Let U be a connected non-singular quasi-projective variety and f : A → U a family of abelian varieties of dimension g. Suppose that the induced map U → Ag is generically finite and there is a compactification Y with complement S = Y \U a normal crossing divisor such that Ω1 Y (log S) is nef and ωY (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 →Ag or by the existence of CM points. More precisely, we show that f : A → U is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem V of R1f∗CA 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 → U is rigid, it is a connected Shimura subvariety of Ag of Hodge type.

We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudohermitian manifold of dimension 2m + 1 ≥ 5. We prove that the equality holds iff the manifold is equivalent to the CR sphere up to a scaling. For this purpose, we establish an Obata-type theorem in CR geometry that characterizes the CR sphere in terms of a nonzero function satisfying a certain overdetermined system. Similar results are proved in dimension 3 under an additional condition.

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov-Poisson system. The schemes are constructed by combing a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

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