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# Wushi Goldring: Which geometric properties of zip-schemes are generated by groups?

Time: Wed 2022-03-09 13.15 - 14.15

Participating: Wushi Goldring (SU)

Abstract:

Building on the theory of G-Zips developed by Pink–Wedhorn–Ziegler (generalizing earlier work by Moonen, Wedhorn, Viehmann), we study the following special case of the very general question "How much of geometry is generated by groups?": Given a scheme X in characteristic p, a connected, reductive $$\mathbb{F}_p$$-group G, a cocharacter $$\mu$$ (over an algebraic closure) and a morphism $$\zeta:X \longrightarrow G-Zip^{\mu}$$ we are interested in understanding how much of the geometry of X is detected by group theoretic-properties of the pair $$(G,\mu)$$ and properties of the morphism $$\zeta$$ (e.g., smoothness). A key example of X's which admit smooth morphisms $$\zeta$$ is given by the special fibers of integral canonical models of Hodge-type Shimura varieties at a good prime p with hyperspecial level at p (and similarly for suitable moduli spaces of K3's). Whether or not X is related to a Shimura variety, it inherits from $$G-Zip^{\mu}$$automorphic vector bundles parameterized by certain dominant weights.

We will describe theorems and conjectures about how the following geometric properties of automorphic bundles on X are controlled by $$(G,\mu)$$ and $$\zeta$$:
(1) The cone of automorphic bundles which admit a nonzero global section,
(2) The positivity (e.g. nefness, ampleness) and corresponding cohomological vanishing properties of automorphic bundles.
(3) The tautological subring of the Chow ring generated by Chern classes of automorphic bundles

Based on a series of joint works with J.-S. Koskivirta and a joint work with Y. Brunebarbe, J.-S. Koskivirta and B. Stroh.

Note: The passcode was sent to the AG and NT mailing lists. If you're not on these lists and would like to attend, or are having trouble accessing the meeting, please email Wushi Goldring at wgoldring@math.su.se . To be added to the AG mailing list, please email Jonas Bergström at jonasb@math.su.se .