Instructor: Chad Schoen
Sheaves will be introduced and used to define various geometric notions (eg. differentiable manifold, complex analytic space, abstract algebraic variety). Sheaf cohomology will be introduced and the connection with line bundles and Cartier divisors will be established. Sheaf cohomology will be used to compute the dimension of linear systems and thus to provide an essential tool in analyzing morphisms between projective algebraic varieties (respectively compact, complex manifolds). Sheaf cohomology will be used to construct important invariants of projective algebraic varieties (respectively compact, complex manifolds) such as the genus. Intersection theory on non-singular, projective, algebraic surfaces will be treated in detail. The sheaf of differential forms will be introduced and versions of the Serre duality theorem and the Riemann-Roch Theorem will be discussed. All these notions will be applied to do concrete computations with particular algebraic varieties.
Lectures lasting an hour and fifteen minutes are given twice each week. In addition, the class meets once a week to discuss homework. Written homework is assigned and graded each week.
Faisceaux algebriques coherents, J.-P. Serre
Topologie algebriques et theorie des faisceaux, R. Godement
Algebraic Geometry, R. Hartshorne
Basic Algebraic Geometry, I, Shafarevich
Lectures on Riemann Surfaces, Forster
Differential Forms in Algebraic Topology, Bott and Tu
Differential Analysis on Complex Manifolds, R. Wells
A course in Homological Algebra, P. Hilton and Stammbach
Introduction to Commutative Algebra, Atiyah and Macdonald