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.

21/10/2025, 17:00-19:00.  Decomposition of the complexified tangent bundle. Forms bi-degree (p,q) and del-bar operator: definitions and first properties.

23/10/2025, 08:00-10:00. Integrable complex structures: definitions and equivalent characterizations. The canonical almost comlex structure of a complex manifold is integrable. Newlander-Nirenberg theorem (without proof).

28/10/2025, 17:00-19:00. Further propeties of the Dolbeault operator. Dolbeault cohomology. Dolbeault operator and Dolbeault complex twisted with a holomorphic vector bundle. Connections on vector bundles: definition and first properties.

30/10/2025, 08:00-10:00.  Basic properties of connections.  Connections on direct sums, tensor products of vector bundles and on the dual bundle.  Local description of a connection: the 1-forms matrix of a connection.

04/11/2025, 17:00-19:00. Behaviour of the connection matrix under a change of frame. Pull-back connection. Curvature of a connection. Local description of the curvature matrix in terms of the connection matrix.

06/11/2025, 08:00-10:00. Behaviour of the curvature matrix under a change of frame. Bianchi identity. Connections compatibl with a metric. Chern connection: existence and uniqueness. Basic properties of the curvature of Chern connection.

11/11/2025, 17:00-19:00. Brief backgroung of Sobolev spaces on R^m. Sobolev spaces of sections on compact manifolds: definition and basic properties. Definition of differential operators acting between the sections of two smooth vector bundles.

13/11/2025, 08:00-10:00. Examples and basic properties of differential operators. Principal symbol of a differential operator. Examples.  Elliptic operators: definition and examples. Formal adjoint of a differential operator.

18/11/2025, 17:00-19:00. Existence and uniqueness of the formal adjoint. Space of symbols of order k. Introduction to pseudodifferential operators and main properties (without proves). Parametrix of an elliptic differential operator.

20/11/2025, 08:00-10:00.  Definition and basic properties of Fredholm  operators. Basic analytic properties of elliptic differential operators: Fredholmness and elliptic regularity. Orthogonal decomposition of the space of smooth section as L2 orthogonal direct sum between the kernel of an elliptic differential operator and the image of its adjoint. Definition of elliptic complex.

27/11/2025. Hodge theorem for elliptic complexes of constant degree. Applications to the de Rham complex.

AVVISO 1: Modalità d' esame. L'esame consiste in una prova scritta contenente sia  esercizi presi  dalle esercitazioni assegnate lungo il corso che da esercizi nuovi. La prova orale è facoltativa.

Primo appello: 26/01/2026.

Secondo appello: 16/02/2026.

E' possibile sostenere lo scritto al primo appello e l'orale al secondo appello. Non è possibile sostenere lo scritto nella sessione di giugno-luglio e l'orale in una sessione successiva.

Terzo appello: 23/06/2026.

Quarto appello: 17/07/2026.

E' possibile sostenere lo scritto al terzo appello e l'orale al quarto appello. Non è possibile sostenere lo scritto nella sessione di settembre e l'orale in una sessione successiva.

Quinto appello: 11/09/2026.