Complex hyperbolic equidistant loci
We describe and study the loci equidistant from finitely many points in the so-called complex hyperbolic geometry, i.e., in the geometry of a holomorphic $2$-ball $\Bbb B$. In particular, we show that the bisectors (= the loci equidistant from $2$ points) containing the (smooth real algebraic) curve...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
23.06.2014
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Subjects | |
Online Access | Get full text |
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Summary: | We describe and study the loci equidistant from finitely many points in the
so-called complex hyperbolic geometry, i.e., in the geometry of a holomorphic
$2$-ball $\Bbb B$. In particular, we show that the bisectors (= the loci
equidistant from $2$ points) containing the (smooth real algebraic) curve
equidistant from given $4$ generic points form a real elliptic curve and that
the foci of the mentioned bisectors constitute an isomorphic elliptic curve.
We are going to use the obtained facts in constructions of (compact)
quotients of $\Bbb B$ by discrete groups.
With similar technique, we also classify up to isotopy generic
$3$-dimensional algebras (i.e., bilinear operations) over an algebraically
closed field $\Bbb K$ of characteristic $\ne2,3$. Briefly speaking, an algebra
is classified by the (plane projective) curve $D$ of its zero divisors equipped
with a nonprojective automorphism of $D$. This classification is almost
equivalent to the classification of the so-called geometric tensors given in
[BoP] by A. Bondal and A. Polishchuk in their study of noncummutative
projective planes. |
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DOI: | 10.48550/arxiv.1406.5985 |