No Arabic abstract
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.
Classically, theorems of Fatou and Julia describe the boundary regularity of functions in one complex variable. The former says that a complex analytic function on the disk has non-tangential boundary values almost everywhere, and the latter describes when a function takes an extreme value at a boundary point and is differentiable there non-tangentially. We describe a class of intermediate theorems in terms of averaged Julia-Fatou quotients. Boundary regularity is related to integrability of certain quantities against a special measure, the so-called Nevanlinna measure. Applications are given to spectral theory.
The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Feffermans invariant hypersurface measure.
In this paper we extend the well known theorem of Angelo Lopez concerning the Picard group of the general space projective surface containing a given smooth projective curve, to the intermediate Neron-Severi group of a general hypersurface in any smooth projective variety.
We investigate the global variation of moduli of linear sections of a general hypersurface. We prove a generic Torelli result for a large proportion of cases, and we obtain a complete picture of the global variation of moduli of line slices of a general hypersurface.
We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.