Complex Geometry


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 this lecture we will take an analytic view on complex manifolds with emphasis on Riemann surfaces. We will start by holomorphic functions of several variables, complex manifolds, holomorphic vector bundles then introduce connections, curvature, and Chern classes. We will prove the Hodge theorem using the theory of elliptic operators. 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 Hormander 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 Kahler 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 are Analysis I-III and Complex Analysis. The lecture will stretch on the whole academic year 2016/17 and there will be one course a week. The examination will take place at the end of summer semester 2017.
The lecture in the summer semester will be given jointly with Dr. Semyon Klevtsov. The mathematics developed in this course has applications to the integer and fractional quantum Hall effects. In particular, the lecturers intend to explain the so-called Fay bosonization formula for Laughlin states on Riemann surfaces. For more details, see the website of the second part.

Prof. Dr. G. Marinescu, Dr. Semyon Klevtsov
Sitz: Weyertal 86-90, Zimmer 112
Sprechstunde: Mo 11 - 12 Uhr

Di. 14.00 - 15.30 Uhr in Seminarraum 1 des Mathematischen Instituts (Raum 005)

Mo. 14.00 - 15.30 Uhr in Übungsraum 1 des Mathematischen Instituts (Raum -119)
Am 24.04.17 findet dort die Vorlesung statt.


1, 2, 3, 4, 5, Summer Semester: 1-4, 5, 6, 7, 8, 9

1, 2, 3, 4, 5, 6, 7, 8, 9