2005-11-07Buch DOI: 10.18452/2647
Quadric-Line Configurations Degenerating Plane Picard Einstein Metrics I-II
We define Picard-Einstein metrics on complex algebraic surfaces as Kähler-Einstein metrics with negative constant sectional curvature pushed down from the complex unit ball 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) with at most 3 cusp points sitting on the double points of the configuration. We determine precisely the uniformizing ball lattices in the case of 3, 2, 1 or 0 cusp(s) respectively. The corresponding orbital planes are (leveled) Shimura surfaces corresponding to Jacobian varieties of certain families of plane genus 3, 6, 5 or 13 genus respectively. We present many examples of plane orbital surfaces with quadrics, and determine for them precisely the uniformizing ball lattices. By the way we check that %precisely these two cases are some of them are Galois quotients of celebrated 27 orbital planes with line arrangements occurring in the PTDM-list (Picard-Terada-Mostow-Deligne) which we will call also BHH-list (Barthel-Hirzebruch-Höfer) because it is most convenient to get it from [BHH]. The others are quotients of Mostow's [M2] half-integral arrangements. Proofs are based on the Proportionality Theorem and classification results for hermitian lattices and algebraic surfaces.
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