- Docente: Enrico Fatighenti
- Credits: 6
- SSD: MAT/03
- Language: Italian
- Moduli: Nicola Tito Pagani (Modulo 2) Enrico Fatighenti (Modulo 1)
- Teaching Mode: Traditional lectures (Modulo 2) Traditional lectures (Modulo 1)
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)
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from Feb 17, 2025 to May 27, 2025
Learning outcomes
At the end of the course, the student will have acquired the fundamentals of the theory of complex manifolds, holomorphic forms, and Hodge theory. They will be able to apply the acquired concepts to solve problems and construct proofs.
Course contents
Sheaf theory and their cohomology. Tools from complex analysis in several variables. Complex structures and complex manifolds, differential forms of type (p,q).
Holomorphic vector bundles, line bundles, exponential sequence, and first Chern class, adjunction formula. Canonical ring and Kodaira dimension, algebraic-geometric examples.
Hodge theory on Kaehler manifolds. Hodge symmetries and Lefschetz theorems. Examples of calculations in the projective case.
Time permitting, the following topics (or some of them) will be covered: Chern classes (axiomatic definition), the Riemann-Roch theorem, Serre duality, Kodaira vanishing.
Readings/Bibliography
The course will follow (alternatively) the following texts. Exact bibliographic references will be provided during the lectures.
Hodge Theory and Complex Algebraic Geometry I, by Claire Voisin (Cambridge University Press)
Complex Geometry, by Daniel Huybrechts (Universitext)
(another optional text)
Principles of Algebraic Geometry, by Phillip Griffiths, Joseph Harris (Wiley)
Teaching methods
Lectures and exercises by the lecturer(s).
Assessment methods
Seminar and oral exam, with possible exercises at the end of the course.
Office hours
See the website of Enrico Fatighenti
See the website of Nicola Tito Pagani