Catalogo dei Corsi di studio

General aim: To get a basic knowledge of projective and Kähler manifolds

Knowledge and understanding: at the end of the lectures the student will be acquainted with basic notions and results from the theory of projective and Kähler manifolds, of holomorphic vector bundles and their characteristic classes, of the correspondence between line bundles and divisors and of the use of cohomological techniques such as the Hirzebruch-Riemann-Roch theorem or the Kodaira theorems.


Applying knowledge and understanding: at the end of the lectures the student will be able to solve simple problems requiring cohomological techniques in the study of the geometry of projective manifolds.


Analytical and judgment abilities: the student will be able to analyse the relations between the topics covered in the lectures and topics in algebraic topology (acquired in the Topologia Algebrica or in the Istituzioni di Geometria Superiore courses), Riemannian and Hermitean geometry (acquired in the Geometria Riemanniana course), and of complex analysis (acquired in the Variabile Complessa course)


Communication abilities: the student will be able to communicate the contents of the lectures and some developements of them in the short presentations that will constitute part of the exam.


Learning abilities: the acquired notions will allow the student to study (either by themselves or in a PhD course) more advanced topics in the theory of projective and Kähler manifolds.

Programme

Complex manifolds
Holomorphic vector bundles
Divisors and line bundles
The projective space
Blow-ups
Differential calculus on complex manifolds
Kähler manifolds
Kähler identities
Hodge theory on Kähler manifolds
Lefschetz theorems
Hermitean vector bundles and Serre duality
Chern classes
The Hirzebruch-Riemann-Roch theorem
The Kodaira vanishing theorem
The Kodaira embedding theorem

Adopted texts

Daniel Huybrechts: Complex Geometry - An Introduction. Springer

Bibliography

Claire Voisin: Hodge Theory and Complex Algebraic Geometry I. Cambridge University Press Phillip Griffiths, Joseph Harris: Principles of Algebraic Geometry. John Wiley and Sons Ltd

Prerequisites

In order to attend lectures for this course, familairity is needed with the topics covered by the lecture courses in Topologia Algebrica or Istituzioni di Geometria Superiore, in Geometria Riemanniana and in Variabile Complessa. In particular, it is required to be familiar with general topology, with the notion of topological and smooth manifold, with Riemannian and Hermitean metrics, with the notion of connection and curvature (at least on the tangent bundle), with singular homology, with singular and de Rham cohomology, with the theory of holomorphic functions in one complex variable, and with basic elements of the theory of holomorphic functions in many complex variables. Being familiar with these topics is essential. No preparatory course is mandatory.

Exam modes

The exam aims to evaluate the student's understanding and confidence of the topics presented in the lectures for the course through exercises and a lecture where the student will explore aspects only minimally touched in the lectures for the course.

In order to pass the exam one needs a mark of no less than 18/30. The student has to prove to have acquired a reasonable knowledge of the topics covered in the lectures and to be able to solve the easiest exercises proposed during the course and to give a proper exposition of the topic which is the subject of their assigned lecture.
In order to get full marks and honour (30/30 e lode), the student has to prove to have acquired an excellent knowledge of the topics covered in the lectures, to be able to connect them in a coherent way, and to be able to solve all of the exercises proposed during the course and to give an excellent exposition of the topic which is the subject of their assigned lecture.

Exam reservation date start	Exam reservation date end	Exam date
19/10/2018	10/06/2019	11/06/2019
19/10/2018	28/06/2019	01/07/2019
19/10/2018	23/07/2019	26/07/2019
19/10/2018	07/09/2019	10/09/2019
19/10/2018	15/09/2019	18/09/2019
19/10/2018	18/11/2019	19/11/2019
19/10/2018	14/01/2020	17/01/2020