Dr. N. Vladov Prof. R.-P. Holzapfel
Quadric-Line Configurations Degenerating Plane Picard Einstein Metrics I-II
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

 11G15 Complex multiplication and moduli of abelian varieties, See also {14K22}

11G18 Arithmetic aspects of modular and Shimura varieties, See also {14G35}

ZDM: 11H56
PACS: 14D05
CR: 11R11
Abstract: 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.
Keywords: algebraic curves, moduli space, Shimura surface, Picard modular group,
monodromy group, arithmetic group, Kähler-Einstein metric, negative constant curvature,
hermitian form, unit ball