Torsion free ball quotients with abelian birational model

Details Torsion free ball quotients with abelian birational model Torsion free ball quotients with abelian birational model

2011-12-30

arxiv.org/abs/1201.0099v1

Mathematics

Algebraic Geometry

Scientific paper

The present note studies the torsion free toroidal compactifications $({\mathbb B} / \Gamma)'$ of ball quotients ${\mathbb B}/ \Gamma$, which are birational to the Cartesian square $E \times E$ of an elliptic curve $E$. The first section is devoted to the elliptic curves $D_i$ on $E \times E$ and the foliations $\Phi_{D_i} : E \times E \rightarrow D'_i$, whose leaves are parallel to $D_i$. Let $\mathcal{O}_{-d}$ be the integers ring of an imaginary quadratic number field ${\mathbb Q} (\sqrt{-d})$ and $E_{-d} = {\mathbb C} / \mathcal{O}_{-d}$. The blow-down $\xi : ({\mathbb B} / \Gamma_{-d} ^{(s,h)})' \rightarrow E_{-d} \times E_{-d}$ of the $s$ smooth rational $(-1)$-curves to the minimal model $E_{-d} \times E_{-d}$ transforms isomorphically the smooth elliptic irreducible components $T_i$ of the compactifying divisor of $({\mathbb B} / \Gamma_{-d} ^{(s,h)})'$ into elliptic curves $D_i \subset E_{-d} \times E_{-d}$. We restrict to $({\mathbb B} / \Gamma_{-d} ^{(s,h)})'$ with minimal fundamental groups $\pi_1 (T_i) = \pi_1 (D_i) = \pi_1 (E)$. For any natural number $s$, relatively prime to 3, the second section constructs a smooth toroidal compactification $({\mathbb B} / \Gamma_{-3}^{(s,4)})'$ with minimal model $E_{-3} \times E_{-3}$. After some technicalities, collected in the third section, the last two sections establish that any $({\mathbb B} / \Gamma_{-d} ^{(s,h)})'$ with $1 \leq s \leq 3$ is GL(2,\mathcal{O}_{-d})$-isomorphic to Hirzebruxh's example $({\mathbb B} / \Gamma_{-3} ^{(1,4)})'$, to Holzapfel's examples $({\mathbb B} / \Gamma_{-3} ^{(3,6)})'$, $({\mathbb B} / \Gamma_{-1} ^{(3,6)})'$ or to the complex conjugate of $({\mathbb B} / \Gamma_{-1} ^{(3,6)})'$.

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