May 2006
Monthly Archive
Tue 16 May 2006
Posted by litlfred under
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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.
Tue 16 May 2006
Posted by litlfred under
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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.
Mon 15 May 2006
Posted by litlfred under
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Deformations of local systems and Eisenstein series.
A. Braverman, D. Gaitsgory
Let be a (smooth and complete) curve and a reductive group. In [BG]we introduced the object that we called “geometric Eisenstein series”. This isa perverse sheaf (or rather a complex of such) on the modulistack of principal -bundles on , which is attached to a localsystem on with respect to the torus , Langlands dual to theCartan subgroup . In loc. cit. we showed thatcorresponds to the -local system induced from , 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 that corresponds to theuniversal deformation of as a local system with respect to the Borelsubgroup ?
We prove, following a conjecture of Drinfeld, that the resulting perversesheaf if the classical, i.e., non-compactified Eisenstein series.
Thu 11 May 2006
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Algebraic Geometry over Lie Algebras.
Ilya Kazachkov
This is a survey paper on Alegbraic Geometry over Lie Algebras
Thu 4 May 2006
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Dimension, multiplicity, holonomic modules, and an analogue of the inequality of Bernstein for rings of differential operators in prime characteristic.
V. V. Bavula
Let be an {\em arbitrary} field of characteristic and be the ring of differential operators on a polynomial algebra in variables. A long anticipated {\em analogue of the inequality of Bernstein} isproved for the ring . 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} -modules. In contrast to the characteristic zerocase where the Geland-Kirillov dimension of a nonzero finitely generated -module can be {\em any natural} number from the interval ,in the prime characteristic, the
(new) dimension can be \underline{{\em any real}} number from theinterval .
Tue 2 May 2006
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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.