Learning objectives
The object of the course is to familiarize the students with the basic language of and some fundamental theorems in Complex Geometry, focusing on cohomological properties and global aspects.
Prerequisites
Analysis 1, 2, 3, Geometry 1, 2, 3, Algebra, Mathematical. Physics
Course unit content
Complex Geometry.
Full programme
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.
2.3 Resolutions.
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.
References:
S.S. Chern, W.H. Chen, K.S. Lam, Lectures on Differential Geometry, Series on University Mathematics - Vol. 1, World Scientific, 2000.
J. Morrow, K. Kodaira, Complex manifolds. Reprint of the 1971 edition with
errata. AMS Chelsea Publishing, Providence, RI, 2006. x+194
Bibliography
S.S. Chern, W.H. Chen, K.S. Lam, Lectures on Differential Geometry, Series on University Mathematics - Vol. 1, World Scientific, 2000.
J. Morrow, K. Kodaira, Complex manifolds. Reprint of the 1971 edition with
errata. AMS Chelsea Publishing, Providence, RI, 2006. x+194
Teaching methods
Theoretical lectures and sessions of oral and written exercises.
Assessment methods and criteria
Homeworks and oral exam.
Other information
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