Non-compact Torsion Free Ball Quotients

Details Non-compact Torsion Free Ball Quotients Non-compact Torsion Free Ball Quotients

2011-12-30

arxiv.org/abs/1201.0096v1

Mathematics

Algebraic Geometry

Scientific paper

The present note specifies the Kodaira-Enriques classification type of a non-compact torsion free ball quotient ${\mathbb B} / \Gamma$. It turns out that ${\mathbb B} / \Gamma$ can be birational to a non-simple abelian surface, to an elliptic fibration of Kodaira dimension 1 with base of genus $\leq 1$ or to a minimal surface of general type. The argument makes use of the embedding of the fundamental group $\pi_1(T_i)$ of a smooth elliptic irreducible component $T_i$ of the toroidal compactifying divisor $T= ({\mathbb B} / \Gamma)' \setminus ({\mathbb B} / \Gamma)$ in the fundamental group $\pi_1 (X')$ of the toroidal compactification $X'= ({\mathbb B} / \Gamma)$. The note elaborates also on various consequences of the Kobayashi non-hyperbolicity of $X'$ and the Kobayashi hyperbolicity of $ {\mathbb B} / \Gamma$.

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