MA954 Complex Geometry

Introductory description

The primary goal of this Module is to present some fundamental techniques from several complex variables, Hermitian differential geometry (and partial differential equations, potential theory, functional analysis), to study the geometry of complex, and in particular, Kaehler manifolds. Hodge theory will be one important major topic of this course.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The primary goal of this Module is to present some fundamental techniques from several complex variables, Hermitian differential geometry (and partial differential equations, potential theory, functional analysis), to study the geometry of complex, and in particular, Kaehler manifolds. Hodge theory will be one important major topic of this course.

-Basics/definitions concerning complex manifolds, vector bundles and sheaf theory

-Some selected topics from several complex variables: the Cauchy integral, the Cauchy-Riemann equations, Hartogs’s principle, plurisubharmonic functions, domains of holomorphy, holomorphic convexity, Riemann extension theorem, Hormander’s L2 estimates …

-Hermitian differential geometry, curvature of Hermitian holomorphic vector bundles, Chern classes

-Some elliptic operator theory, Kaehler manifolds, Hodge decomposition, Kodaira embedding,

• Outlook on the topology of varieties, Morse theory, Lefschetz pencils, variation of Hodge structures, Clemens-Schmid exact sequences, etc.

Learning outcomes

By the end of the module, students should be able to:

• By the end of the module students should be able to use the differential geometry and global analysis viewpoint to study complex manifolds. As the module provides the foundations of several different branches of mathematics, the module would be an important preparation for PhD students in related areas to read research papers.

Indicative reading list

Reading lists can be found in Talis

Subject specific skills

Develop a deep understanding and applicability of the following topics:

• fundamental techniques from several complex variables,
• Hermitian differential geometry (and partial differential equations, potential theory, functional analysis),
• geometry of complex, and in particular, Kaehler manifolds.

By the end of the module students should be able to use the differential geometry and global analysis viewpoint to study complex manifolds. The module would be an important preparation for PhD students in related areas to read research papers.

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)