Kartick Ghosh
TIFR, Mumbai
November 16, 2023
On some canonical metrics on holomorphic vector bundles over K$\ddot{\mathrm{a}}$hler manifolds: I will talk about two PDEs: Coupled K$\ddot{\mathrm{a}}$hler-Einstein and Hermitian-Yang-Mills equations and vortex-type equations.
TIFR, Mumbai
November 16, 2023
On some canonical metrics on holomorphic vector bundles over K$\ddot{\mathrm{a}}$hler manifolds: I will talk about two PDEs: Coupled K$\ddot{\mathrm{a}}$hler-Einstein and Hermitian-Yang-Mills equations and vortex-type equations.
Firstly, I will introduce the coupled K$\ddot{\mathrm{a}}$hler-Einstein and Hermitian-Yang-Mills equations. I will then talk about two obstructions to the existence of solutions of these equations: Matsushima-Lichnerowicz type obstruction and Futaki inavariant type obstruction. We will solve the equations is some cases using deformation. Using Calabi ansatz, we will solve the equations on some projective bundles.
In the second part, I will introduce vortex-type equation and derive a priori estimates for it. Then I will give some applications.