Talk by Dr. G. Sudarshan

[Ugo Bruzzo]

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SISSA GEOMETRY SEMINARS
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Dr. Gurjar Sudarshan (ICTP) wiill give a talk entitled

Schematic Harder-Narasimhan stratication 

on Thursday, June 6th, at 2:30 pm in Room 136, SISSA
Building A.

Abstract:
For any at family of pure-dimensional coherent sheaves on a fam-
ily of projective schemes, the Harder-Narasimhan type (in the sense
of Gieseker semistability) of its restriction to each ber is known to
vary semicontinuously on the parameter scheme of the family. This
denes a stratication of the parameter scheme by locally closed sub-
sets, known as the Harder-Narasimhan stratication. In a paper of
Nitin Nitsure written few years ago, he showed how to endow each
Harder-Narasimhan stratum with the structure of a locally closed
subscheme of the parameter scheme, which enjoys the universal prop-
erty that under any base change the pullback family admits a relative
Harder-Narasimhan ltration with a given Harder-Narasimhan type
if and only if the base change factors through the schematic stratum
corresponding to that Harder-Narasimhan type. This was then gen-
eralized by Nitsure and myself to the case of principal bundles with
a reductive algebraic group as structure group over a smooth family
of curves parametrized by a noetherian k-scheme of characteristic 0.
The above stratications have the consequence that coherent sheaves
(resp. principal bundles) with a xed Harder-Narasimhan type ( form
an algebraic stack in the sense of Artin. I will explain these results
and give an idea of the proofs.

U. Bruzzo