Marco Gualtieri / Publications

In this paper we propose a noncommutative generalization of the relationship between compact Kähler manifolds and complex projective algebraic varieties. Beginning with a prequantized Kähler structure, we use a holomorphic Poisson tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented for all compact toric Kähler manifolds equipped with an R-matrix holomorphic Poisson structure, resulting in what could be called noncommutative toric varieties. To define the homomorphisms between generalized complex branes, we propose a method which involves lifting each pair of generalized complex branes to a single coisotropic A-brane in the real symplectic groupoid of the underlying Poisson structure, and compute morphisms in the A-model between the Lagrangian identity bisection and the lifted coisotropic brane. This is done with the use of a multiplicative holomorphic Lagrangian polarization of the groupoid.

We solve the problem of determining the fundamental degrees of freedom underlying a generalized Kähler structure of symplectic type. For a usual Kähler structure, it is well-known that the geometry is determined by a complex structure, a Kähler class, and the choice of a positive (1,1)-form in this class, which depends locally on only a single real-valued function: the Kähler potential. Such a description for generalized Kähler geometry has been sought since it was discovered in 1984. We show that a generalized Kähler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on only a single real-valued function, which we call the generalized Kähler potential. Our solution draws upon, and specializes to, the many results in the physics literature which solve the problem under the assumption (which we do not make) that the Poisson structures involved have constant rank. To solve the problem we make use of, and generalize, two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds; the second is Donaldson's interpretation of a Kähler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.

We give a new characterization of generalized Kaehler structures in terms of their corresponding complex Dirac structures. We then give an alternative proof of Hitchin’s partial unobstructedness for holomorphic Poisson structures. Our main application is to show that there is a corresponding unobstructedness result for arbitrary generalized Kaehler structures. That is, we show that any generalized Kaehler structure may be deformed in such a way that one of its underlying holomorphic Poisson structures remains fixed, while the other deforms via Hitchin’s deformation. Finally, we indicate a close relationship between this deformation and the notion of a Hamiltonian family of Poisson structures.

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve the prequantization problem, showing that under the appropriate integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibres degenerate along the divisor. We give a geometric description of such bundles and establish a classification theorem for them, showing that they correspond to a subgroup of the holomorphic line bundles on the divisor complement. Importantly, these holomorphic line bundles need not be algebraic. Finally, we provide concrete methods for explicitly constructing examples of meromorphic line bundles, such as for a smooth cubic divisor in the projective plane.

The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an L-infinity algebra instead. We develop a simplified method for describing this L-infinity algebra and use it to prove that the L∞ algebras corresponding to different transversals are canonically L-infinity-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.

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.

We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a weakly holomorphic symplectic groupoid, which is a real symplectic groupoid with a compatible complex structure defined only on the associated stack, i.e., only up to Morita equivalence. We explain how such objects differentiate to give generalized complex manifolds, and we show that a generalized complex manifold is integrable in this sense if and only if its underlying real Poisson structure is integrable. Crucial to our solution are several new technical tools which are of independent interest, namely, a reduction procedure for Lie groupoid actions on Courant algebroids, as well as certain local-to-global extension results for multiplicative forms on local Lie groupoids. Finally, we implement our generalized complex integration procedure in several concrete examples.

We give a gauge-theoretic description of the natural Dirac structure on a Lie Group. Our insight is that the formal Poisson structure on the space of connections on the circle is not an actual Poisson structure, but is itself a Dirac structure, due to the fact that it is defined by an unbounded operator.

A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold. We introduce a Lie algebroid which allows us to view such structures as symplectic forms. We use this to define period maps for deformations in which the background three-form flux is either fixed or not, proving the unobstructedness of both deformation problems.

We develop the theory of toric log symplectic manifolds with normal crossing degeneracy loci. The appropriate notion of tropical moment map has codomain which is a welding of tropical domains and can have nontrivial topology.

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.

Explicit construction and classification of symplectic groupoids for log symplectic manifolds. Techniques are applicable to many other algebroids.

Established Bondal's conjecture for Fano 4-folds, and developed new geometric invariants of Poisson modules.

Showed that Hitchin's GK structure on the moduli of instantons can be obtained by a generalized Kähler reduction. Also, we show that the reduction gives a geometric interpretation of Donaldson's μ-map on degree 3 cohomology classes.

A classification of solution types for first-order ODE.

Summarizes our work on the fundamental role of T-duality in generalized geometry.

Develops a blow-up procedure for bi-Hermitian manifolds and generalized Kahler metrics.

Develops generalized Kahler structure and its connection to holomorphic Courant brackets.

Develops blow-up and blow-down operations and shows that 3CP² is generalized complex.

Using the theory of Poisson modules to construct examples of bi-Hermitian metrics on Poisson varieties.

An article based on the thesis, significantly condensed (no generalized Kahler) and with several new ideas. Prepared while teaching the topics course below.

With Vestislav Apostolov. Completes the classification of generalized Kähler 4-manifolds.

Presents the first generalized complex 4-manifold which is neither symplectic nor complex.

Develops a theory of reduction for Courant algebroids and related geometrical structures.

Proves a Hodge decomposition for generalized Kähler structures.

Some of the first nontrivial examples of generalized complex structures.

My 2004 doctoral thesis. Develops the basic structure theory of generalized complex geometry as well as generalized Kähler geometry

Explains why the golf ball sometimes emerges from the hole.