In this talk, we shall talk about two equations: Coupled Kähler-Einstein and Hermitian-Yang-Mills equations, and Vortex-type equations.
In the first part, we shall talk about coupled Kähler-Einstein and Hermitian-Yang-Mills equations. We shall give a moment map interpretation of these equations. We shall produce some nontrivial examples of solutions of these equations on some projective bundles using Calabi ansatz. Another class of nontrivial examples shall be produced using deformation. We then identify a Futaki-type invariant as an obstruction to the existence of solutions to these equations. We shall also show a Matsushima-Lichnerowicz-type theorem as another obstruction.
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-Ampère vortex equation. We shall establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We shall also investigate the Kählerness of the symplectic form which arises in the moment map interpretation of Calabi-Yang-Mills equations.
Zoom link: https://icts-res-in.zoom.us/j/84136629627?pwd=NThDVEtTRUdXTFh2RU1nQWI2dUJ0QT09
Meeting ID: 841 3662 9627
Passcode: 272723