Introductory Workshop: Special Geometric Structures and Analysis: Constant scalar curvature metrics and semistable vector bundles
Presenter
September 6, 2024
Keywords:
- Kähler manifolds
- Kahler metrics
- Einstein metrics
- canonical metrics
- special holonomy
- Calabi-Yau
- geometric elliptic and parabolic PDEs
- Pluripotential Theory
- variational approach
- Monge-Ampère equation
- area minimizing currents
- semicalibrated currents
- minimal surfaces
MSC:
- 32Q15 - Kähler manifolds
- 32Q20 - Kähler-Einstein manifolds
- 32Q25 - Calabi-Yau theory (complex-analytic aspects)
- 32Q57 - Classification theorems for complex manifolds
- 32U05 - Plurisubharmonic functions and generalizations
- 32W20 - Complex Monge-Ampère operators
- 35B65 - Smoothness and regularity of solutions to PDEs
- 35J47 - Second-order elliptic systems
- 49Q05 - Minimal surfaces and optimization
- 49Q15 - Geometric measure and integration theory
- integral and normal currents in optimization
- 49Q20 - Variational problems in a geometric measure-theoretic setting
- 53A10 - Minimal surfaces in differential geometry
- surfaces with prescribed mean curvature
- 53C07 - Special connections and metrics on vector bundles (Hermite-Einstein
- Yang-Mills)
- 53C38 - Calibrations and calibrated geometries
- 53C55 - Global differential geometry of Hermitian and Kählerian manifolds
Abstract
In this talk, we present a construction of Kaehler metrics with constant scalar curvature on the projectivisation of certain holomorphic vector bundles. When the vector bundle is slope-stable and the base admits a constant scalar curvature metric, it is a classical result of Hong that the total space of the projectivisation admits a constant scalar curvature metric in adiabatic classes. We extend their result to slope-semistable vector bundles: we show that if $E \to B$ is slope-semistable and the total space of the projectivisation is K-polystable then it admits a constant scalar curvature metric in adiabatic classes.