- Docente: Enrico Fatighenti
- Credits: 6
- SSD: MAT/03
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 6730)
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 - (1,1) and (time permitting) Hard Lefschetz. Examples of calculations in the projective case.
Chern classes (axiomatic definition), the Riemann-Roch theorem, Serre duality, Kodaira vanishing.
Prerequisites:
Necessary courses include fundamental BSc courses in geometry (in particular Geometria 3 - 28377), and the MSc course Geometria Differenziale 00474 .
The course contents covered in Topologia Algebrica 28446, Varietà Algebriche B9037 will be extremely useful.
It will be useful to be familiar with the course contents of Geometria Proiettiva 54777 and Algebra commutativa 06689.
The course Teoria degli Schemi 96734 can be followed independently from this course.
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)
3264 and All That: A Second Course in Algebraic Geometry (David Eisenbud, Joe Harris)
Teaching methods
Lectures and exercises by the lecturer.
Assessment methods
Seminar and oral exam, with exercises at the end of the course.
Office hours
See the website of Enrico Fatighenti