96728 - Complex Geometry

Learning outcomes

At the end of the course, the students understand the foundations of the theory of complex manifolds, of holomorphic functions among them and the basis of Hodge Theory. They are able to apply these knowledge in the solution of problems and in devising proofs.

Course contents

Local theory of holomorphic functions, Hartogs theorem, Weierstrass preparation theorem, Riemann extension theorem. Weierstrass division theorem and the sheaf of holomorphic functions.

Complex structures, orientation, fundamental form, Lefschetz operators, Hodge-Riemann relations.

Complex manifolds, quotients of complex manifolds, meromorphic functions, Siegel's theorem. Holomorphic vector bundles, line bundles, exponential sequence and first chern class. Adjunction, canonical ring and Kodaira dimension.

Divisors and line bundles, blow-ups, differential calculus on complex manifolds, Dolbeault complex.

Hermitian and Kähler manifolds, Fubini-Study metric, Kähler identities, Harmonic functions and Hodge theory. 1,1 Lefschetz theorem, Albanese map and Lefschetz hard theorem.

Readings/Bibliography

Complex Geometry, by Daniel Huybrechts (Universitext)

Optional books:

Hodge Theory and Complex Algebraic Geometry I, by Claire Voisin (Cambridge University Press)

Teaching methods

Lecture and exercises in class

Assessment methods

A seminar on optional topics, plus an oral exam after it.

Office hours

See the website of Giovanni Mongardi

See the website of Luca Battistella