## Stability of Hodge bundles and a numerical characterization of Shimura varieties.(English)Zbl 1435.14025

Summary: 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 $$S = Y \setminus U$$ 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 $$R^1 f_\ast \mathbb{C}_A$$ 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 $$\mathbb V$$ 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.

### MSC:

 14G35 Modular and Shimura varieties 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 14D07 Variation of Hodge structures (algebro-geometric aspects) 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14K10 Algebraic moduli of abelian varieties, classification
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