Practical information

Lectures:

Mondays from 13:15 to 15:00 at MIN014 (first half) and BBG017 and HFG611 (second half) scheduled as follows

Exercise classes

TBD

Main reference: Well’s Differential Analysis on Complex Manifolds.

Alternative reference: Huybrecht’s Complex Geometry.

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.

About the subject

In Complex Geometry we study manifolds with an atlas whose transition functions are holomorphic. The reasons to study those are many: they arise naturally when studying zeros of polynomials, are closely related to symplectic structrures and appear left and right in physics.

The switch from smooth to holomorphic has many consequences as we have already seen when studying functions of one complex variable. Among the striking features in higher dimensions, Whitney's embedding theorem fails: no compact complex manifold of positive dimension can be embedded in Cⁿ. This raises the question: is there a natural space where compact complex manifolds live? Also the behaviour of holomorphic singularities is much more tightly controlled than that of smooth singularities.

On the plus side, new information comes to light: besides usual de Rham cohomology of smooth manifolds, complex manifolds have a cohomology theory associated to the complex structure: Dolbeault cohomology and in the most interesting cases (Kähler manifolds) there is a close relation between these two cohomology theories.

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.

Pre-requisites

I will assume that you are familiar with

• the contents of a typical bachelor course on differentiable manifolds, including exterior forms, exterior derivative, de Rham cohomology, integration and Stokes theorem.

• the mastermath course on differential geometry, including vector bundles, connections on vector bundles and curvature.

• some analysis will show up, including Sobolev spaces, but I will aim to state without proof the main results about them as we go along.

7 February 2022 (Week 6)

Organisation and outlook of the course, Chapter 1, sessions 1, 2 and 3.

14 February 2022 (Week 7)

Chapter 1, sessions 4 and 5; Chapter 2, sessions 1 and 2.

28 February 2022 (Week 8)

Chapter 2, session 3.

7 March 2022 (Week 8)

Chapter 3, session 1,2 and 3.

14 March 2022 (Week 8)

Chapter 3, session 4 and Chapter 4, session 2.

21 March February 2022 (Week 8)

Chapter 4, session 1 and Chapter 5, sessions 1 and 2.

28 March 2022 (Week 8)

Chapter 5, sessions 3, 4 and 5.

4 April 2022 (Week 8)

Chapter 6, sessions 1 and 2.

11 April 2022 (Week 8)

Midterm exam. Material: Chapters 1 to 5 (inclusive). Open book exam.

25 April 2022 (Week 8)

Chapter 6, sessions 3 and 4.