Non-hyperbolic complex space with a hyperbolic normalization

Abstract

We construct an example of a non-hyperbolic singular projective surface X whose normalization V is the square of a genus 3 C and hence, hyperbolic. Let C be a smooth irreducible projective of genus g > 2. Then the smooth projective surface V = C x C is Kobayashi hyperbolic, that is, the Kobayashi pseudodistance on it is a distance [2]. Let V ___ pN be a projective embedding. Consider a generic projection ir V __ p3. By Bertini's theorem, the singular locus S (i.e. the closure of the set of points) of the image surface X = ir(V) c is an irreducible curve, and ir: V -* X is a normalization map (see [4]). The question arises whether the surface X is also hyperbolic. The answer is positive [5], and hence by the stability of hyperbolicity theorem [6], any (smooth) surface X' in I3 close enough to X is hyperbolic, as well. In that way examples of degree 32 smooth hyperbolic surfaces in p3 were produced [5]. By Proposition 1.1 in [5], hyperbolicity of a (singular) surface X as above is equivalent to hyperbolicity of its S. Actually, in [5] it is shown that the geometric genus of the S is > 225, which provides that X is hyperbolic. On the other hand, by the Kobayashi-Kwack theorem [2, 3], a normalization of a hyperbolic complex space is also hyperbolic. In this note we give an example which shows that in general, the converse is not true. To describe this example, denote by C the Fermat quartic x4 + y4 + z4 0 O in P2. Then the Cartesian square V = C X C C Ip2 X Ip2 c-_ p8 (the Segre embedding) is a smooth surface of degree 32 in Ip8. We construct a singular projective surface X whose normalization is V, which has a fibration X -* C over C with general fibre isomorphic to C and with four degenerate fibres Ci, i = 1, . . ., 4, isomorphic to Pl1. The double curve S = Cl U ... U C4 C X of X is neither irreducible nor hyperbolic, in contrast with the situation studied in [5]. Thus, the assumption in [5] that the projection ir is generic, is likely to be essential to provide hyperbolicity of the image surface X = 7r(V). Actually, in our example the surface X does not appear as a projection of V = C x C; but it has a natural embedding into a four-dimensional Brauer-Severi variety Y (see [1]) which is a smooth projective fiber bundle over Ip2 with general fibre Ip2. Received by the editors July 30, 1999. 2000 Mathematics Subject Classification. Primary 32H15, 32H20. (?2000 American Mathematical Society