Complex Manifolds
A.Y. 2020/2021
Learning objectives
Learn some basic tools and methods in the theory of complex manifolds.
Expected learning outcomes
Students will learn some basic tools and results in the theory of complex manifolds including vector bundles, sheaves and their cohomology. Special attention will be given to 1 dimensional tori and the adjunction formula.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
The lectures will be by video (not live), and they will be put on "Ariel" shortly before the scheduled lectures. Occasionally there will be meetings on Skype or Zoom or Teams for further information and to discuss any questions the students may have.
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].
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].
Prerequisites for admission
Basic concepts from the theory of (real) differential geometry and complex analysis.
Teaching methods
Probably only video lectures, maybe live, see the updated information on the web site of the course http://www.mat.unimi.it/users/geemen/VarietaComplesse20E.html and on the "Ariel" site of the course.
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/VarietaComplesse20E.html
[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/VarietaComplesse20E.html
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.
Professor(s)