Title: On some canonical metrics on holomorphic vector bundles over Kähler manifolds
Speaker: Kartick Ghosh (IISc Mathematics)
Date: 06 July 2023
Time: 3 pm
Venue: Hybrid - Google Meet (online) and LH-3, Mathematics Department
This thesis consists of two parts. In the first part, we introduce coupled K¨ahler-Einstein and Hermitian-Yang-Mills equations.
It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an
obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another
obstruction. Using Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles.
Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for
vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence
and uniqueness result concerning solutions of Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex
equation and the Monge-Amp`ere vortex equation. We establish a correspondence result between Gieseker stability and the
existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the K¨ahlerness of the symplectic
form which arises in the moment map interpretation of Calabi-Yang-Mills equations.