Yau manifolds. We introduce a cohomo... more We study type one generalized complex and generalized Calabi-Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the generalized complex structure, the twisting class. We prove that in a compact, type one, 4n-dimensional generalized complex manifold the Euler characteristic must be even and equal to the signature modulo four. The generalized Calabi-Yau condition places much stronger constrains: a compact type one generalized Calabi-Yau fibers over the 2-torus and if the structure has one compact leaf, then this fibration can be chosen to be the fibration by the symplectic leaves of the generalized complex structure. If the twisting class vanishes, one can always deform the structure so that it has a compact leaf. Finally we prove that every symplectic fibration over the 2-torus admits a type one generalized Calabi-Yau structure.
Journal of Differential Geometry
We introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold c... more We introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold containing a symplectic 2-torus with trivial normal bundle and whose output is a 4-manifold endowed with a generalized complex structure exhibiting type change along a 2-torus. Performing this surgery on a K3 surface, we obtain a generalized complex structure on 3CP 2 #19CP 2 , which has vanishing Seiberg-Witten invariants and hence does not admit complex or symplectic structure.
Generalized K�hler Manifolds, Commuting Complex Structures, and Split Tangent Bundles
Commun Math Phys, 2007
Communications in Mathematical Physics, Feb 13, 2007
We study generalized Kähler manifolds for which the corresponding complex structures commute and ... more We study generalized Kähler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kähler four-manifolds for which the induced complex structures yield opposite orientations.
M.: Generalized Kähler manifolds with split tangent bundle
We study generalized Kähler manifolds for which the corresponding complex structures commute and ... more We study generalized Kähler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kähler four-manifolds for which the induced complex structures yield opposite orientations. 1.
Contemporary Mathematics, 2008
We develop a theory of reduction for generalized Kähler and hyper-Kähler structures which uses th... more We develop a theory of reduction for generalized Kähler and hyper-Kähler structures which uses the generalized Riemannian metric in an essential way, and which is not described with reference solely to a single generalized complex structure. We show that our construction specializes to the usual theory of Kähler and hyper-Kähler reduction, and it gives a way to view usual hyper-Kähler quotients in terms of generalized Kähler reduction.
We study a class of Poisson manifolds which are symplectic away from a collection of hypersurface... more We study a class of Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We impose the analogue of toric Hamiltonian symmetries, which in this case means that there is a momentum map to a generalization of affine space called a log affine manifold. We also extend the Delzant construction to this setting, obtaining a classification of such manifolds in terms of decorated log affine polytopes.
In this lecture, delivered at the string theory and geometry workshop in Oberwolfach, we review s... more In this lecture, delivered at the string theory and geometry workshop in Oberwolfach, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the dd c-lemma of Kähler geometry.
The Many Facets of Geometry, 2010
We first extend the notion of connection in the context of Courant algebroids to obtain a new cha... more We first extend the notion of connection in the context of Courant algebroids to obtain a new characterization of generalized Kähler geometry. We then establish a new notion of isomorphism between holomorphic Poisson manifolds, which is nonholomorphic in nature. Finally we show an equivalence between certain configurations of branes on Poisson varieties and generalized Kähler structures, and use this to construct explicitly new families of generalized Kähler structures on compact holomorphic Poisson manifolds equipped with positive Poisson line bundles (e.g. Fano manifolds). We end with some speculations concerning the connection to noncommutative algebraic geometry.
Communications in Mathematical Physics, 2014
We prove that Hitchin's generalized Kähler structure on the moduli space of instantons over a com... more We prove that Hitchin's generalized Kähler structure on the moduli space of instantons over a compact, even generalized Kähler four-manifold may be obtained by generalized Kähler reduction, in analogy with the usual Kähler case.
We study generalized Kaehler manifolds for which the corresponding complex structures commute and... more We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.
Proceedings of the London Mathematical Society, 2013
In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geom... more In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geometry. In particular, we define a new notion of "residue" for a Poisson module, analogous to the Poincaré residue of a meromorphic volume form. Of particular interest is the interaction between the residues of the canonical line bundle of a Poisson manifold and its degeneracy loci-where the rank of the Poisson structure drops. As an application, we provide new evidence in favour of Bondal's conjecture that the rank ≤ 2k locus of a Fano Poisson manifold always has dimension ≥ 2k + 1. In particular, we show that the conjecture holds for Fano fourfolds. We also apply our techniques to a family of Poisson structures defined by Feȋgin and Odesskiȋ, where the degeneracy loci are given by the secant varieties of elliptic normal curves.
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Papers by Marco Gualtieri