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Schematic Harder-Narasimhan stratication

Gurjar Sudarshan
Thursday, June 6, 2013 - 14:30 to 16:00

For any at family of pure-dimensional coherent sheaves on a family 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 subsets, 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 property 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 generalized 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.

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