Advanced Topics in Geometry  a.a. 2025/2026

Advanced Topics in Geometry, a.a. 2025/2026,  Corso di Laurea Magistrale in Matematica.

Timetable: Tuesday 17:00-19:00 room G, Thursday 08:00-10:00 room G.

Recommended textbook: Differential Analysis on Complex Manifolds (third edition), Raymond O. Wells, Jr., Springer 2008.

Further readings:  D. Huybrechts, "Complex Geometry: An Introduction"; P. Griffith, J. Harris, "Principles of Algebraic Geometry".

Office hours: Wednesday 10:00-12:00.

Logbook:

25/09/2025, 08:00-10:00.  Smooth manifolds and smooth maps: definitions and examples. Rank of a smooth maps: immersions, submersions, embedding and diffeomorphisms. Vector bundles: definitions and first examples.

30/09/2025, 17:00-19:00. Tangent bundle and cotangent bundle. Transintion functions of a vector bundle. Construction of a vector bundle with an assigned family of transiction functions. Direct sums, tensor products, exterior product and dual. Pull-back bundle.

02/09/2025, 08:00-10:00. Transition functions of a pull-back bundles. Sections of vector bundles: definition and basic properties. Morphisms of vector bundles: definition and basic properties. Vector subbundles: definition and equivalent characterizations.

07/10/2025, 17:00-19:00. Vector bundles associated to a VB morphism with constant rank. Quotient vector bundle, complexification of real vector bundle and conjugate vector bundle. Riemannian (Hermitian) metrics on real (complex) vector bundles.

09/10/2025, 08:00-10:00. Orthogonal vector subbubdle. Short exact sequences of vector bundles. Basic properties of holomorphic functions in several complex variables.

14/10/2025, 17:00-19:00. Holomorphic maps: basic properties. Complex manifolds: definition and examples.

16/10/2025, 08:00-10:00. Holomorphic vector bundles: definition and basic properties. Holomorphic tangent bundle of a complex manifold. Almost complex manifolds: definition and basic properties. A complex manifold is canonically an almost complex manifold.