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Università degli Studi di Parma Università degli Studi di Parma
Second cycle degree course in
Mathematics
Università degli Studi di Parma
Department of Mathematical, Physical and Computer Sciences

COMPLEX GEOMETRY
cod. 23980

Academic year 2013/14
1° year of course - First semester
Professor
Adriano TOMASSINI
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
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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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2030 agenda goals for sustainable development

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Contacts

Toll-free number

800 904 084

Segreteria studenti

E. segreteria.scienze@unipr.it
T. +39 0521 905116

Quality assurance office

Education manager
dott.ssa Giulia Bonamartini
T. +39 0521 906968
Office E. smfi.didattica@unipr.it
Manager E.giulia.bonamartini@unipr.it

President of the degree course

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Faculty advisor

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Career guidance delegate

Prof. Francesco Morandin
E. francesco.morandin@unipr.it

Tutor Professors

Prof.ssa Alessandra Aimi
E. alessandra.aimi@unipr.it

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Prof. Adriano Tomassini
E. adriano.tomassini@unipr.it

 

Erasmus delegates

Prof. Leonardo Biliotti
E. leonardo.biliotti@unipr.it

Quality assurance manager

Prof.ssa Alessandra Aimi
E. alessandra.aimi@unipr.it

Internships

Prof. Costantino Medori
E. costantino.medori@unipr.it

Tutor students

Dott.ssa Fabiola Ricci
E. fabiola.ricci1@studenti.unipr.it