Complex Manifolds (Varietà Complesse)

Academic year 2014-2015.
The first six hours of the course are `tutorato' (Tutor: Dr. Mongardi) where the basics of complex analysis are treated, based on these notes (in italian, .pdf).


Prerequisites:
Basic concepts from the theory of (real) differential geometry and complex analysis.


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.

Program (6cfu):
Complex differentiable manifolds, holomorphic tangent bundle, holomorphic maps and their differential, differential forms of type (p,q) ([H], [W]).
Elliptic curves: The meromorphic Weierstrass "p" function, plane cubic curves, addition law, j-invariant [K], [S].
Vector bundles, the tangent bundle, the canonical bundle, the normal bundle, divisors and line bundles, the adjunction formula [H].
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].

Exercises for Algant students only, for the course "Complex manifolds" (6cfu, accademic year 2013-2014). An Algant student should do at least one of these homework exercises and consign the exercises a few days before the oral exam (which is on appointment).


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.
[H] D. Huybrechts, Complex geometry, an introduction. Berlin Springer-Verlag 2005.
[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).
Notes from the course "Complex manifolds" (9cfu, accademic year 2011-2012, author: Marco Ramponi).
Notes from the course "Complex manifolds" (6cfu, accademic year 2012-2013, author: Michele Ferrari).