Topics

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, blowup, divisors). We then present
the machinery of sheaves and cohomology of sheaves.
The differentialgeometric 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 CauchyRiemann 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 IIII, Complex Analysis and Algebra.



Professor

Prof. Dr. G. Marinescu
Tel.: 470 2661
Sitz: Weyertal 8690, Zimmer 110
Sprechstunde: Di 13  14 Uhr



Lectures

Tue 89.30, Großer Hörsaal der Botanik und Do. 1011.30, Seminarraum 1 des Mathematischen
Instituts (Raum 005)



Tutorials

Thu 1213:30 S2 (Room 204)



References




Homeworks
