Mathematics | University of Parma

Professor
Academic discipline: Geometria (MAT/03)
Field: Formazione teorica avanzata
Type of training activity: Characterising

72 hours
of face-to-face activities

9 credits

hub: PARMA

course unit
in - - -

lectures timetable unavailable

exam calls

Learning objectives

Introduction to complex geometry. Cohomological aspects of complex manifolds.

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.

Bibliography

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

Theoretical lectures and sessions of oral and written exercises.

Other information

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ADVANCED GEOMETRY 1 cod. 18873