Algebraic Geometry Seminar
Wern Yeen Yeong
Notre Dame
Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces
Abstract: A complex algebraic variety is said to be hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. Demailly introduced algebraic hyperbolicity as an algebraic version of this property, and it has since been well-studied as a means for understanding Kobayashi’s conjecture, which says that a generic hypersurface in projective space is hyperbolic whenever its degree is large enough. In this talk, we study the algebraic hyperbolicity of very general hypersurfaces of high bi-degrees in Pm x Pn and completely classify them by their bi-degrees, except for a few cases in P3 x P1. We present three techniques to do that, which build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As another application of these techniques, we improve the known result that very general hypersurfaces in Pn of degree at least 2n − 2 are algebraically hyperbolic when n is at least 6 to when n is at least 5, leaving n = 4 as the only open case.
Monday April 4, 2022 at 3:00 PM in Zoom