Positivity in varieties of maximal Albanese dimension

Abstract

Abstract Given a generically finite morphism f from a smooth projective variety X to an abelian variety A, we show that f * ⁢ ω X {f_{*}\omega_{X}} is “sufficiently positive” on A. As an application, we prove that when X is of general type, the global sections of ω X 2 {\omega_{X}^{2}} define a generically finite map of X. We also study the structure of X when X is of general type and satisfies χ ⁢ ( X , ω X ) = 0 {\chi(X,\omega_{X})=0} . We formulate a conjectural characterization of such X and prove the conjecture when A has exactly three simple factors.