Second Level Degree in Mathematics
Jump second level menu- Courses
- Timetable
- Exams
- Individual study program
- Student seminars
- Thesis
- Thesis Archive
- Graduation dates
- Contacts
ALGEBRAIC GEOMETRY 2 - 6 CFU
Teacher
Scheduled Period
I Year - 2 Semester | 28/02/2022 - 11/06/2022
Hours: 48 (24 esercitazione, 24 lezione)
Prerequisites
Basics in topology and differential geometry. It is recommended to have some knowledge on the theory of Riemann Surfaces.
Target skills and knowledge
Good knowledge of the algebraic and differential tools used in complex and algebraic geometry.
Examination methods
Seminar with questions.
Assessment criteria
The student will be evaluated on his/her understanding of the topics, on the acquisition of concepts and methodologies proposed and on the ability to apply them in full independence and awareness.
Course contents
The course is thought about as a continuation in higher dimension of the ideas developed in the theory of Riemann Surfaces.
- Preliminaries: holomorphic functions in complex variables, the difference with holomorphic functions in one complex variable.
- Complex varieties: definitions and first properties.
- Complex tangent space, and complex differential calculus.
- Vector Bundles: complex, hermitian, holomorphic vector bundles. Connections and curvature, the Chern connection. The example of line bundles. Chern forms.
- Divisors, line bundles, and Picard group. Weil Divisors and Cartier divisors.
- Kaehler manifolds: examples, volume form. Fubini-Study metric, smooth projective varieties as Kaehler manifolds. Examples of non-Kaehler varieties.
- Introduction to cohomology: de Rham cohomology, Dolbeault cohomology. Short overview on singular cohomology and sheaf cohomology. The computation of some cohomology groups will be shown, this will serve as a motivation for more general theories (to be studied in other classes).
- Analytic tools: integration, L^2 metric, operators on Kaehler manifolds, Laplacian.
- Hodge identities, Hodge decomposition, Hodge symmetry.
- Hodge decomposition in the algebraic case (projective varieties), Hodge conjecture.
- 2 important theorems: Kodaira embedding theorem, Torelli?s theorem.
Planned learning activities and teaching methods
Lectures and recommended exercises.
Additional notes about suggested reading
Further material will be available in the moodle page of the course.
Main textbook:
Claire Voisin "Hodge Theory and Complex Algebraic Geometry", Cambridge University Press.
Supplementary textbooks:
Daniel Huybrechts "Complex Geometry: An Introduction", Springer.
Phillip Griffiths, Joseph Harris "Principles of Algebraic Geometry", Wiley.