**Date:**- October 12th - October 18th, 2008
**Organizers:**- Christopher Hacon, Salt Lake City
- Sándor Kovács, Seattle
**Programme:**-
**Extension theorems and the existence of flips**

8 lectures, Christopher HaconThe 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 KovácsIn 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.

**Prerequisites:**-
- Hartshorne, Robin, Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp. ISBN: 0-387-90244-9
- Beauville, Arnaud, Complex algebraic surfaces. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. Second edition. London Mathematical Society Student Texts, 34. Cambridge University Press, Cambridge, 1996. x+132 pp. ISBN: 0-521-49510-5; 0-521-49842-2

**Recommended reading:**-
- Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties. (English summary) With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998. viii+254 pp. ISBN: 0-521-63277-3
- Lazarsfeld, Robert Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 49. Springer-Verlag, Berlin, 2004. xviii+385 pp. ISBN: 3-540-22534-X

**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
- full name and address, including e-mail address
- present position, university
- name of supervisor of Ph.D. thesis
- a short summary of previous work and interest

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