Zero-Cycles and Measures of Irrationality for Abelian Varieties

In this thesis we make several advances in the study of the birational geometry of complex abelian varieties. We are mainly concerned with two birational invariants: the degree of irrationality and the covering gonality. The degree of irrationality of a projective variety $X/k$ is the minimal degree of a dominant rational map $\varphi: X\dashrightarrow \mathbb{P}^{\dim X}$. The covering gonality of $X$ is the minimal $g\in \mathbb{Z}_{\geq 0}$ such that $X$ is birationally covered by a family of $g$-gonal curves, or equivalently such that a generic $x\in X$ is contained in a $g$-gonal curve. The degree of irrationality and the covering gonality measure respectively the failure of $X$ to be rational and uniruled and are thus called measures of irrationality. By studying rational equivalence of zero-cycles on abelian varieties, this work contributes new lower bounds on the covering gonality and the degree of irrationality of very general abelian varieties. In Chapter 2, we provide the reader with preliminary background on algebraic cycles and measures of irrationality. In Chapter 3, we present our first contribution: a proof of a conjecture of Voisin on the covering gonality of very general abelian varieties. In fact, in collaboration with E. Colombo, J. C. Naranjo, and G. P. Pirola, we prove an extension of this result which provides lower bounds on the degree of irrationality of subvarieties of very general abelian varieties. In Chapter 4, we derive a cohomological obstruction to the existence of low degree dominant rational maps from an abelian $g$-fold to a $g$-fold admitting a cohomological decomposition of the diagonal. In particular, using this obstruction we show that the degree of irrationality of a $(1,d)$-polarized abelian surface with Picard number one is $4$ if $d$ does not divide $6$. This theorem and its generalization settle a conjecture of Chen, answer questions of Yoshihara, and provide the best known lower bound on the degree of irrationality of very general abelian varieties of large degree in all dimensions. Finally, in the Appendix C we present some new identities for zero-cycles on abelian varieties