Complex Varieties
Academic year 2011-2012.
Prerequisites:
Basic concepts from the theory of (real) differential geometry and complex analysis.
Program (6cfu):
Complex differentiable varieties, 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, 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].
Program (+3cfu):
The Picard group, the exponential sequence, the first Chern class, Kaehler varieties
[H], [W].
Exercises (for Algant students)
References:
[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.
[W] R.O. Wells, Differential Analysis on Complex Manifolds. Prentice Hall 1973 (Springer-Verlag 2008).