Shafarevich's conjecture for canonically polarized varieties revisited

Ariyan Javanpeykar (Johannes Gutenberg University of Mainz)

05-May-2022, 10:00-11:00 (23 months ago)

Abstract: Abstract: Arakelov-Parshin proved Shafarevich's conjecture for the (actual) moduli space M_g of curves of genus g (g>1). Namely, for every curve C, the set of isomorphism classes of non-isotrivial morphisms C \to M_g is finite. This finiteness result is similar to the theorem of De Franchis: for every hyperbolic curve X, the set of non-constant morphisms C->X is finite. Interestingly, M_g is hyperbolic (in any reasonable sense of the word "hyperbolic"). Is maybe the theorem of Arakelov-Parshin and De Franchis an instance of a more general finiteness property for hyperbolic varieties/stacks? The answer is no. The conclusion of the theorem of De Franchis fails for hyperbolic surfaces (such as X x X) and the conclusion of Arakelov-Parshin's theorem fails for the moduli space of canonically polarized surfaces (because it contains copies of X x X). How to remedy this? Guided by finiteness properties of compact hyperbolic varieties, we establish new finiteness properties for the moduli space CanPol of canonically polarized varieties by proving rigidity properties of pointed maps. Applications include bounds on dimensions of moduli spaces of maps to CanPol. Joint work with Steven Lu, Ruiran Sun, and Kang Zuo.

algebraic geometry

Audience: researchers in the topic


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