Complex hyperbolic equidistant loci

TL;DR

This paper explores the geometry of loci equidistant from points in complex hyperbolic space, revealing elliptic curve structures, and classifies certain algebraic structures over algebraically closed fields.

Contribution

It describes the structure of equidistant loci in complex hyperbolic geometry and classifies 3-dimensional algebras via geometric and automorphism data.

Findings

Bisectors containing equidistant curves form real elliptic curves.

Foci of bisectors constitute an isomorphic elliptic curve.

Classification of 3D algebras via zero divisor curves and automorphisms.

Abstract

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.