Canonical metrics on Kähler manifolds and holomorphic vector bundles

Third-cycle level | 7.5 credits | Course code: NFMV025
HT 2022
Study period: 2022-09-26 - 2022-12-09
LANGUAGE OF INSTRUCTION: The course is given in English
Application period: 2022-09-07 - 2022-09-14

Course description

Main contents of the course:

The Hodge decomposition theorem on Kähler manifolds

The Uniformisation theorem

The existence of Kähler-Einstein metrics on canonically polarised/Calabi-Yau manifolds

The classical Futaki invariant as an obstruction to the existence of constant scalar curvature Kähler (cscK) metrics

The LeBrun-Simanca openness theorem for cscK/extremal metrics

The infinite dimensional moment map picture for the scalar curvature and K-stability as an obstruction to the existence of cscK metrics

The Hitchin-Kobayashi correspondence for holomorphic vector bundles 

Requirements and Selection

Entry requirements

Some basic differential geometry and complex geometry. Some experience with (elliptic) PDEs preferable, but we will go over the key points needed for our purposes in the lectures. Some parts of the course will feature some algebraic geometry, but this will be more focused on the ideas rather than precise proofs, so no real background with algebraic geometry is necessary.


Not relevant

Course syllabus



Department of Mathematical Sciences


Natural Science and Mathematics

Type of course

Subject area course


Kähler manifolds, metrics

CONTACTLars Martin Sektnan