Algebraic cycles on genus-2 modular fourfolds (Q1732982)

The article's main purpose is to provide new evidence for the Hodge and Tate conjectures, by showing that they hold for the case of certain smooth projective fourfolds. The hypothesis is that the fourfolds $X$ are birationally equivalent to some universal family of stable genus-2 curves with level structure, in such a way that there is a map $ f: X \to Y $, where the threefold $Y$ is a desingularization of the Deligne-Mumford compactification. It is also requested that $f$ satisfies a list of properties, in which case $ f: X \to Y $ is said to be a good model of the universal curve map. In particular, there are normal crossing boundary divisors $E$ and $D$ so that $f:(X,E)\to (Y,D)$ is a semistable map in this paper's sense. A couple of basic lemmas form the core of the argument of proof. Recall that there is a mixed Hodge structure on $H^i(U, R^jf_*\mathbb Q)$ where $U= Y-D$, see the author's [Invent. Math. 160, No. 3, 567--589 (2005; Zbl 1083.14011)] . One lemma says that given a semistable map of smooth projective varieties $f:(X,E) \to (Y,D)$ such that $dim Y=3$ and $dim X=4$, then the vanishing condition $Gr_F^2H^3(U, R^1f_* \mathbb C)=0$ implies the Hodge conjecture for $X$. The second lemma, which requires the same vanishing in addition to other conditions, gives the Tate conjecture. \par In order to prove the requested vanishing for the situation at hand, the article's method is to exploit the result that the graded quotients are computed by means of the cohomology of a complex $K_{\mathcal X /\mathcal Y}(j,p)$ of logarithmic sheaves on $Y$, namely one has $Gr_F^pH^i(U, R^jf_*\mathbb C)\cong H^i(K_{\mathcal X /\mathcal Y}(j,p))$. The crucial step is performed by showing that $K_{\mathcal X /\mathcal Y}(1,2)$ is quasi-isomorphic to $0$. The proof of this fact depends on the properties of the moduli space in genus $2$ and it relies among other things on an isomorphism due to \textit{G. Faltings} and \textit{C.-L. Chai}, which computes the logarithmic sheaf $\Omega_{Y}^1(\log D)$, see [Degeneration of abelian varieties. Berlin etc.: Springer-Verlag (1990; Zbl 0744.14031)]. \par The paper contains also the result that the Mordell-Weil rank of $Pic^0 (X \to Y)$ is zero and that divisor classes span $H^{1,1} (X).$