000002295 001__ 2295
000002295 005__ 20240523045515.0
000002295 0247_ $$2doi$$a10.6082/uchicago.2295
000002295 041__ $$aen
000002295 245__ $$aZero-Cycles and Measures of Irrationality for Abelian Varieties
000002295 260__ $$bThe University of Chicago
000002295 269__ $$a2020-06
000002295 300__ $$a131
000002295 336__ $$aDissertation
000002295 502__ $$bPh.D.
000002295 520__ $$aIn 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.
000002295 542__ $$fCC BY-NC-ND
000002295 650__ $$aMathematics
000002295 653__ $$aAbelian varieties
000002295 653__ $$aAlgebraic cycles
000002295 653__ $$aBirational geometry
000002295 653__ $$aMeasures of irrationality
000002295 690__ $$aPhysical Sciences Division
000002295 691__ $$aMathematics
000002295 7001_ $$aMartin, Olivier$$uUniversity of Chicago
000002295 72012 $$aAlexander A. Beilinson
000002295 72012 $$aMadhav V. Nori
000002295 72014 $$aAlexander A. Beilinson
000002295 72014 $$aMadhav V. Nori
000002295 8564_ $$9718f51c3-ed66-43d3-bc83-848d8ee89c8f$$ePublic$$s565865$$uhttps://knowledge.uchicago.edu/record/2295/files/Martin_uchicago_0330D_15254.pdf
000002295 909CO $$ooai:uchicago.tind.io:2295$$pDissertations$$pGLOBAL_SET
000002295 983__ $$aDissertation