Stability of Hodge bundles and a numerical characterization of Shimura varieties

Abstract

Let $U$ be a connected non-singular quasi-projective variety and $f : A \to U$ a family of abelian varieties of dimension $g$. Suppose that the induced map $U \to \mathcal{A}_g$ is generically finite and there is a compactification $Y$ with complement $S = Y \backslash U$ a normal crossing divisor such that $\Omega_Y^1 (\log S)$ is nef and $\omega_Y (S)$ is ample with respect to $U.

We characterize whether $U$ is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map $U \to \mathcal{A}_g$ or by the existence of CM points.

More precisely, we show that $f : A \to U$ is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem $\mathbb{V}$ of $R_1 f_* \mathbb{C}_A$ is either unitary or satisfies the Arakelov equality. Second, for each factor $M$ in the universal cover of $U$ whose tangent bundle behaves like that of a complex ball, an iterated Kodaira-Spencer map associated with $V$ has minimal possible length in the direction of $M$. If in addition $f : A \to U$ is rigid, it is a connected Shimura subvariety of $\mathcal{A}_g$ of Hodge type.