Oberwolfach-Seminar: Higher Dimensional Algebraic Geometry

October 12th - October 18th, 2008
Christopher Hacon, Salt Lake City
Sándor Kovács, Seattle

Extension theorems and the existence of flips
8 lectures, Christopher Hacon

The main goal of this series of lectures is to give a proof of the existence of flips which is the main ingredient in the inductive proof of the minimal model program for varieties of log general type.

We will begin by giving some background on the main features of the minimal model program explaining in particular what singularities are allowed, what are flips and divisorial contractions, how to run a minimal model program with scaling and how the existence of flips fits in to the inductive proof of the minimal model program for varieties of of log general type.

Next, we will discuss vanishing theorems and multiplier ideal sheaf techniques that lead to the proof of results concerning the extensions of log-pluricanonical forms from a divisor to the ambient variety. These results are a generalisation of Siu's celebrated theorem on the invariance of plurigenera for varieties of general type.

Using ideas of Shokurov and the above mentioned extension theorems, we will then prove that assuming that the minimal model program for varieties of log general type in dimension n-1 holds, then flips exist in dimension n.

Moduli of higher dimensional varieties
7 lectures, Sándor Kovács

In these lectures we will sketch the main ideas of the construction of moduli spaces of higher dimensional varieties. After a general overview of the classification of higher dimensional varieties and of moduli theory, we will review moduli problems in more detail and take a brief look at Hilbert schemes. We will then discuss the definition and the most important properties of moduli functors. Each new observation will lead us to reconsider our objectives and along the way we will have to accept that it is necessary to work with singular varieties. Because of this, the particular type of singularities that one needs to be able to deal with will be reviewed and then finally the moduli functors of higher dimensional canonically polarized varieties are defined in the form that is currently believed to be the “right” one.

During these lectures it will become clear how closely moduli theory is related to the minimal model program and how much it benefits from the recent advances achieved there. In particular, the connections to the parallel lecture series will be emphasized. It will be shown how the results mentioned in that lecture series influence the results mentioned in this one.

Recommended reading:
Deadline for applications:
September 1st, 2008

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. The Institute covers accommodation and food. Travel expenses cannot be reimbursed. Applications including

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel
Universität Kaiserslautern
Fachbereich Mathematik
Erwin Schrödingerstr.
67663 Kaiserslautern, Germany

Mathematisches Forschungsinstitut Oberwolfach   updated: May 30th, 2008