Let X be an n-dimensional connected compact complex manifold and A be an analytic subset of X. We say that the pair (X, A) is a complex analytic compactification of Cn if X − A is biholomorphic to Cn. If X admits a Kähler metric, we shall say that (X, A) is a (non-singular) Kähler compactification of Cn . For n = 1, it is easy to see that (X, A) ≃ (P 1, ∞). For n = 2, Remmert-Van de Ven [17] proved that (X, A) ≃ (P 2, P 1) if A is irreducible, where A = P 1 is linearly embedded in P 2. Morrow [15] gave more detailed classifications of complex analytic compactifications of C 2 For n = 3, Brenton-Morrow showed the following

THEOREM ([5]). Let (X, A) be a non-singular Kähler complex analytic compactification of C 3such that the analytic subset A has only isolated singular points. Then X is projective algebraic and A is birationally equivalent to a ruled surface over an algebraic curve of genus g = b 3(X)/2.

Further, Brenton [3] classified the possible types of singular points of A in the case that the canonical line bundle KA of A is not trivial.

A probability measure μ is called unimodal if there is a point α such that the distribution function of μ is convex on (− ∞, α) and concave on (α, ∞). The point α is called a mode of μ. When μ is unimodal, the mode of μ is not always unique; the set of modes is a one point set or a closed interval. If μ is a unimodal distribution with finite variance, Johnson and Rogers [6] give a bound

where m and v are mean and variance of μ (see also [11]).

Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp -extension F ∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F ∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp -extension (which is generated by torsion points of Gm ).

The aim of this note is to construct “involutive” Markov families for geodesic flows of negative curvature. Roughly speaking, a Markov family for a flow is a finite family of local cross-sections to the flow with fine boundary conditions. They are basic tools in the study of dynamical systems. In 1973, R. Bowen [5] constructed Markov families for Axiom A flows. Using these families, he reduced the problem of counting periodic orbits of an Axiom A flow to the case of hyperbolic symbolic flows.

Let P(x, D) be a linear partial differential operator with real analytic coefficients and let C ⊂ Rn be a germ of closed subset, say at the origin. We say that C is (the locus of) an irremovable singularity of a real analytic solution u of P(x, D)u = 0 if u is defined outside C on a neighborhood Ω of 0 but cannot be extended to the whole neighborhood Ω even as a hyperfunction solution of P(x, D)u = 0. This usage of the word “singularity” is the same as the one for the analytic functions in complex analysis, and is different of the usual usage of “singularities of solutions” in the theory of partial differential equations.

Let A be a noetherian ring (all the rings are supposed here to be commutative with identity), a ⊂ A a proper ideal and  the completion of A in the α-adic topology. We consider the following conditions

(WAP) Every finite system of polynomial equations over A has a solution in A iff it has one in Â.

Let X be a compact space and f be a continuous map from X into itself. The topological entropy of f, h(f), was defined by Adler, Konheim and McAndrew [1]. After that Bowen [4] defined the topological entropy for uniformly continuous maps of metric spaces, and proved that the two entropies coincide when the spaces are compact. The definition of Bowen is useful in calculating entropy of continuous maps.

Let b0 be a positive real number and

be a Jacobi matrix. We can associate with them a Jacobi continued fraction, which will be abbreviated to a J fraction from the next section, as follows

where An (z)/Bn (z) is the n-th Padé approximant of φ(z).

We denote by M(n) the space of all n × n-matrices with their coefficients in the complex number field C and by G the group of invertible matrices GL(n, C). Let W = M(n)i be the vector space of l-tuples of n × ra-matrices. We denote by ρ: G → GL(W) a rational representation of G defined as follows:

if S ∈ G, A(i) ∈ M(n) (i = 1, 2, …, l).

A Hadamard matrix of order n is an n by n matrix of 1’s and − 1’s such that HHt − nI. In such a matrix n is necessarily 1, 2 or a multiple of 4. Two Hadamard matrices H 1 and H 2 are called equivalent if there exist monomial matrices P, Q with PH 1 Q = H 2. An automorphism of a Hadamard matrix H is an equivalence of the matrix to itself, i.e. a pair (P, Q) of monomial matrices such that PHQ = H. In other words, an automorphism of H is a permutation of its rows followed by multiplication of some rows by − 1, which leads to reordering of its columns and multiplication of some columns by − 1. The set of all automorphisms form a group under composition called the automorphism group (Aut H) of H. For a detailed study of the basic properties and applications of Hadamard matrices see, e.g. [1], [7, Chap. 14], [8].

In this paper, we deal with the first part of an application of our Riemann-Roch type inequalities (cf. [13], [23]) toward deformations of polarized normal varieties. In Chapter I, we discuss the problem of eliminating fixed components from complete linear systems defined by multiples of a given divisor. Let Un be a complete normal variety and Y0 an ample Cartier divisor on U. The main result of the chapter is that there is a positive integer c0, predicted by the first two leading coefficients of the polynomial X(U, O(m Y 0)), such that the complete linear system Λ(rc 0 Y 0) has no fixed component whenever r is a positive integer (which is essentially contained in Theorem 1.1 (cf. [13], Lemma 5.2)). An easy consequence of this result is that when n = 2, we can find another positive integer c 1 predicted by the same coefficients as above, such that rc 1 c 0 Y 0 is very ample on U whenever r is a positive integer. Even though this has been generalized to n = 3 by J. Kollár (cf. [12]), we have included this in Section 3 since it is very simple.