Kobayashi-Ochiai's theorem states that the set of dominant rational maps from a complex variety to a complex variety of general type is finite. Kazuya Kato conjectured a similar result in the category of log schemes. Our main theorem of this paper is a solution to his conjecture.

We study splitting densities of primitive elements of a discrete subgroup of a connected non-compact semisimple Lie group of real rank one with finite center in another larger such discrete subgroup. When the corresponding cover of such a locally symmetric negatively curved Riemannian manifold is regular, the densities can be easily obtained from the results due to Sarnak or Sunada. Our main interest is a case where the covering is not necessarily regular. Specifically, for the case of the modular group and its congruence subgroups, we determine the splitting densities explicitly. As an application, we study analytic properties of the zeta function defined by the Euler product over elements consisting of all primitive elements which satisfy a certain splitting law for a given lifting.

Study of stochastic differential equations on the field of $p$-adic numbers was initiated by the second author and has been developed by the first author, who proved several results for the $p$-adic case, similar to the theory of ordinary stochastic integral with respect to Lévy processes on Euclidean spaces. In this article, we present an improved definition of a stochastic integral on the field and prove the joint (time and space) continuity of the local time for $p$-adic stable processes. Then we use the method of random time change to obtain sufficient conditions for the existence of a weak solution of a stochastic differential equation on the field, driven by the $p$-adic stable process, with a Borel measurable coefficient.

We determine the local structure of all pseudo-Riemannian manifolds of dimensions greater than 3 whose Weyl conformal tensor is parallel and has rank 1 when treated as an operator acting on exterior 2-forms at each point. If one fixes three discrete parameters: the dimension, the metric signature (with at least two minuses and at least two pluses), and a sign factor accounting for semidefiniteness of the Weyl tensor, then the local-isometry types of our metrics correspond bijectively to equivalence classes of surfaces with equiaffine projectively flat torsionfree connections; the latter equivalence relation is provided by unimodular affine local diffeomorphisms. The surface just mentioned arises, locally, as the leaf space of a codimension-two parallel distribution on the pseudo-Riemannian manifold in question, naturally associated with its metric. We construct examples showing that the leaves of this distribution may form a fibration with the base which is a closed surface of any prescribed diffeomorphic type.

Our result also completes a local classification of pseudo-Riemannian metrics with parallel Weyl tensor that are neither conformally flat nor locally symmetric: for those among such metrics which are not Ricci-recurrent, the Weyl tensor has rank 1, and so they belong to the class discussed in the previous paragraph; on the other hand, the Ricci-recurrent ones have already been classified by the second author.

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired by recent results, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is given. Also, we describe under which condition a Lagrangian orbit is isolated. If $M$ is a compact Kähler manifold, we give a necessary and sufficient condition for an isometric action to admit a Lagrangian orbit. Then we investigate homogeneous Lagrangian submanifolds on the symplectic cut and on the symplectic reduction. As an application of our results, we exhibit new examples of homogeneous Lagrangian submanifolds on the blow-up at one point of the complex projective space and on the weighted projective spaces. Finally, applying our result which may be regarded as Lagrangian slice theorem for a Hamiltonian group action with a fixed point, we give new examples of homogeneous Lagrangian submanifolds on irreducible Hermitian symmetric spaces of compact or noncompact type.