Catalogo dei Corsi di studio

General goals: to acquire basic notions and skills in algebraic geometry.

Specific goals:

Knowledge and comprehension: at the end of the course the student will have learnt basic results on Hodge Theory and sheaf cohomology,
as applied to algebraic varieties.

Applied knowledge and comprehension: at the end of the course the student will be able to understand and appreciate part of the current literature on complex algebraic geometry.

Critical development: the student will appreciate the interaction between different fields such as differential geometry, global analysis, algebraic topology and algebaric geometry.

Communication skills: the ability to make a clear presentation of parts of the theory
introduced in the course.

Learning skills: the knowledge acquired will be useful in studying more specialized courses in algebraic and complex geometry.

Programme

Affine and projective varieties. Holomorphic functions of several variables. Complex manifolds. Vector bundles. Differential forms on complex manifolds. Harmonic forms. Kaehler manifolds. Kaehler identities. Hodge decomposition, Lefschetz decomposition. Hodge-Riemann bilinear relations. Sheaf cohomology. Period map. Mixed Hodge structure on a smooth quasi-projective variety.

Adopted texts

Claire Voisin: Hodge theory and complex algebraic geometry, vols I and II.

Jean-Pierre Demailly: Complex analytic and differential geometry. (Freely available on the authors' website)

Prerequisites

The course Istituzioni di Geometria Superiore of the laurea magistrale

Exam modes

The exam is oral. In order to pass the exam (the minimum score is 18/30) the student is required to know the main theoretical results, and to be able to solve simple exercises. In order to get the maximum vote, i.e. 30/30 e lode, the student is required to have a deep knowledge of the theoretical results, and to be able to solve difficult exercises.