ADVANCED GEOMETRY 2
Learning outcomes of the course unit
Learn advanced notions of complex analysis and geometry.
Deal with open problems.
Learn basic notions on complex algebraic manifolds. Understanding which complex manifold is isomorphic to an algebraic subvariety of some complex projective space.
Prerequisites
Holomorphic functions of one and several complex variables.
Dolbeault cohomology.
Holomorphic functions of one complex variable. Complex manifolds. Hodge theory on Kaehler manifolds.
Course contents summary
Holomorphic functions of several variables. Kobayashi and Caratheodory metrics. Algebras of holomorphic functions. Extension problems and the boundary problem.
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.
Course contents
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).
Recommended readings
Della Sala, Saracco, Simioniuc, Tomassini: Lectures on complex and analytic geometry, Edizioni della Normale 2006.
Abate: Iteration theory of holomorphic maps on taut manifolds, Mediterranean Press 1989.
Saracco: Extension problems in complex and CR geometry, Edizioni della Normale 2008.
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
Teaching methods
Standard blackboard lectures.
Standard blackboard lectures.
Assessment methods and criteria
Final exam will be an expository talk on a subject assigned by the theacher.
Homeworks during the course. Final exam will be an expository talk on a subject assigned by the theacher.
