Complex hyperbolic orbifolds and hybrid lattices

A class of complex hyperbolic lattices in PU (2, 1) called the Deligne-Mostow lattices has been reinterpreted by Hirzebruch (see [ 1 , 10 ] and [ 24 ]) in terms of line arrangements. They use branched covers over a suitable blow up of the complete quadrilateral arrangement of lines in P 2 to constru...

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Bibliographic Details
Published in:Geometriae dedicata Vol. 217; no. 2; p. 28
Main Authors: Falbel, Elisha, Pasquinelli, Irene
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.04.2023
Springer Nature B.V
Subjects:
Algebraic Geometry
Construction
Convex and Discrete Geometry
Differential Geometry
Hyperbolic Geometry
Inequality
Lattices
Mathematics
Mathematics and Statistics
Original Paper
Projective Geometry
Quadrilaterals
Symmetry
Topology
ISSN:0046-5755, 1572-9168
Online Access:Get full text
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Summary:A class of complex hyperbolic lattices in PU (2, 1) called the Deligne-Mostow lattices has been reinterpreted by Hirzebruch (see [ 1 , 10 ] and [ 24 ]) in terms of line arrangements. They use branched covers over a suitable blow up of the complete quadrilateral arrangement of lines in P 2 to construct the complex hyperbolic surfaces over the orbifolds associated to the lattices. In [ 18 ] and [ 19 ], fundamental domains for these lattices have been built by Pasquinelli. Here we show how the fundamental domains can be interpreted in terms of line arrangements as above. This parallel is then applied in the following context. Wells in [ 25 ] shows that two of the Deligne-Mostow lattices in PU (2, 1) can be seen as hybrids of lattices in PU (1, 1). Here we show that he implicitly uses the line arrangement and we complete his analysis to all possible pairs of lines. In this way, we show that three more Deligne-Mostow lattices can be given as hybrids.
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ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-022-00762-y