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.