Algebraic cycles on genus two modular fourfolds

TL;DR

This paper investigates algebraic cycles on genus two modular fourfolds, establishing key properties of their cohomology and confirming the Hodge and Tate conjectures for these varieties.

Contribution

It demonstrates that the (1,1) cohomology is generated by divisor classes and proves the absence of certain cycles in the third cohomology, leading to the verification of the Hodge and Tate conjectures.

Findings

(1,1) part spanned by divisor classes

No (2,2) cycles in third cohomology

Hodge and Tate conjectures hold for these varieties

Abstract

We study universal families of stable genus two curves with level structure. Among other things, it is shown that the (1,1) part is spanned by divisor classes, and that there are no cycles of type (2,2) in the third cohomology of the first direct image. Using this, we deduce the Hodge and Tate conjectures hold for these varieties.