TOPICS IN GEOMETRY - Complex Geometry

Mathematics and Statistics

About this course

Selected topics in geometry and its applications. This course will study complex manifolds, differentiable manifolds with an atlas whose transition functions are holomorphic. These manifolds arise naturally when studying zeros of polynomials and are ubiquitous throughout physics. As one sees in a course on complex analysis, the transition from smooth functions to holomorphic ones has numerous consequences and imposes rigid restrictions on the geometry of complex manifolds, often giving the subject a very algebraic flavor.

Contact Hours: 3 per week This provides new information about familiar invariants of manifolds: besides usual de Rham cohomology of smooth manifolds, complex manifolds have a cohomology theory associated to their complex structure: Dolbeault cohomology. In the most interesting cases, that of Kähler manifolds, there is in fact a close relation between these two cohomology theories embodied in the Hodge decomposition. On the other hand, in higher dimension, another manifestation of this rigidity is the failure of Whitney's embedding theorem: no compact complex manifold of positive dimension can be embedded in Cn. This raises the fundamental question: Is there a natural space where compact complex manifolds live?

We will start with the foundations of almost complex structures and integrability, sheaf theory, sheaf cohomology, Hermitian and holomorphic vector bundles and line bundles in particular. We then move on to study elliptic operator theory and its application to the Hodge decomposition of cohomology on Kähler manifolds and proceed to proving Kodaira's embedding theorem, which characterises which Kähler manifolds can be embedded in CPn, giving a partial answer to the fundamental question above.

Expected learning outcomes

A student completing the course in good standing will be able to understand the myriad of applications of complex geometry throughout mathematics and physics. Most importantly, the student will have the requisite background to study complex algebraic geometry, mirror symmetry, and symplectic geometry, in addition to going further in the geometry of complex manifolds.