Complex hyperbolic equidistant loci

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.