One-cusped complex hyperbolic 2-manifolds

2024/09/12 by Deraux, Martin, Stover, Matthew
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2409.08028

Abstract

This paper builds one-cusped complex hyperbolic 2-manifolds by an explicit geometric construction. Specifically, for each odd d ≥ 1 there is a smooth projective surface Z_d with c₁²(Z_d) = c₂(Z_d) = 6d and a smooth irreducible curve E_d on Z_d of genus one so that Z_d \smallsetminus E_d admits a finite volume uniformization by the unit ball \mathbbB² in ℂ². This produces one-cusped complex hyperbolic 2-manifolds of arbitrarily large volume. As a consequence, the 3-dimensional nilmanifold of Euler number 12d bounds geometrically for all odd d ≥ 1.

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