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Sommersemester 2015

Complex Geometry

Gerasim Kokarev (Mathematisches Institut der LMU)

Course description:
The course is an introduction to the differential geometry of complex manifolds. Complex geometry lies at the intersection of a number of fields, such as Differential Geometry, Complex Analysis, PDE, Topology, and Algebraic Geometry. It also has important connections and serves as a primary language to many problems in mathematical physics.

The course is oriented on students in Mathematics and Physics, and is one of the modules in the Mathematics Master Programme as well as the Master Programme in Theoretical and Mathematical Physics (TMP). It is worth 9 ECTS points.

The core module "Differenzierbare Mannigfaltigkeiten/Differential geometry".

Lectures schedule:
Lectures will be given in English twice a week: 10.00-12.00 Tue and 12.00-14.00 Thu, Room B 046

Course outline:
The course covers the standard material on complex manifolds, holomorphic forms, connections on holomorphic vector bundles, elements of Chern-Weyl theory, Kähler metrics and their properties. More advanced material includes examples and properties of Kähler metrics with various curvature constraints; in particular, at the end of the course we plan to discuss the existence of Kähler-Einstein metrics.

Reading list:

Main material:
1. Moroianu, A. Lectures on Kähler geometry. London Mathematical Society Student Texts, 69. Cambridge University Press, Cambridge, 2007. x+171 pp.
2. Tian, G. Canonical metrics in Kähler geometry. Notes taken by Meike Akveld. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2000. vi+101 pp.

Additional material:
3. Ballmann, W. Lectures on Kähler manifolds. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2006. x+172 pp.
4. Chern, S. Complex manifolds without potential theory (with an appendix on the geometry of characteristic classes). Revised printing of the second edition. Universitext. Springer-Verlag, New York, 1995. vi+160 pp.
5. Huybrechts, D. Complex geometry. An introduction. Universitext. Springer-Verlag, Berlin, 2005. xii+309 pp.
6. Kobayashi, S., Nomizu, K. Foundations of differential geometry. Vol. II. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969 xv+470 pp.

Exercise classes (Übungen):
The exercise classes will hold at 14.00-16.00 Fri, Room B046.

The problem sets for the exercise classes are posted on this web-page on Fridays and are due a week later by Friday 12am. Students are encouraged to do as many problems in these sets as possible, since they comprise an important part of the course and similar problems are likely to appear on the exam.

Problem Set 1 Problem Set 2 Problem Set 3 Problem Set 4
Problem Set 5 Problem Set 6 Problem Set 7 Problem Set 8
Problem Set 9 Problem Set 10 Problem Set 11 Problem Set 12

Final Exam:
There will be an oral exam at the end of July. If you would like to take an exam please send an email to register and arrange the date and time. The registration deadline is 10 July.

Course Programme:

1. Complex Differential Geometry

1.0. Introduction I: holomorphic functions of several variables, extension theorems, zero sets, holomorphic maps and the implicit function theorem.
1.1. Introduction II: complex and Hermitian structures on vector spaces.
1.2. Complex manifolds: examples and constructions, blowing up points; holomorphic tangent bundle.
1.3. Almost complex structures, integrability and torsion, complex vector fields, the Newlander-Nirenberg theorem; a non-integrable almost complex structure (on a 6-dimensional sphere).
1.4. Differential forms on almost complex manifolds; exterior derivative, Cauchy-Riemann operator, and integrability; Dolbeault-Grothendieck lemma and its consequences; holomorphic vector fields and forms.
1.5. Complex vector bundles: Hermitian structures, linear connections and their curvature, invariant polynomials and Chern classes, line bundles and hypersurfaces.
1.6. Holomorphic vector bundles: basic definitions and examples, the Koszul-Malgrange theorem, Chern connections.

2. Kähler metrics and their properties

2.1. Hermitian and Kähler metrics on complex manifolds: basic hypotheses and characteristic properties, examples, properties of the curvature tensor.
2.2. Holomorphic sectional curvature, spaces of constant holomorphic sectional curvature and uniformisation, the notion of bisectional curvature.
2.3. Laplace operator on differential forms on Riemannian and complex Hermitian manifolds; the Hodge and the Dolbeault-Hodge decomposition theorems, topological applications.
2.4. Differential operators on Kähler manifolds and elements of Hodge theory; the Hodge diamond, relations between Betti and Hodge numbers, other applications.
2.5. Prescribing Ricci curvature: Calabi-Yau theorem and its applications. Existence of Kähler-Einstein metrics: Aubin-Yau theorem, outline of a proof.

Last modified: 2 April 2015