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Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a Q-variation of Hodge structures of weight one on Y with Higgs bundle E1,0 ⊕ E0,1, coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the slopes satisfy μ(E1,0) − μ(E0,1) ≤ μ(­1 Y ). We prove a similar inequality in the higher dimensional case. If the latter is an equality, and if the discriminant of E1,0 or the one of E0,1 is zero, one hopes that Y is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in case the universal covering of Y does not contain factors of rank> 1. Part of the results extend to variations of Hodge structures over quasi-projective manifolds U.

Strongly pseudoconvex CR manifolds are boundaries of Stein varieties with isolated normal singularities.We prove that any nonconstant CR morphism between two (2n.1)-dimensional strongly pseudoconvex CR manifolds lying in an n-dimensional Stein variety with isolated singularities are necessarily a CR biholomorphism. As a corollary, we prove that any nonconstant self map of (2n . 1)-dimensional strongly pseudoconvex CR manifold is a CR automorphism. We also prove that a finite ′etale covering map between two resolutions of isolated normal singularities must be an isomorphism.

We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of Conn’s linearization theorem.

Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min–max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to achieve a strong “Colding–Minicozzi” type min–max approximation result.