We give a version in characteristic $p>0$ of Mumfords theorem characterizing a smooth complex germ of surface $(X,x)$ by the triviality of the topological fundamental group of $U=Xsetminus {x}$. This note relies on discussions the authors had during the Christmas break 2009/10 in Ivry. They have been written down by Hel`ene in the night when Eckart died, as a despaired sign of love.
Consider a family f:A --> U of g-dimensional abelian varieties over a quasiprojective manifold U. Suppose that the induced map from U to the moduli scheme of polarized abelian varieties is generically finite and that there is a projective manifold Y, containing U as the complement of a normal crossing divisor S, such that the sheaf of logarithmic one forms is nef and that its determinant 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 from U to the moduli scheme or by the existence of CM points. More precisely, we show that U is a Shimura variety, if and only if two conditions hold. First, each irreducible local subsystem V of the complex weight one variation of Hodge structures is either unitary or satisfies the Arakelov equality. Secondly, for each factor M in the universal cover of U whose tangent bundle behaves like the one of a complex ball, an iterated Kodaira-Spencer map associated with V has minimal possible length in the direction of M.
This is a small note meant to be published in a Conference Proceedings. We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by a finite group. Part of it reproduces the content of a letter dated September 27, 2008 addressed to Johannes Nicaise
Unlike Grothendiecks etale fundamental group, Noris fundamental group does not fulfill the homotopy exact sequence in general. We give necessary and sufficient conditions which force exactness of the sequence.
We consider Kobayashi geodesics in the moduli space of abelian varieties A_g that is, algebraic curves that are totally geodesic submanifolds for the Kobayashi metric. We show that Kobayashi geodesics can be characterized as those curves whose logarithmic tangent bundle splits as a subbundle of the logarithmic tangent bundle of A_g. Both Shimura curves and Teichmueller curves are examples of Kobayashi geodesics, but there are other examples. We show moreover that non-compact Kobayashi geodesics always map to the locus of real multiplication and that the Q-irreducibility of the induced variation of Hodge structures implies that they are defined over a number field.
We discuss several numerical conditions for families of projective varieties or variations of Hodge structures.
Given a projective symplectic manifold $M$ and a non-singular hypersurface $X subset M$, the symplectic form of $M$ induces a foliation of rank 1 on $X$, called the characteristic foliation. We study the question when the characteristic foliation is algebraic, namely, all the leaves are algebraic curves. Our main result is that the characteristic foliation of $X$ is not algebraic if $X$ is of general type. For the proof, we first establish an etale version of Reeb stability theorem in foliation theory and then combine it with the positivity of the direct image sheaves associated to families of curves.
We show that the M-canonical map of an n-dimensional complex projective manifold X of Kodaira dimension two is birational to an Iitaka fibration for a computable positive integer M. M depends on the index b of a general fibre F of the Iitaka fibration and on the Betti number of the canonical covering of F, In particular, M is a universal constant if the dimension n is smaller than or equal to 4.
