Complex manifolds 1000-135ROZ

https://www.mimuw.edu.pl/~jjelisiejew/uw/202324-cplx-manifolds/index.html (term 2023Z)

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.

Course coordinators

Term 2024Z:

Agnieszka Kisielińska

Term 2023Z:

Aleksandra Sus

Type of course

elective courses

Mode

Classroom

Requirements

Algebra I
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

Algebraic Geometry
Differential geometry
Algebraic methods in geometry and topology
Algebraic topology

Prerequisites (description)

Differential manifolds, differential forms, cohomology, theory of complex functions in one variable and algebraic methods of topology and geometry.

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