## Geometry Seminar - Abstracts

### Talk

Tuesday 13 December 2022, 16:00 - 17:00 in HG00.308

**Ariyan Javanpeykar** (RU)

*Finiteness properties of hyperbolic varieties*

### Abstract

I will talk about finiteness properties of hyperbolic varieties (some
old ones, some new ones, and some we expect to be true, but can't prove yet).
Our starting point is the theorem of de Franchis: given a variety \(Y\) and a
hyperbolic Riemann surface \(C\), the set of non-constant maps from \(Y\) to \(C\) is
finite. The main question I'd like us to ask is simply "to what extent does
this finiteness statement hold for higher-dimensional hyperbolic targets"? It
obviously fails for surfaces (take \((C \times C)\), but it surprisingly holds for the
(orbifold) moduli space of compact hyperbolic Riemann surfaces of genus \(g\) (\(g>
1\)). I will propose an (in my honest opinion) reasonable conjecture for *all*
hyperbolic varieties. This finiteness conjecture is proven in joint work with
Steven Lu, Ruiran Sun and Kang Zuo for moduli spaces of polarized varieties.

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