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The Integral Hodge Conjecture and Universality of the Abel-Jacobi Maps

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posted on 01.05.2020, 00:00 by Fumiaki Suzuki
The rational Hodge conjecture states that rational Hodge classes are algebraic. This longstanding heavily studied conjecture has remained widely open since it was proposed in the nineteen fifties. In contrast, the integral Hodge conjecture is known to fail in general. To better understand the rational Hodge conjecture, it is important to ask how the integral Hodge conjecture can fail. In this thesis, we prove that there exists a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question posed by Murre on the universality of the Abel-Jacobi maps in codimension three.

History

Advisor

Ein, Lawrence

Chair

Ein, Lawrence

Department

Mathematics, Statistics and Computer Science

Degree Grantor

University of Illinois at Chicago

Degree Level

Doctoral

Degree name

PhD, Doctor of Philosophy

Committee Member

Coskun, Izzet Riedl, Eric Tucker, Kevin Zhang, Wenliang

Submitted date

May 2020

Thesis type

application/pdf

Language

en

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