Lectures on birational geometry

Le cours de Caucher Birkar, lauréat 2010 du Prix de la Fondation, s'est tenu les lundis et les mercredis du 31 janvier au 27 avril 2011 au Collège de France. Il s'intitulait Lectures on birational geometry et se tiendra de 11h à 12h30.

Notes de cours

Cliquez ici pour télécharger les notes du cours Lectures on birational geometry.

Vidéos des séances

Vidéos

Vidéo par séance

Lundi 31 janvier 2011 General overview, birational geometry of curves and surfaces
Mercredi 2 février 2011 The Neron‑Severi space and numerical properties of divisors
Lundi 7 février 2011 Kodaira dimension
Mercredi 9 février 2011 The minimal model program and its main conjectures
Lundi 28 février 2011  
Mercredi 2 mars 2011  
Lundi 7 mars 2011  
Mercredi 9 mars 2011 Global generation of centain line bundles: the base point free
Lundi 14 mars 2011 Shokurov nonvanishing
Mercredi 16 mars 2011 Existence of flips
Lundi 21 mars 2011 Extension theorems and flips
Mercredi 23 mars 2011 Extension theorems and flips II
Lundi 28 mars 2011 From nonvanishing to minimal models
Mercredi 30 mars 2011 Finiteness of minimal models
Lundi 4 avril 2011  
Mercredi 6 avril 2011  
Mercredi 27 avril 2011 de 11:00 à 12:30  

Résumé du cours

Birational geometry, that is, the classication of algebraic varieties up to birational equivalence occupies a central position in algebraic geometry. The machinery for carrying out birational geometry is the so‑called minimal model program (MMP for short). The MMP is a specific process designed to identify nice elements in a birational class and then to classify them. These nice elements are of two very different types: minimal models and Mori fibre spaces. Minimal models have a certain positivity property and Mori fibre spaces have a certain negativity property. Curves of genus at least one and Calabi‑Yau varieties are examples of the first type but a Fano variety is an example of the second type. Some of the general ideas for finding minimal models and Mori fibre spaces go back to the early 20th century but it was in the 1970's and afterward that modern points of view emerged which accumulated in the proof of some of the main conjectures of the field in recent years.

Birational geometry is primarily concerned with smooth projective varieties (over the complex numbers for much of this course). However, to apply the MMP we first have to deal with singularities as they appear in the study of extremal rays and contractions. Second, we need to treat flipping contractions and prove that their corresponding flips exist. Third, we have the problem of proving that sequences of flips terminate. Fourth comes the study of pluricanonical systems on minimal models and the abundance problem. In addition, there are several other problems concerning the behavior of singularities and Fano varieties.

In this course, by starting from scratch I will try to give a detailed account of many of the main tools, methods, and fundamental results in the subject including the progress made in recent years, and of what remains to be solved. Here is a list of some of the topics that I hope to cover in the course: birational geometry of curves and surfaces, rational curves, adjunction, pairs and the various classes of singularities, Kodaira‑type vanishing theorems, extension theorems, extremal rays, minimal models, Mori fibre spaces, nonvanishing, base point freeness, the cone and contraction, flips, termination, MMP, Shokurov algebras, finite generation of canonical rings, ACC conjectures on singularities, Sarkisov program, rationally connected varieties, Iitaka conjecture, and abundance.

In algebraic geometry, one usually has to learn a lot of material before feeling a sense of direction. A pedagogical advantage of birational geometry is that one can see advanced techniques of algebraic geometry applied to concrete geometric problems.

Prérequis

Familiarity with basic algebraic geometry is assumed (ideally on the level of Hartshorne's book).

Problèmes inverses hybrides
Stochastic homogenization of Hamilton-Jacobi equat...

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