Introductory Workshop: Special Geometric Structures and Analysis: Hodge cohomology of complete hyperKähler manifolds
Presenter
September 4, 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
After reviewing the notion of L^2-cohomology and presenting various conjectures and recent results about the Hodge cohomology (in other words reduced L^2-cohomology) of some complete hyperKähler metrics, I will focus on a result of Hausel-Hunsicker-Mazzeo of 2004 describing the Hodge cohomology of most types of gravitational instantons. In particular, I will explain how this result crucially relies on a decay at infinity of L^2-harmonic forms which will be the object of the next lecture.