Algebraic Hyperbolicity of Very General Hypersurfaces and Their Complements

A complex algebraic variety is said to be (Brody) hyperbolic if it does not admit any non-constant holomorphic maps from the complex line, which are called entire curves. Conjectures by Kobayashi, Green, Griffiths, Lang, and others predict that the hyperbolicity of a variety is controlled by the positivity of its canonical bundle. More precisely, this circle of conjectures predicts that a variety of general type is hyperbolic outside of a proper subvariety called an exceptional locus, which may be empty if its canonical bundle has enough positivity. In this dissertation, we make several advances to the study of these conjectures with respect to the notion of algebraic hyperbolicity as defined by Demailly.

We obtain a complete classification of very general hypersurfaces in Pm x Pn by their bidegrees, except in the case of threefolds in P3 x P1. We present three techniques to do so, which build on past work by Ein, Voisin, Pacienza, Coskun, Riedl, and others. As another application of these techniques, we solve the penultimate case of degree 8 fourfolds in the widely-studied question of the algebraic hyperbolicity of very general hypersurfaces in Pn, leaving sextic threefolds as the only open case.

In a collaboration with X. Chen and E. Riedl, we develop some of the above techniques to the setting of quasi-projective varieties. We prove that curves in the complement of a very general hypersurface in Pn of degree 2n that may possibly violate algebraic hyperbolicity lift to the universal line via the associated line map. Assuming an additional hypothesis on the associated line map, we show that algebraic hyperbolicity holds for such varieties outside of a proper exceptional locus when n is at least 2. Moreover, for the complement of a very general quartic plane curve, we show that the exceptional locus is precisely the union of the bitangent and flex lines to the curve, given this additional hypothesis.