2014/06/23 by Sasha Anan’in, Sasha Anan'in, Anan'in, Sasha
Mathematics · #17A99 #51M10 #57S30 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Holomorphic and Operator Theory #math.DG #math.GT #msc:17A99 #msc:51M10 #msc:57S30
paper · pdf · doi:10.48550/arxiv.1406.5985
27 pages
arxiv created 2014/06/23 · openalex publication_date 2014/06/23 · arxiv updated 2014/06/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 ≠2,3. 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.