Bogomolov-65: talks and abstracts
Home | Venue | Schedule | Program | Poster | PDF
Ekaterina Amerik (HSE and Universite Paris-Sud)
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.
On iteration of algebraic points under a rational self-map
Jaume Amoros (UPC, Barcelona)
The birational classification of projective manifolds with holomorphic tangent
vector fields was developed by F. Severi, R. Hall and D. Lieberman.
Compact Kaehler and projective manifolds with holomorphic tangent vector fields
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)
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
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.
Stable cohomology of alternating groups
Frederic Campana (Nancy Universite)
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.
Birational stability of the cotangent bundle and generic
semipositivity for orbifolds pairs
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
Paolo Cascini (Imperial College, London)
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.
Effective log geography
Jean-Louis Colliot-Thélène (Universite Paris-Sud)
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.
Galois descent on the Brauer group of varieties
(joint work with A. N. Skorobogatov)
Dima Kaledin (Steklov Institute)
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
Ludmil Katzarkov (University of Miami)
We will establish connection between some categorical invariants.
Wallcorssings and degenerations
Michael McQuillan (Universita degli Studi di Roma "Tor Vergata")
I'll explain how to solve the equation:
in holomorphic functions.
Tony Pantev (University of Pennsylvania)
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
Integral transforms in non-abelian Hodge theory
Miles Reid (University of Warwick)
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.
Rings and varieties
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)
We will look at effective solutions of Shafarevich problem (1956) from elliptic curves to dynamical systems.
Discriminant, Conductor, Dynamical systems
Mina Teicher (Emmy Noether Institute)
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
On Topology of Line 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)
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.
The Chern number inequality for minimal varieties
Yura Tschinkel (Courant Institute)
I will explain the notion of "balanced" emerging in
(joint work with B. Hassett and S. Tanimoto).
Balanced line bundles
Misha Verbitsky (HSE)
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.
Global Torelli theorem for hyperkahler manifolds
Home | Venue | Schedule | Program | Poster | PDF