Intrinsic construction of moduli spaces via affine grassmannians

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Arithmetic Geometry Seminar
January 18, 2022
3:00PM - 4:00PM
Location
UH 74

Date Range
Add to Calendar 2022-01-18 15:00:00 2022-01-18 16:00:00 Intrinsic construction of moduli spaces via affine grassmannians Speaker:  Andres Fernandes Herrero (Cornell)   Title:  Intrinsic construction of moduli spaces via affine grassmannians   Abstract:  For a projective variety X, the moduli problem of coherent sheaves on X is naturally parametrized by a geometric object M called an "algebraic stack". In this talk I will explain a GIT-free construction of the moduli space of Gieseker semistable pure sheaves which is intrinsic to the moduli stack M. This approach also yields a Harder-Narasimhan stratification of the unstable locus of the stack. Our main technical tools are the theory of Theta-stability introduced by Halpern-Leistner, and some recent techniques developed by Alper, Halpern-Leistner and Heinloth. In order to apply these results, one needs to prove some monotonicity conditions for a polynomial numerical invariant on the stack. We show monotonicity by defining a higher dimensional analogue of the affine grassmannian for pure sheaves. If time allows, I will also explain some applications of these ideas to some other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones. UH 74 Department of Mathematics math@osu.edu America/New_York public
Description
Speaker:  Andres Fernandes Herrero (Cornell)
 
Title:  Intrinsic construction of moduli spaces via affine grassmannians
 
Abstract:  For a projective variety X, the moduli problem of coherent sheaves on X is naturally parametrized by a geometric object M called an "algebraic stack". In this talk I will explain a GIT-free construction of the moduli space of Gieseker semistable pure sheaves which is intrinsic to the moduli stack M. This approach also yields a Harder-Narasimhan stratification of the unstable locus of the stack. Our main technical tools are the theory of Theta-stability introduced by Halpern-Leistner, and some recent techniques developed by Alper, Halpern-Leistner and Heinloth. In order to apply these results, one needs to prove some monotonicity conditions for a polynomial numerical invariant on the stack. We show monotonicity by defining a higher dimensional analogue of the affine grassmannian for pure sheaves. If time allows, I will also explain some applications of these ideas to some other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones.

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