Algebraic Geometry Seminar

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Hodge Classes on Prym Varieties

Speaker: Yilong Zhang, Purdue University

Location: Warren Weaver Hall 512

Date: Tuesday, March 26, 2024, 3:30 p.m.

Synopsis:

 Let C->X be a cyclic cover between smooth curves. There is an abelian subvariety of the Jacobian of C that has CM structure over the cyclotomic field. Such an abelian variety has interesting Hodge classes that are not from intersections of divisors, which are called Weil Hodge classes. Chad Schoen proved that at a certain degree, these Weil Hodge classes are represented by algebraic cycles. As a consequence, he proved Hodge conjecture of abelian 4-folds with CM structure Q(\sqrt{-3}) and Q(i). In this talk, I will first give an introduction to the subject, with an emphasis on reviewing Schoen's approach. Then, we recover Schoen's construction of algebraic cycles using geometric class field theory. Finally, we generalize Schoen's result to abelian covers.