Schedule. Mondays 13.15-15.00.
Aims and contents. The aim of this course is to study complex manifolds, paying particular attention to Kähler manifolds. We will start with the general theory including almost complex structures, integrability sheaf theory, sheaf cohomology, Hermitian and holomorphic vector bundles and line bundle in particular. We then move on to study elliptic operator theory and its application to the Hodge decomposition of cohomology on Kähler manifolds. We then will move on to prove Kodaira's embedding theorem, which characterises which Kähler manifolds can be embedded in CPn.
References.
- R. O. Wells, Differential Analysis on Complex Manifolds.
- Daniel Huybrechts, Complex Geometry, An Introduction.
Prerequisites. The Mastermath course Differential Geometry.
Format. The course will consist of lectures and discussion time.
Evaluation. Examination will be done in two moments (midterm 40% and final exam 60%). Students who fail to achieve passing grade will do a retake exam for 100% of the final mark.
Learning goals and evaluation matrix.
- be able to construct complex manifolds and check integrability of almost complex manifolds (midterm)
- be able to do concrete and abstract computations on vector bundles (midterm)
- understand Chern-Weil theory for characteristic classes (midterm)
- work with familiar operators on complex and symplectic manifolds (midterm & final)
- use elliptic operator theory (midterm & final)
- use line bundles to produce interesting maps on manifolds (midterm & final)
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