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.


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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