Speakers

Provisional Schedule

Titles of the talks, schedule

Tuesday, February 16

Wednesday, February 17

Thursday, February 18

Friday, February 19

Saturday, February 20

Abstracts

Dan Abramovich: Varieties with a twist

I will discuss a number of questions and results where the geometry and moduli of algebraic stacks help in understanding the geometry and moduli of algebraic varieties. In particular: work with Hassett gives an approach to understanding aspects of the boundary of moduli of polarized varieties; and questions of Campana which would shed light on the birational geometry of varieties with quotient singularities.

Alexander Beilinson: On the structure of the space of rational functions on a curve

I will discuss a contractibility theorem conjectured by V. Drinfeld and proved recently by D. Gaitsgory.

Patrick Brosnan: Admissible normal functions and Hodge theory

Let H be a variation of pure Hodge structure of negative weight on a smooth complex algebraic variety S. Attached to H is a family J(H) over S of Griffiths intermediate Jacobians. Normal functions are certain sections of this family. The admissible normal functions are a subset of the normal functions formally introduced and studied by M. Saito in a 1996 Journal of Algebraic Geometry paper. Since normal functions are essentially analytic objects, it is clear that the zero locus of a normal function on S is an analytic subset of S. The main theorem of my talk will be a result recently obtained in joint work with Greg Pearlstein: that the zero locus is, in fact, algebraic. If time permits, I will also mention the independent approaches to this theorem by C. Schnell and by K. Kato, C. Nakayama and S. Usui.

Jean-Louis Colliot-Thélène: Dritte unverzweigte Kohomologie und ganzzahlige Hodge Vermutung

Ganzzahlige Kohomologie Klassen von Grad 4 auf einer komplexen, glatten, projektiven Mannigfaltigkeit, die von Hodge Typ (2,2) sind, brauchen nicht unbedingt Bilder von algebraischen Zyklen zu sein, wie bekannte Beispiele von Atiyah und Hirzebruch, und andere Beispiele von Koll'ar, zeigen. In einer gemeinsamen Arbeit mit Claire Voisin werden solche Beispiele unter unirationalen Varietäten gefunden. Dafür wird eine Beziehung zwischen diesem Problem und dem Studium der dritten unverzweigten Kohomologie mit endichen Koeffizienten festgestellt. Dann können alte Beispiele von Ojanguren und dem Sprecher einberufen sein.

Bas Edixhoven: Computational aspects of 2-dimensional Galois representations

A deterministic algorithm that computes the two-dimensional Galois representations associated to modular forms over finite fields, in time polynomial in the weight and in the cardinality of the field, will be described. Applications to computing Hecke operators T_p with p large, and to theta functions of lattices will be given. This is joint work with J-M. Couveignes, R. de Jong, F. Merkl, J. Bosman, and P. Bruin.

Barbara Fantechi: Twists instead of logs

Enumerative problems often require to compactify a moduli space while preserving the obstruction theory. E.g. Jun Li's proof of the degeneration formula for GW invariants uses a delicate analysis to do this (later simplified by Kim using log geometry). In this joint work with D. Abramovich we show how to use added stackiness (or twisting) instead. As an application, we extend the GW degeneration formula to orbifolds.

Thomas Geisser: Some remarks on Suslin homology

Suslin and Voevodsky proved that Suslin homology of a variety over an algebraically closed field, with coefficients prime to the characteristic of the field, is dual to etale cohomology. In the first half of the talk we discuss properties of Suslin homology with rational coefficients, and coefficients a power of the characteristic. In the second half we focus on finite base fields. We define a candidate for a well-behaved integral homology theory, which is related to tamely ramified class field theory (even for singular schemes).

Haruzo Hida: Characterization of abelian components of the 'big' Hecke algebra

In the first 20 minutes, we try to explain why such a characterization is important. Then, we state a (conjectural) example of such characterization, and at the end, we explain a proof of one of such statements.

Uwe Jannsen: Strong resolution of singularities for two-dimensional schemes

For varieties over fields of characteristic zero there are very strong results (by Hironaka and others) on resolution of singularities: a canonical and to a certain extent also functorial procedure by so-called permissible blow-ups. Over fields of positive characteristic or for schemes with mixed characteristic very little is known, and only in small dimension - more precisely in dimensions 2 and 3. Even in dimension 2, where one has several methods (e.g., a procedure by Lipman using alternately normalization and blow-ups in points), there were until recently no results on strong resolution as described above. I will report on joint work with V. Cossart and S. Saito, in which we fill this gap. We can even treat non-irreducible or non-reducible schemes, as well as 'boundaries' on them.

Minhyong Kim: Diophantine Geometry and Galois Theory

In his manuscripts from the 1980's Grothendieck proposed ideas that have been interpreted variously as embedding the theory of schemes into either -group theory and higher-dimensional generalizations ; -or homotopy theory. It was suggested, moreover, that such a framework would have profound implications for the study of Diophantine problems. In this talk, we will discuss mostly the little bit of progress made on this last point using some mildly non-abelian motives associated to hyperbolic curves.

Jürg Kramer: An arithmetic Riemann-Roch theorem for log-singular metrics on modular curves

In our joint work with J. Burgos and U. Kühn we generalized arithmetic intersection theory originally introduced by H. Gillet and C. Soule to the extent that hermitian vector bundles equipped with logarithmically singular metrics can be taken into account. This is of particular interest when applying methods of arithmetic intersection theory to automorphic vector bundles equipped with their natural invariant metric on Shimura varieties of non-compact type. The last key element missing in the development of our theory is the establishing of an arithmetic Riemann-Roch theorem. In order to state such a theorem the definition of a direct image of the hermitian vector bundles under consideration for proper morphisms between arithmetic varieties is needed. In our talk we will present joint work with T. Hahn in which first results in this direction have been established for the Hodge bundle on modular curves equipped with the hyperbolic metric. Using different techniques a similar result has been obtained by G. Freixas.

Robert Lazarsfeld: Positivity of cycles on abelian varieties

The cones of divisors and curves defined by various positivity conditions on a smooth projective variety have been the subject of a great deal of work in algebraic geometry, and by now they are quite well understood. However the analogous cones for cycles of higher codimension have started to come into focus only recently. I will discuss a couple of computations on abelian varieties where one can work out the picture fairly completely -- already here one sees some non-classical phenomena. I will also discuss at some length some of the many open problems that present themselves around this circle of ideas. (This is joint work in progress with Olivier Debarre, Lawrence Ein and Claire Voisin.)

Takuro Mochizuki: Betti structure of holonomic D-modules

We would like to give a report on the rather recent progress in the study of holonomic D-modules. Then, we will describe an attempt to define an appropriate notion of ``Betti structure'' for holonomic D-modules with nice functoriality.

Johannes Nicaise: A proof of the motivic monodromy conjecture for abelian varieties

We formulate a global form of Denef and Loeser's motivic monodromy conjecture, and we prove it for tamely ramified abelian varieties A over a discretely valued field. More precisely, we show that the motivic zeta function of A has a unique pole, which coincides with Chai's base change conductor c(A), and that this pole corresponds to a monodromy eigenvalue on the tame ell-adic cohomology of A. This is joint work with Lars Halvard Halle (Hannover).

Otmar Venjakob: On the noncommutative Iwasawa Main Conjecture for CM-elliptic curves

We discuss under which assumptions the (commutative) 2-variable Main Conjecture for CM-elliptic curves (due to Rubin, Yager, Katz etc.) implies the non-commutative Main Conjecture as formulated together with Coates, Fukaya, Kato and Sujatha. This is joint work with Thanasis Bouganis.

Angelo Vistoli: The essential dimension of a curve

Let k be a base field of characteristic 0. The essential dimension of a projective curve C over an extension K of k is the largest transcendence degree over k of a field of definition of C. Fix a genus g; a reasonable question to ask is: what is the least upper bound for the essential dimension of a smoth curve C of genus g over some extension of k? In other words, how many parameters are needed to write a smooth curve of genus g? The answer to this question is known, due to work of Brosnan, Reichstein and myself. I will discuss this result, and some recent extensions of it to singular curves.