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May 2006


Higher and derived stacks: a global overview.

B. Toen

These are expended notes of my talk at the summer institute in algebraicgeometry (Seattle, July-August 2005), whose main purpose is to present a globaloverview on the theory of higher and derived stacks. This text is far frombeing exhaustive but is intended to cover a rather large part of the subject,starting from the motivations and the foundational material, passing throughsome examples and basic notions, and ending with some more recent developmentsand open questions.

On deformations of flag manifolds with b_2=1.

Norbert Kuhlmann

Each global deformation of a flag manifold F with Picardgroup one isbiholomorphic to F. The same result holds for del Pezzo manifolds of degree 5and dimension >2.

Deformations of local systems and Eisenstein series.

A. Braverman, D. Gaitsgory

Let X be a (smooth and complete) curve and G a reductive group. In [BG]we introduced the object that we called “geometric Eisenstein series”. This isa perverse sheaf Eis E (or rather a complex of such) on the modulistack Bun G (X) of principal G -bundles on X , which is attached to a localsystem E on X with respect to the torus T ¯ , Langlands dual to theCartan subgroup T G . In loc. cit. we showed that Eis E corresponds to the G ¯ -local system induced from E , in the sense ofthe geometric Langlands correspondence.

In the present paper we address the following question, suggested by V.Drinfeld: what is the perverse sheaf on Bun G (X) that corresponds to theuniversal deformation of E as a local system with respect to the Borelsubgroup B ¯ G ¯ ?

We prove, following a conjecture of Drinfeld, that the resulting perversesheaf if the classical, i.e., non-compactified Eisenstein series.

Algebraic Geometry over Lie Algebras.

Ilya Kazachkov

This is a survey paper on Alegbraic Geometry over Lie Algebras

Dimension, multiplicity, holonomic modules, and an analogue of the inequality of Bernstein for rings of differential operators in prime characteristic.

V. V. Bavula

Let K be an {\em arbitrary} field of characteristic p >0 and CD (P n) be the ring of differential operators on a polynomial algebra P n in n variables. A long anticipated {\em analogue of the inequality of Bernstein} isproved for the ring CD (P n) . On the way, analogues of the concepts of

(Gelfand-Kirillov) {\em dimension, multiplicity, holonomic modules} are foundin prime characteristic (giving answers to old questions of finding suchanalogs).An analogue of the {\em Quillen’s Lemma} is proved for simple {\emfinitely presented} CD (P n) -modules. In contrast to the characteristic zerocase where the Geland-Kirillov dimension of a nonzero finitely generated CD (P n) -module M can be {\em any natural} number from the interval [ n,2n] ,in the prime characteristic, the

(new) dimension Dim (M) can be \underline{{\em any real}} number from theinterval [ n,2n] .

Quasi-Holonomic Modules in Positive Characteristic.

Anatoly N. Kochubei

We study modules over the Carlitz ring, a counterpart of the Weyl algebra inanalysis over local fields of positive characteristic. It is shown that somebasic objects of function field arithmetic, like the Carlitz module, Thakur’shypergeometric polynomials, and analogs of binomial coefficients arising in thefunction field version of umbral calculus, generate quasi-holonomic modules.This class of modules is, in many respects, similar to the class of holonomicmodules in the characteristic zero theory.