Bogomolov-65: talks and abstracts

September 1-4, 2011,
Laboratory of Algebraic Geometry,
Steklov Math Institute,

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Ekaterina Amerik (HSE and Universite Paris-Sud)
On iteration of algebraic points under a rational self-map

Let X be an algebraic variety and f: X ---> X a rational self-map, both defined over a number field K. One would like to compare the iterated orbits of "sufficiently general" algebraic points of X with those of sufficiently general complex points. In particular, as the first step in this direction, I shall prove that as soon as f is of infinite order, most of algebraic points on X are non-preperiodic.

Jaume Amoros (UPC, Barcelona)
Compact Kaehler and projective manifolds with holomorphic tangent vector fields

The birational classification of projective manifolds with holomorphic tangent vector fields was developed by F. Severi, R. Hall and D. Lieberman.

Adapting Bogomolov's technique for the study of varieties with trivial canonical bundle, one may use the Albanese mapping to obtain the biholomorphic classification of non-uniruled projective manifolds with holomorphic tangent vector fields.

These results may be extended to compact Kaehler manifolds, using small deformations of the complex structure. They show that the study of the dynamics of holomorphic vector fields in them reduces to the case of rational varieties.

Christian Boehning (University Hamburg)
Stable cohomology of alternating groups

The notion of stable cohomology of groups (and algebraic varieties) was introduced by Fedor Bogomolov in the nineties in the course of his birational anabelian geometry program, and the analysis of rationality properties of linear group quotients $V/G$; in fact the subring of unramified elements in the stable cohomology of $G$ provides obstructions to the stable rationality of $V/G$. Closely related notions have also been studied by Jean-Pierre Serre, and even earlier Grothendieck. In the talk we will discuss some methods for and problems with the effective computation of stable group cohomology, and then focus on the computation of the stable cohomology of the alternating groups $A_n$, which is recent joint work with Bogomolov.

Frederic Campana (Nancy Universite)
Birational stability of the cotangent bundle and generic semipositivity for orbifolds pairs

Smooth orbifold pairs $(X,\Delta)$ play a major role in the birational classification of projective manifolds, both in the LMMP and through the multiple fibres of fibre spaces.

For orbifold pairs, the usual geometric invariants of manifolds can be defined (cotangent sheaf, morphisms and birational maps in particular).

A major problem of birational classification consists in deriving positivity properties of the cotangent bundle from those of the canonical bundle. A central result in this direction is Miyaoka's generic positivity of the cotangent bundle for non-uniruled manifolds.

The aim of the talk is to extend this result to the orbifold context. The original proof of Miyaoka cannot however be adapted. Instead, a combination of Bogomolov-Mc Quillan and orbifold additivity theorem parmit to show that if $K_X+\Delta$ is pseudo-effective, then $\Omega^1(X,\Delta)$ is generically semi-positive. This is joint work with M. Paun.

Paolo Cascini (Imperial College, London)
Effective log geography

The Minimal Model Program is the attempt to classify projective varieties from a birational geometry point of view. Some of its main conjectures are equivalent to the finite generation of some adjoint rings, which is known to hold in the case of big boundaries. The aim of this talk is to present a new plan to obtain an effective version of some of these results. In particular, we will use Shokurov's log geography and its applications.

Jean-Louis Colliot-Thélène (Universite Paris-Sud)
Galois descent on the Brauer group of varieties (joint work with A. N. Skorobogatov)

For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions on k, e.g. over number fields, we give estimates for the order of this cokernel. We emphasise the role played by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves modulo numerical equivalence.

Dima Kaledin (Steklov Institute)
Cyclic K-theory

Cyclic K-theory is a variant of algebraic K-theory introduced by Goodwillie 25 years ago and more-or-less forgotten by now. I will try to convince the audience that cyclic K-theory is actually quite useful -- in particular, it can be effectively computed for varieties over a finite field.

Ludmil Katzarkov (University of Miami)
Wallcorssings and degenerations

We will establish connection between some categorical invariants.

Michael McQuillan (Universita degli Studi di Roma "Tor Vergata")

I'll explain how to solve the equation: $$f'(z)+f(z)=g(z)$$ in holomorphic functions.

Tony Pantev (University of Pennsylvania)
Integral transforms in non-abelian Hodge theory

In order to construct integral transforms and Fourier-Mukai functors for variations of twistor structures one must have very strong functoriality properties of the non-abelian Hodge correspondence. I will discuss the problem of compatibility of non-abelian Hodge theory with Grothendieck's six operations and will report on a recent joint work with R.Donagi and C.Simpson. Our main result is an explicit algebraic formula which, in the tamely ramified case, captures the interaction of the Hodge correspondence with pushforwards.

Miles Reid (University of Warwick)
Rings and varieties

I leave the title and abstract as vague as possible, so that I can talk about whatever I feel like on the day. Many varieties of interest in the classification of varieties are obtained as Spec or Proj of a Gorenstein ring. In codimension <= 3, the well known structure theory provides explicit methods of calculating with Gorenstein rings. In contrast, there is no useable structure theory for rings of codimension >= 4. Nevertheless, in many cases, Gorenstein projection (and its inverse, Kustin-Miller unprojection) provide methods of attacking these rings. These methods apply to sporadic classes of canonical rings of regular algebraic surfaces, and to more systematic constructions of Q-Fano 3-folds, Sarkisov links between these, and the 3-folds flips of Type A of Mori theory.

For introductory tutorial material, see my website + surfaces + Graded rings and the associated homework.

For applications of Gorenstein unprojection, see "Graded rings and birational geometry" on my website + 3-folds, or the more recent paper

Gavin Brown, Michael Kerber and Miles Reid, Fano 3-folds in codimension 4, Tom and Jerry (unprojection constructions of Q-Fano 3-folds), Composition to appear, arXiv:1009.4313

Lucien Szpiro (City University of New York)
Discriminant, Conductor, Dynamical systems

We will look at effective solutions of Shafarevich problem (1956) from elliptic curves to dynamical systems.

Mina Teicher (Emmy Noether Institute)
On Topology of Line arrangements

In the talk I shall give some background on the question of topology vs combinatorics of line arrangements, the application to classification of algebraic surfaces and will report on two results.The first one is a roof of a conjecture by Fan - characterization of arrangements with no cycles and the second on conjugation-free geometric presentation of fundamental groups of arrangements

Fan proved that Line arrangements whose graph has no cycles then the fundamental group is a sum of free groups each in the order of one the multiple points minus 1 ( no of groups equal to number of multiple points) plus a commutative free to the power that will make the total group of base l minus 1 when l is the number of lines. We proved the opposite - from fundamental group to a property of the conjugation ( joint work with Eliyahu, Liberman and Schaps)

A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators and the cyclic relations: with no conjugations on the generators. We study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We actually show that a large family of these presentations satisfy a completeness property in the sense of Dehornoy. The completeness property is a powerful property which leads to many nice properties concerning the presentation (as the left-cancellativity in the associated monoid and yields some simple criterion for the solvability of the word problem in the group. ( joint work with Eliyahu, and Garber)

Gang Tian (Peking University and Princeton University)
The Chern number inequality for minimal varieties

In this talk, I will discuss the Chern number inequality on minimal varieties of general type which may have canonical singularities of complex codimension at least 3. This inequality extends the famous Bogomolov-Miyaoka-Yau inequality for smooth manifolds. I will also exam the case when the equality holds and show a new uniformization theorem for singular varieties. This is a joint work with B. Wang.

Yura Tschinkel (Courant Institute)
Balanced line bundles

I will explain the notion of "balanced" emerging in arithmetic geometry (joint work with B. Hassett and S. Tanimoto).

Misha Verbitsky (HSE)
Global Torelli theorem for hyperkahler manifolds

The global Torelli theorem for hyperkaehler manifolds is a statement about a diffeomorphism between the "birational moduli space", obtained by gluing together certain birationally equivalent points in the coarse moduli space, and a quotient of the period domain by the mapping class group. I will state the theorem and explain how one computes the mapping class group using the Sullivan's theory of rational homotopy.

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Laboratory of Algebraic Geometry and its Applications