ADVANCED GEOMETRY 2
Learning outcomes of the course unit
Learn basic notions on complex algebraic manifolds. Understanding which complex manifold is isomorphic to an algebraic subvariety of some complex projective space.
Holomorphic functions of one complex variable. Complex manifolds. Hodge theory on Kaehler manifolds.
Course contents summary
Holomorphic functions of several variables. Sheaf theory and sheaf cohomology. Holomorphic vector bundles and divisors. Blow-ups. Hermitian vector bundle, connections, curvature and Chern classes. Applications of cohomology.
Holomorphic functions of several variables (Hartogs' Teorem, Weierstrass' Theorems, Riemann' extension Theorem, Nullstellensatz). Sheaf theory and sheaf cohomology (rudiments of homological algebra, abstract de Rham Theorem, de Rham and Dolbeault Theorems). Holomorphic vector bundles (canonical bundle, adjunction formula, Kodaira dimension) and divisors (relations with line bundles, Kobaira map, divisors on curves). Blow-ups (canonical bundle of a blow-up). Hermitian vector bundle, connections, curvature and Chern classes (Serre duality, Bianchi identity, Chern connection, positive vector bundles). Applications of cohomology (Kodaira vanishing Theorem, Kodaira embedding Theorem, Riemann-Roch theorem on curves and Hirzebruch-Riemann-Roch formula).
D. Huybrechts, COMPLEX GEOMETRY (AN INTRODUCTION), Springer 2005
J.-P. Demailly, COMPLEX ANALYTIC AND DIFFERENTIAL GEOMETRY, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
R. Hartshorne, ALGEBRAIC GEOMETRY, Springer 1977
C. Voisin, Hodge theory and complex algebraic geometry, Cambridge 2002
Standard blackboard lectures.
Assessment methods and criteria
Homeworks during the course. Final exam will be an expository talk on a subject assigned by the theacher.