Picard-Einstein Metrics and Class Fields Connected with Apollonius Cycle

Abstract

We define Picard-Einstein metrics on complex algebraic surfaces as Kähler-Einstein metrics with negative constant sectional curvature pushed down from the unit ball via Picard modular groups allowing degenerations along cycles. We demonstrate how the tool of orbital heights, especially the Proportionality Theorem presented in [H98], works for detecting such orbital cycles on the projective plane. The simplest cycle we found on this way is supported by a quadric and three tangent lines (Apollonius configuration). We give a complete proof for the fact that it belongs to the congruence subgroup of level 1+i of the full Picard modular group of Gauß numbers together with precise octahedral- symmetric interpretation as moduli space of an explicit Shimura family of curves of genus 3. Proofs are based only on the Proportionality Theorem and classification results for hermitian lattices and algebraic surfaces.