Caucher Birkar's class
Caucher Birkar, laureate in 2010 of the Foundation's Prize, is going to give a course entitled Lectures on birational geometry the mondays and wednesdays from January 31th 2011, from 11 a.m. to 12:30 p.m. at the Collège de France, 3 rue d'Ulm, 75005 Paris.
Click here to read the notes of Lectures on birational geometry.
Videos of the lectures
Monday 31th of January 2011 : General overview, birational geometry of curves and surfaces
Wednesday 2nd of February2011 : The Neron-Severi space and numerical properties of divisors
Monday 7th of February 2011 : Kodaira dimension
Wednesday 9th of February 2011 : The minimal model program and its main conjectures
Wednesday 9th of March 2011 : Global generation of centain line bundles: the base point free
Monday 14th of March 2011 : Shokurov nonvanishing
Wednesday 23rd of march 2011 : Extension theorems and flips II
Monday 28th of march 2011 : From nonvanishing to minimal models
Wednesday 30th of march 2011 : Finiteness of minimal models
Wednesday 27th of april 2011 from 11 a.m. to 12:30 p.m.
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.
Familiarity with basic algebraic geometry is assumed (ideally on he level of Hartshorne's book).