Algebraic cycles on genus-2 modular fourfolds

Abstract

This paper studies universal families of stable genus-2 curves with level structure. Among other things, it is shown that the $\left(1,1\right)$-part is spanned by divisor classes, and that there are no cycles of type $\left(2,2\right)$ in the third cohomology of the first direct image of $ℂ$ under projection to the moduli space of curves. Using this, it shown that the Hodge and Tate conjectures hold for these varieties.