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 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 CauchyRiemann 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 IIII 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 socalled Fay bosonization formula for Laughlin states on Riemann surfaces. For more details, see
the
website of the second part.



Instructors

Prof. Dr. G. Marinescu, Dr. Semyon Klevtsov
Sitz: Weyertal 8690, Zimmer 112
Sprechstunde: Mo 11  12 Uhr



Lectures

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



Tutorials

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


References




Notes




Homeworks




Presentations
