Wern Yeong, UCLA

Title: A hyperbolicity conjecture for adjoint bundles

Abstract: A complex manifold X is said to be Brody hyperbolic if it admits no entire curves, which are non-constant holomorphic maps from the complex numbers. When X is a smooth complex projective variety, Demailly introduced an algebraic analogue of this property known as algebraic hyperbolicity. We propose a conjecture on the algebraic hyperbolicity of generic sections of adjoint bundles on X, motivated by Fujita’s freeness conjecture and recent results by Bangere and Lacini on syzygies of adjoint bundles. We present some old and new evidence supporting this conjecture, including when X is any smooth projective toric variety or Gorenstein toric threefold. This is based on joint work with Joaquín Moraga.