Special subvarieties of non-arithmetic ball quotients and Hodge Theory

2020/05/07 by Baldi, Gregorio, Ullmo, Emmanuel · 3 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2005.03524

Abstract

Let Γ⊂ PU(1,n) be a lattice, and S_Γ the associated ball quotient. We prove that, if S_Γ contains infinitely many maximal totally geodesic subvarieties, then Γ is arithmetic. We also prove an Ax-Schanuel Conjecture for S_Γ, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise S_Γ inside a period domain for polarised integral variations of Hodge structures and interpret totally geodesic subvarieties as unlikely intersections.

Cited by

• Properness and finiteness of totally geodesic submanifolds in the convex core
• Cohomological nonvanishing for algebraic fundamental groups of ball quotients
• Real geometric transcendence for uniformization maps of algebraic Riemann surfaces

Related

• The Ax-Schanuel conjecture for variations of Hodge structures
• Arithmeticity, Superrigidity, and Totally Geodesic Submanifolds
• Tame topology of arithmetic quotients and algebraicity of Hodge loci
• The Hodge conjecture and arithmetic quotients of complex balls
• Moduli spaces as ball quotients I, local theory