ADVANCED GEOMETRY 1
Learning outcomes of the course unit
Introduction to complex geometry. Cohomological aspects of complex manifolds.
At the end of the course, students will be familiar with the theory of holomorphic function of several complex variables, with the differential calculus on complex maniflds and with the Hodge theory on compact Hermitian manifolds. They will also be able to tackle the resolution of problems of a theoretical and practical nature in the area of complex differential geometry.
Course contents summary
1. Complex manifolds.
1.1 Introduction to the theory of holomorphic functions of several complex variables.
1.2 Complex structures. Complex projective spaces. Complex tori.
1.3 Almost complex structures. Newlander-Nirenberg theorem.
1.4 (p,q)-forms on complex manifolds. del-bar operator.
1.5 Dolbeault complex.
2. Sheaves and cohomology.
2.1 Pre-sheaves and sheaves.
2.2 Cech cohomolgy.
3. Kaehler manifolds.
3.1 Hermitian and Kaehler metrics.
3.2. Kaehler metrics in local coordinates. Examples of Kaehler manifolds.
3.3. Curvature of Kaehler manifolds.
3.4 Cohomological properties of compact Kaehlermanifolds.
3.5 The del-del-bar Lemma.
3.6 Formality of compact Kaehler manifolds.
3.7 Massey products.
4. Introduction to the theory of deformations of complex structures
4.1 Complex analytic families of compact complex manifolds.
4.2 Infinitesimal deformations.
4.3 Differential Graded Algebras.
4.4 del-bar operator and Maurer-Cartan equation.
4.5. Kodaira and Spencer Stability Theorem.
J. Morrow, K. Kodaira, Complex manifolds. Reprint of the 1971 edition with
errata. AMS Chelsea Publishing, Providence, RI, 2006. x+194.
D.Huybrechts, Complex Geometry: An Introduction, Springer Universitext, 2014
Lectures and classroom exercises.
During lectures in traditional mode, the
topics will be formally and rigorously presented. The course will give particular emphasis to application and computation aspects, while not letting out
the theoretical aspect. The classroom exercises are aimed at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying.
Assessment methods and criteria
Final exam consists of theoretical lectures and sessions of oral and written exercises.
Exercises and problems to be performed outside of class hours will be assigned by the teacher.
Notes from the teacher will be distributed. The course notes in PDF format and all the material used during the lessons and exercises are made available to the students on the Elly educational platform.