ADVANCED GEOMETRY 1
Learning outcomes of the course unit
Introduction to complex geometry. Cohomological aspects of complex manifolds.
Analysis 1, 2, 3, Geometry 1, 2, 3, Algebra, Mathematical. Physics
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
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
Theoretical lectures and sessions of oral and written exercises.