# Marco Gualtieri / Publications

## 21. Dirac Geometry of the holonomy fibration

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.

## 20. Stable generalized complex structures

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.

## 19. Tropical moment maps for toric log symplectic manifolds

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.

## 18. The Stokes Groupoids

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.

## 17. Symplectic groupoids on log symplectic manifolds

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

## 16. Poisson modules and degeneracy loci

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

## 15. Generalized Kähler geometry of instanton moduli spaces

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.

## 14. Orbits of the centralizer of a linear operator

A classification of solution types for first-order ODE.

## 13. Generalized complex geometry and T-duality

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

## 12. Blowing up generalized Kähler 4-manifolds

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

## 11. Generalized Kähler geometry

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

## 10. Blow-up of generalized complex 4-manifolds

Develops blow-up and blow-down operations and shows that 3CP^{2} is generalized complex.

## 9. Branes on Poisson varieties

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

## 8. Generalized complex geometry

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

## 7. Generalized Kähler manifolds, commuting complex structures, and split tangent bundles

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

## 6. A surgery for generalized complex structures on 4-manifolds

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

## 5. Reduction of Courant algebroids and generalized complex structures

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

## 4. Generalized geometry and the Hodge decomposition

Proves a Hodge decomposition for generalized Kähler structures.

## 3. Generalized complex structures on nilmanifolds

Some of the first nontrivial examples of generalized complex structures.

## 2. Generalized complex geometry

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

## 1. Golfer's Dilemma (PDF)

Explains why the golf ball sometimes emerges from the hole.