Complex geometry studies the geometry of complex manifolds, that is, manifolds possessing an atlas whose transition maps are holomorphic. Connected complex manifolds of dimension one are called Riemann surfaces, they were studied thoroughly on the previous semester. The existence of a holomorphic atlas implies several interesting restrictions on the manifold. For example, a complex submanifold of the complex projective space has to be algebraic, that is, can be described as zero set of polynomials. This builds the bridge to classical algebraic geometry.

If we look at complex manifolds from the point of view of Riemannian geometry, we find distinguished Riemannian metrics related to the complex structure, called Kähler metrics. The existence of a Kähler metric on a compact manifold imposes special structures on the cohomology of the manifold, namely the Hodge and Lefschetz decompositions. On the other hand, Kähler manifolds are special cases of symplectic manifolds, and their study leads to interesting insights in symplectic geometry.

In the introduction we study holomorphic functions of several variables, holomorphic convexity and pseudoconvexity. Next we deal with complex manifolds and holomorphic vector bundles, and provide some important examples (projective spaces, blow-up, divisors). We then present the machinery of sheaves and cohomology of sheaves.

The differential-geometric aspects are also discussed: connections, curvature, and Chern classes. With this tools at hand we define the notion of positivity for vector bundles and prove basic vanishing theorems for their cohomology. An upshot of the Kodaira vanishing theorem is the charaterization of projective submanifolds in terms of positive line bundles (also due to Kodaira).

We introduce a powerful analytical method, the L2 method of Hörmander for solving the Cauchy-Riemann equation. Another related analytic tool is the asymptotic expansion of the Bergman kernel. We show how the Bergman kernel yields deep results about Kähler metrics on a projective manifold. Another application of the Bergman kernel which we cover is the equidistribution of zeros of random polynomials or holomorphic sections. Holomorphic random sections provide a model for quantum chaos and have been intensively studied by physicists.

Prerequisities for this lecture are Analysis I-III, Complex Analysis and Algebra.

Detailed Contents

Prof. Dr. G. Marinescu

Tel.: 470 2661

Sitz: Weyertal 86-90, Zimmer 110

Sprechstunde: Di 13 - 14 Uhr

Tue 8-9.30, Großer Hörsaal der Botanik und Do. 10-11.30, Seminarraum 1 des Mathematischen Instituts (Raum 005)

Lecture 1

Thu 12-13:30 S2 (Room 204)

• P. Griffiths, J. Harris, Principles of Algebraic Geometry. Wiley (1978)
• J.-P. Demailly, Complex analytic and algebraic geometry
• X. Ma, G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels. Birkhäuser (2007)

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