Videos

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.