The degree of irrationality of most abelian g-folds is at least 2g

Abstract: The degree of irrationality of a complex projective n-dimensional variety X is the minimal degree of a dominant rational map from X to n-dimensional projective space. It is a birational invariant that measures how far X is from being rational. Accordingly, one expects the computation of this invariant in general to be a difficult problem. Alzati and Pirola showed in 1993 that the degree of irrationality of any abelian g-fold is at least g+1 using inequalities on holomorphic length. Tokunaga and Yoshihara later proved that this bound is sharp for abelian surfaces and Yoshihara asked for examples of abelian surfaces with degree of irrationality at least 4. Recently, Chen and Chen-Stapleton showed that the degree of irrationality of any abelian surface is at most 4. In this work, I provide the first examples of abelian surfaces with degree of irrationality 4. In fact, I show that most abelian g-folds have degree of irrationality at least 2g. We will present the proof of the case g=2 and indicate how to obtain the result in general. The argument rests on Mumford's theorem on rational equivalence of zero-cycles on surfaces with p_g>0 along with (new?) results on integral Hodge classes on self-products of abelian varieties.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic