Showing papers in "Tohoku Mathematical Journal in 1998"

The strong rigidity theorem for non-Archimedean uniformization

Abstract: In this paper, we present a purely algebraic proof of the strong rigidity for non-Archimedean uniformization, in case the base ring is of characteristic zero. In the last section, we apply this result to Mumford's construction of fake projective planes. In view of recent result on discrete groups by Cartwright, Mantero, Steger and Zappa, we see that there exist at least three fake projective planes.

The functor of a toric variety with enough invariant effective cartier divisors

Abstract: The homogeneous coordinate ring of a toric variety was first introduced by Cox. In this paper, we study that of a toric variety with enough invariant effective Cartier divisors in detail. Here a toric variety is said to have enough invariant effective Cartier divisors if, for each nonempty affine open subset stable under the action of the torus, there exists an effective Cartier divisor whose support equals its complement. Both quasi-projective toric varieties and simplicial toric varieties have enough invariant effective Cartier divisors. In terms of the homogeneous coordinate ring, we describe the data needed to specify a morphism from a scheme to such a toric variety. As a consequence, we generalize a result of Cox, one of Oda and Sankaran, and one of Guest concerning data on morphisms. Introduction. Let A: be a field, N a free Z-module of rank r, M the Z-module dual to N, T: = Gm®N the algebraic torus of dimension r corresponding to N9 and Δ a (finite) fan of NQ. Let XΔ be the toric variety associated to Δ, Dp the closure of the Γ-orbit corresponding to a one-dimensional cone peΔ, σ(l) the set of one-dimensional cones contained in a cone σeΔ, and Pic(zl)>0 the monoid of linear equivalence classes of invariant effective Cartier divisors. A toric variety XΔ is said to have enough invariant effective Cartier divisors if, for each cone σeΔ, there exists an effective Γ-invariant Cartier divisor D with SuppD=(Jp ί έ σ ( 1 )Z)p. Both quasi-projective toric varieties and simplicial toric varieties have enough invariant effective Cartier divisors (cf. Remark 1.6(3)). Cox [1] introduced two homogeneous coordinate rings of a toric variety XΔ\\ one is the monoid algebra S of the monoid of effective Γ-invariant Weil divisors with Chow-grading, while the other is the subring SΔ of S with Pic-grading (see [1, p. 19, p. 35]). He constructed in [1] the toric variety XΔ as the quotient of an open subscheme of Spec S, and described in [2, Theorem 1.1] the data needed to specify a map from a scheme to an arbitrary smooth toric variety in terms of its homogeneous coordinate ring. The purpose of this paper is to generalize Cox's description to one for an arbitrary toric variety with enough invariant effective Cartier divisors by studying the latter homogeneous coordinate ring in detail (cf. Theorem 3.4 and Theorem 4.3). The contents of this paper are as follows: 1991 Mathematics Subject Classification. Primary 14M25; Secondary 14E99. Partly supported by the Grants-in-Aid for Encouragement of Young Scientists, the Ministry of Education, Science, Sports and Culture, Japan.

Compact spacelike surfaces with constant mean curvature in the Lorentz-Minkowski $3$-space

Abstract: In this paper we develop some integral formulas for compact spacelike surfaces (necessarily with non-empty boundary) with constant mean curvature in the Lorentz-Minkowski three-space. As an application of this, when the boundary is a circle, we prove that the only such surfaces are the planar discs and the hyperbolic caps. By means of an appropriate maximum principle, we also obtain a uniqueness result for compact spacelike surfaces with constant mean curvature whose boundary projects onto a planar Jordan curve contained in a spacelike plane.

Bochner-Riesz means on symmetric spaces

Abstract: We combine results of Giulini and Mauceri and our earlier work to obtain an almost-everywhere convergence result for the Bochner-Riesz means of the inverse spherical transform of bi-invariant L functions on a noncompact rank one Riemannian symmetric space. Following a technique of Kanjin, we show that this result is sharp. 1. Notation. Suppose that G/K is a noncompact rank one Riemannian symmetric space of dimension d. Here functions on G/K can be viewed as being right-^-invariant functions on G, and ^-invariant functions on G/K are identified with bi-7£-invariant functions on G. Denote by — Δo the Laplace-Beltrami operator on G/K, and — Δ its self-adjoint extension to L\\G/K). Its spectral resolution is Δ = tdE(t), J\\P\\ where the constant \\p\\ depends on the geometry of G/K. For every ze C with there are the Bochner-Riesz mean operators

Calibrations and lagrangian submanifolds in the six sphere

Abstract: We study the first and second variations of volume for Lagrangian submanifolds in the six dimensional sphere.

Rough isometry and the asymptotic Dirichlet problem

Abstract: We propose a new asymptotic Dirichlet problem for harmonic functions via the rough isometry on a certain class of Riemannian manifolds. We prove that this problem is solvable for naturally defined class of functions. This result generalizes those of Schoen and Yau and of Cheng. 1. Introduction. The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. It has a long history, and by now, it is well understood by the works of the first author, M. Anderson, D. Sullivan, R. Schoen and others, when M is a Cartan-Hadamar d manifold with sectional curvature — b2

On the complexity of smooth projective toric varieties

Abstract: In this paper we answer a question posed by V.V. Batyrev. The question asked if there exists a complete regular fan with more than quadratically many primitive collections. We construct a smooth projective toric variety associated to a complete regular fan $\Delta$ in R^d with $n$ generators where the number of primitive collections of $\Delta$ is at least exponential in $n-d$. We also exhibit the connection between the number of primitive collections of $\Delta$ and the facet complexity of the Gr\"obner fan of the associated integer program.