Complex manifolds 1000-135ROZ
- https://www.mimuw.edu.pl/~jjelisiejew/uw/202324-cplx-manifolds/index.html (term 2023Z)
- https://moodle.mimuw.edu.pl/course/view.php?id=2264 (term 2024Z)
1. Local theory: holomorphic functions in many variables.
2. Almost complex structure, Newlander–Nirenberg theorem.
3. Holomorphic differential forms and smooth forms of the type (p,q).
4. Complex manifolds. Examples: curves (i.e. Riemann surfaces), projective space, grassmannians, complex tori, projective varieties.
5. Hermitian and Kaehler structure. Fubini-Study metric.
6. Complex Dolbeault and its cohomology. Holomorphic Poincare Lemma.
7. Laplasian and Hodge decomposition. Hard Lefschetz theorem for Kaehler manifolds.Hodge diamond.
8. Complex vector bundles, connections, differential definition of Chern classes.
Type of course
Mode
Requirements
Mathematical analysis II.1
Mathematical analysis II.2
Analytic Functions of One Complex Variable
Linear algebra and geometry I
Linear algebra and geometry II
Topology I
Prerequisites
Prerequisites (description)
Course coordinators
Term 2024Z: | Term 2023Z: |
Learning outcomes
Students knows basic notions of contemporary complex geometry and Kaehler
geometry. In particular understands the topics listed in the description of
the course. The course is a starting point to further studies in complex
geometry.
Assessment criteria
oral exam
Bibliography
1. D. Arapura, Algebraic Geometry over the complex numbers.
2. D. Huybrechts: Complex geometry. An introduction.
3. B. Shabat, An introduction to complex analysis
4. P. Griffiths, J. Harris: Principles of algebraic geometry.
5. M. De Cataldo: Lectures on the Hodge theory of projective manifolds.
6 S. S. Chern: Complex Manifolds without Potential Theory
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: