Complex manifolds

A.Y. 2019/2020
9
Max ECTS
69
Overall hours
SSD
MAT/03
Language
Italian
Learning objectives
The aim of the course is to provide an introduction to the theory of complex manifolds.
Expected learning outcomes
Basic knowledge of modern complex geometry.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
Prerequisites for admission
Basic concepts from the theory of (real) differential geometry and complex analysis.
Assessment methods and Criteria
The final examination consists of an oral exam on appointment. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding the matter covered in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Varietà complesse (prima parte)
Course syllabus
Program with references:
Complex differentiable manifolds, holomorphic tangent bundle, holomorphic maps and their differential, differential forms of type (p,q) [Hu], [Hö] [W], [A].
Vector bundles, the tangent bundle, the canonical bundle, the normal bundle, divisors and line bundles, the adjunction formula [Hu], [Hö].
Sheaves and presheaves of abelian groups, homomorphisms of sheaves, exact sequences of sheaves, cohomology with coefficients in a sheaf of abelian groups, acyclic resolutions, the De Rham theorem [W], [A].
Elliptic curves: The meromorphic Weierstrass "p" function, plane cubic curves, addition law, j-invariant [K], [Si].
Teaching methods
Traditional lectures at the black board.
Teaching Resources
[A] D. Arapura, Algebraic geometry over the complex numbers. Springer-Verlag 2012.
[GH] Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley and Sons 1978.
[Hö] A. Höring, Kähler geometry and Hodge theory.
[Hu] D. Huybrechts, Complex geometry, an introduction. Berlin Springer-Verlag 2005.
[K] A.W. Knapp, Elliptic curves, Mathematical notes 40. Princeton University Press 1993.
[Sc] C. Schnell, Complex manifolds.
[Si] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106. Springer-Verlag 1986.
[W] R.O. Wells, Differential Analysis on Complex Manifolds. Prentice Hall 1973 (Springer-Verlag 2008).

Web site: http://www.mat.unimi.it/users/geemen/VarietaComplesse19E.html
Varietà complesse mod/2
Course syllabus
Program with references:
Cech cohomology, the Picard group, the first Chern class, basics of Hodge theory, Kähler manifolds [A], [H], [Hö], [W].
Teaching methods
Traditional lectures at the black board.
Teaching Resources
[A] D. Arapura, Algebraic geometry over the complex numbers. Springer-Verlag 2012.
[Hu] D. Huybrechts, Complex geometry, an introduction. Berlin Springer-Verlag 2005.
[Hö] A. Höring, Kähler geometry and Hodge theory.
[K] A.W. Knapp, Elliptic curves, Mathematical notes 40. Princeton University Press 1993.
[S] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106. Springer-Verlag 1986.
[W] R.O. Wells, Differential Analysis on Complex Manifolds. Prentice Hall 1973 (Springer-Verlag 2008).

Web page: http://www.mat.unimi.it/users/geemen/VarietaComplesse19E.html
Varietà complesse (prima parte)
MAT/03 - GEOMETRY - University credits: 6
Practicals: 10 hours
Lessons: 35 hours
Varietà complesse mod/2
MAT/03 - GEOMETRY - University credits: 3
Practicals: 10 hours
Lessons: 14 hours