The Cohomology of Courant algebroids and their characteristic classes

Speaker: Miquel Cueca Ten, IMPA.

Date: 05 jul 2019, 14h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: Courant algebroids originated over 20 years ago motivated by constrained mechanics but now play an important role in Poisson geometry and related areas. Courant algebroids have an associated cohomology, which is hard to describe concretely. Building on work of Keller and Waldmann, I will show an explicit description of the complex of a Courant algebroid where the differential satisfies a Cartan-type formula. This leads to a new viewpoint on connections and representations of Courant algebroids and allows us to define new invariants as secondary charcateristic classes, analogous to what Crainic and Fernandes did for Lie algebroids.

This is joint work with R. Mehta.

 

Hyperpolygons and parabolic Higgs bundles

Speaker: Alessia Mandini, PUC-Rio.

Date: 07 jun 2019, 11h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: Hyperpolygons spaces are a family of (finite dimensional, non-compact) hyperkaehler spaces, that can be obtained from coadjoint orbits by hyperkaehler reduction. Jointly with L. Godinho, we show that these space are diffeomorphic (in fact, symplectomorphic) to certain families of parabolic Higgs bundles. In this talk, I will describe this relation and use it to analyze the fixed points locus of a natural involution on the moduli space of parabolic Higgs bundles. I will show that each connected component of the fixed point locus of this involution is identified with a moduli space of polygons in Minkowski 3-space.

This is based on

  • L.Godinho, A. Mandini, "Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles" Adv. Math. 244 (2013), 465–532
  • I.Biswas, C.Florentino, L.Godinho, A.Mandini, "Polygons in the Minkowski three space and parabolic Higgs bundles of rank two on CP^1, Transfom. Groups 18 (2013)
  • I.Biswas, C.Florentino, L.Godinho, A.Mandini, "symplectic form on hyperpolygon space", Geom. Dedicata 179 (2015)
 

On deformations of symplectic groupoids

Speaker: Cristian Cardenas, UFF.

Date: 31 may 2019, 11h.

Place: Room 409, Bloco H, Campus Gragoatá, UFF.

Abstract: In this talk, I will show how a subcomplex of the Bott-Shulman complex controls deformations of symplectic groupoids. I will also explain a Moser argument in the context of symplectic groupoids by using this subcomplex.

Moreover, I will describe the global counterpart of the map defined by Crainic and Moerdijk (2004) which relates the Poisson cohomology of a Poisson manifold to the deformation cohomology of its underlying algebroid.
Joint work with Ivan Struchiner and João Mestre.

 

Genus Integration, Abelianization and Extended Monodromy

Speaker: Rui Loja Fernandes, UIUC.

Date: 10 may 2019, 11h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: I will describe the genus integration of a Lie algebroid A, i.e., the space of A-paths modulo A-homology, and how it is related to a version of the classical Hurewicz homomorphism for Lie algebroids/groupoids. I will also explain that the smoothness of the genus integration is controlled by the so-called "extended monodromy" of a Lie algebroid A, introduced first by I. Marcut. This talk is based on joint work with Ivan Contreras.

 

Hyperholomorphic bundles on hyperkähler manifolds

Speaker: Misha Verbitsky, IMPA.

Date: 05 apr 2019, 14h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: Let M be a compact hyperkahler manifold, that is, a Riemannian manifold equipped with a parallel action of an algebra of quaternions in its tangent bundle. Each complex structure I in quaternions defines a complex structure on M, called induced complex structure. The quaternionic action on $TM$ defines an action of the group $SU(2)$ of unitary quaternions on cohomology. A hyperholomorphic bundle is a polystable bundle on $(M,I)$ with first and second Chern classes $SU(2)$-invariant. The category of such bundles is equivalent to the category of vector bundles with Hermitian connection and $SU(2)$-invariant curvature. Therefore, this category is independent from the choice of an induced complex structure $I$. Hyperholomorphic bundles have many nice properties. For a general complex structure $J$ of hyperkahler type, any polystable holomorphic bundle is hyperholomorphic, and this category is in fact independent from the choice of $J$. Deformations of hyperholomorphic vector bundles have no higher obstructions, and their deformation spaces are also hyperkahler. I will give all definitions and explain some of the proofs. This construction might lead to new examples of hyperkahler manifolds; I will explain what is known in this direction.

The talk relies on Kobayashi-Hitchin correspondence (Donaldson-Uhlenbeck-Yau), and I would try to remind how it works and state all results that I am using.

 

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