Complex Manifolds (Varietà Complesse)
Academic year 2023-2024.
Prof. B. van Geemen (35 hours of the 6 cfu), Prof. L. Tasin (12 hours of the 6 cfu + (all) 26 hours of the +3cfu).
Prerequisites:
Basic concepts from the theory of (real) differential geometry and complex analysis (see for example these notes (in Italian)).
The course provides an introduction to complex manifolds for students who are familiar with (real) differentiable manifolds (as exposed for example in the book `An Introduction to Manifolds' by Loring W. Tu). The emphasis is on geometrical aspects, rather than on the basics of complex analysis in several variables (which one can learn from the books [GH] and [H] below). In particular, we discuss elliptic curves,
a very classical topic with many applications in physics and number theory. We define the basic notions of
vector bundles, sheaves and the cohomology of sheaves.
Since the 1950's these are the main instruments in the study of the geometry of complex manifolds. We provide various examples
and applications like the adjunction formula for the canonical bundle of a submanifold and the De Rham theorem, which identifies the cohomology groups of the sheaf of locally constant real-valued functions with the De Rham cohomology groups for a differentiable manifold.
In the (optional) second part we introduce a method to compute cohomology groups of
sheaves, Cech cohomology. Using this, and some results from Hodge theory,
we determine the Picard group (defined in terms of line bundles) of a complex manifold.
We conclude with an introduction to Kähler manifolds and their cohomology.
Program (6cfu, van Geemen + Tasin):
Complex differentiable manifolds, holomorphic tangent bundle, holomorphic maps and their differential,
differential forms of type (p,q) [Hu], [Hö] [W], [A].
Elliptic curves: The meromorphic Weierstrass "p" function, plane cubic curves, addition law, j-invariant [K], [Si].
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,
Cech cohomology, the Picard group [W], [A], [Tu].
Program (+3cfu, Tasin, see this
site):
the first Chern class, basics of Hodge theory, Kähler manifolds.
Exam: oral exam, one exam with Prof. B. van Geemen for the 6cfu and (optionally, only if you want to do the 6+3cfu), an additional oral exam with Prof. L. Tasin for the +3cfu-part.
The exams are on appointment by email (and you have to register on `SIFA' in order to be able to register the (final) vote).
Exercises
for Algant students only, for the course "Complex manifolds" (6cfu).
An Algant student should do at least one (and three is certainly enough) of these homework exercises and consign the exercises a few days before the oral exam
(which is on appointment by email).
Exercises for Algant students for the +3cfu: Exercises,
do one or two of these.
References
(Math library catalogue):
[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
(unpublished notes).
[Hu] D. Huybrechts, Complex geometry, an introduction. Berlin Springer-Verlag 2005.
[K] A.W. Knapp, Elliptic curves, Mathematical notes 40. Princeton University Press 1993.
[L] U. Lim, Harmonic Forms, Hodge Theory and the Kodaira Embedding Theorem (arXiv:2210.07952).
[Sc] C. Schnell, Complex manifolds (unpublished notes).
[Si] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106. Springer-Verlag 1986.
[Tu] L. Tu, Introduction to sheaf cohomology (arXiv:2206.07512).
[W] R.O. Wells, Differential Analysis on Complex Manifolds. Prentice Hall 1973 (Springer-Verlag 2008).