Positivities of varieties with maximal Albanese dimension

Given a variety of maximal Albanese dimension, there is a generically finite morphism to its Albanese variety. It is known that Euler characteristic is non-negative. It is tempting to characterize varieties with vanishing Euler characteristic.
Inspired by the Fourier–Mukai transform, one can have some weaker notion of positivity of sheaves, such as M-regular or GV. By some study of the push-forward of canonical sheaf, one can encode various geometric properties of the variety in question by the cohomological property of the canonical sheaf. In recent joint work with Debarre and Jiang, we obtained some characterization, for example in dimension three. We will explain these work in this talk.