The quotient of a Hermitian symmetric space of non-compact type by a torsion-free cocompact arithmetic subgroup of the identity component of the group of isometries of the symmetric space is called an arithmetic fake compact Hermitian symmetric space if it has the same Betti numbers as the compact dual of the Hermitian symmetric space. This is a natural generalization of the notion of “fake projective planes” to higher dimensions. Study of arithmetic fake compact Hermitian symmetric spaces of type A_n with even n has been completed in [PY1], [PY2]. The results of this paper, combined with those of [PY2], imply that there does not exist any arithmetic fake compact Hermitian symmetric space of type other than A_n, n ≤ 4 (see Theorems 1 and 2 in the Introduction below and Theorem 2 of [PY2]). The proof involves the volume formula given in [P], the Bruhat-Tits theory of reductive p-adic groups, and delicate estimates of various number theoretic invariants.

The discrete second Painlevé equation dP_II is mapped to the second Painlevé equation P_II by its continuous limit, and then, as shown by Kajiwara et al., a rational solution of dP_II also reduces to that of P_II. In this paper, regarding dP_II as a difference equation, we present a certain asymptotic solution that reduces to a triply-truncated solution of P_II in this continuous limit. In a special case our solution corresponds to a rational one of dP_II. Furthermore we show the existence of families of solutions having sequential limits to truncated solutions of P_II.

In this paper we study (strongly) locally o-minimal structures. We first give a characterization of the strong local o-minimality. We also investigate locally o-minimal expansions of (R, +, <).

We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain the sharp estimate for weakly complete case. As an application of this result, we provide a new and simple proof of the parametric affine Bernstein problem for improper affine spheres. Moreover, we get the same estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.

We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another. Based on this multiplicity, we define a pseudo distance on the class of objects. We define and study several multiplicities in the category of topological spaces and continuous maps, the category of groups and homomorphisms, the category of finitely generated R-modules and R-linear maps over a principal ideal domain R, and the neighbourhood category of oriented knots in the 3-sphere.

We derive an energy decay estimate for solutions to the initial-boundary value problem of a semilinear wave equation in exterior domains with a nonlinear localized dissipation. Our equation includes an absorbing term like |u|^αu, α ≥ 0, and can be regarded as a generalized Klein-Gordon equation at least if α is closed to 0. This observation plays an essential role in our argument.

Let L be a non-negative self adjoint operator on L²(X) where X is a space of homogeneous type. Assume that L generates an analytic semigroup e^-tL whose kernel satisfies the standard Gaussian upper bounds. By the spectral theory, we can define the spectral multiplier operator F(L). In this article, we show that the commutator of a BMO function with F(L) is bounded on L^p(X) for 1 < p < ∞ when F is a suitable function.

It is shown that all uniquely homogeneous spaces are connected. We characterize the uniquely homogeneous spaces that are semitopological or quasitopological groups. We identify two properties of homogeneous spaces called skew-2-flexibility and 2-flexibility that are useful in studying unique homogeneity. We also construct a large family of uniquely homogeneous spaces with only trivial continuous maps.

Some basic results on compact affine Nash groups related to their Nash representations are given. So, first a Nash version of the Peter-Weil theorem is proved and then several more results are given: for example, it is proved that an analytic representation of such a group is of class Nash and that the category of the classes of isomorphic embedded compact Nash groups is isomorphic with that of the classes of isomorphic embedded algebraic groups. Moreover, given a compact affine Nash group G, a closed subgroup H and a homogeneous Nash G-manifold X, it is proved that the twisted product G ×_H X is a Nash G-manifold which is Nash G-diffeomorphic to an algebraic G-variety; besides, this algebraic structure is unique.

In this paper we study smooth complex projective polarized varieties (X,H) of dimension n ≥ 2 which admit a covering family V of rational curves of degree 3 with respect to H such that two general points of X may be joined by a curve parametrized by V, and such that there is a covering family of rational curves of H-degree one.

We prove that the Picard number of these manifolds is at most three, and that, if equality holds, (X,H) has an adjunction theoretic scroll structure over a smooth variety.

We study the global behavior of trajectories for Kähler magnetic fields on a connected complete Kähler manifold M of negative curvature. Concerning these trajectories we show that theorems of Hadamard-Cartan type and of Hopf-Rinow type hold: If sectional curvatures of M are not greater than c (< 0) and the strength of a Kähler magnetic field is not greater than $\sqrt{|c|}$, then every magnetic exponential map is a covering map. Hence arbitrary distinct points on M can be joined by a minimizing trajectory for this magnetic field.

In the homogeneous space Sol₃, a translation surface is parametrized by x(s,t) = α(s) ∗ β(t), where α and β are curves contained in coordinate planes and ∗ denotes the group operation of Sol₃. In this paper we study translation surfaces in Sol₃ whose mean curvature vanishes.

Let X₀ be an affine variety with only normal isolated singularity at p. We assume that the complement X₀ \ {p} is biholomorphic to the cone C(S) of an Einstein-Sasakian manifold S of real dimension 2n − 1. If there is a resolution of singularity π: X → X₀ with trivial canonical line bundle K_X, then there is a Ricci-flat complete Kähler metric for every Kähler class of X. We also obtain a uniqueness theorem of Ricci-flat conical Kähler metrics in each Kähler class with a certain boundary condition. We show there are many examples of Ricci-flat complete Kähler manifolds arising as crepant resolutions.