Consider the complex of differential forms on an open affine subvariety U of AN with differential where d is the usual exterior derivative and ø is a fixed 1-form on U. For certain U and ø, we compute the cohomology of this complex.

We use the method of Scholz and Reichardt and a transfer principle from finite fields to pseudo finite fields in order to prove the following result. THEOREM Let G be a group of order ln, where l is a prime number. Let K0be either a finite field with |K0| > l4n+4or a pseudo finite field. Suppose that l ≠ char(K0) and that K0does not contain the root of unity ζl of order l. Let K = K0(t), with t transcendental over K0. Then K has a Galois extension L with the following properties: (a) (L/K) ≅ G; (b) L/K0is a regular extension; (c) genus(L) < ; (d) K0[t] has exactly n prime ideals which ramify in L; the degree of each of them is [K0: K0]; (e) (t)∞totally decomposes in L; (f) L = K(x), withand deg(ai(t)) < deg(a1(t)) for i = 1,…,ln.

The classification of Riemann surfaces has largely reached its completion. The purpose of the present paper is to lay the foundation for a new intriguing field in the classification theory: Riemannian spaces with Euclidean metrics. The paper is self-contained, both for the Riemann surface expert and the reader whose main interest is with higher dimensions.

Let us consider soluble groups whose Sylow subgroups are all abelian. Such groups we call A-groups, following P. Hall. A-groups were investigated thoroughly by P. Hall and D. R. Taunt from the view point of the structure theory. In this note, we want to give some remarks concerning representation theoretical properties of A-groups.

Let R = k[x1,…,xn] be a polynomial ring over a field k. Let J = {j1,…,jt} be a subset of {1,…, n}, and let mJ ⊂ R denote the ideal (xj1,…,xjt). Given subsets J1,…,Js of {1,…, n} and positive integers a1,…,as, we study ideals of the form These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s ≤ 3 or when Ji ∪ Jj = [n] for all i ≠ j. When s ≥ 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s = 2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k) = 0, our work also yields new cases in which this conjecture holds.

In this article, we present a concavity property of the minimal $L^{2}$ integrals related to multiplier ideal sheaves with Lebesgue measurable gain. As applications, we give necessary conditions for our concavity degenerating to linearity, characterizations for 1-dimensional case, and a characterization for the holding of the equality in optimal $L^2$ extension problem on open Riemann surfaces with weights may not be subharmonic.

A toric face ring, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Römer and their coauthors recently. In this paper, under the “normality” assumption, we describe a dualizing complex of a toric face ring R in a very concise way. Since R is not a graded ring in general, the proof is not straightforward. We also develop the square-free module theory over R, and show that the Cohen-Macaulay, Buchsbaum, and Gorenstein* properties of R are topological properties of its associated cell complex.

We show the twisted cohomology on has a natural polarized Hodge structure and hence derive the analogues of Riemann’s equality and inequailty.

Let G be a connected linear algebraic group, p a rational representation of G on a finite-dimensional vector space V, all defined over C.

Let BSPL be the classifying space of the stable oriented PL micro bundles. The purpose of this paper is to determine H*(BSPL:Zp) as a Hopf algebra over Zp, where p is an odd prime number. In this chapter, p is always an odd prime number.

In this note we study the estimate from above and below and the asymptotic behaviour of the class number of positive definite integral quadratic forms.

The purpose of the present note is to reform Mr. K. Nomizu’s result on the group of all affine transformations of an affinely connected manifold. We shall prove the following.

Recently different authors have studied the conormal modules I/I2 of almost complete intersections in local Gorenstein rings (c. t. Aoyama [1], Herzog [8], Kunz [13], Matsuoka [16]). An essential tool in these papers is the theory of canonical modules and the fact that these modules are easy to handle in the case of almost complete intersections.

S. Kondō defined a birational period map between the moduli space of genus three curves and a moduli space of polarized K3 surfaces. In this paper we give a resolution of the period map, providing a surjective morphism from a suitable compactification of 3 to the Baily-Borel compactification of a six dimensional ball quotient.

Let Rn+1 = Rn × R denote the (n + 1)-dimensional Euclidean space (n ≧ 1). For X ∊ Rn+1 we write X = (x, t) with x ∊ Rn and t ∊ R. In this case, Rn is called the x-space of Rn+1 = Rn × R.

For an α with 0 < α < 1, we write

One of the most powerful tools in homotopy theory is the homotopy groups of a triad introduced by Blakers and Massey in [1]. Our aim here is to develop systematically the formal, elementary aspects of the theory of a generalized triad and the mapping track associated with it. This will be used in §5 to deduce a result (Theorem 5.5) which seems to be closely related to an exact sequence established by Brown [2].

Our setting for this paper is projective 3-space over an algebraically closed field K. By a curve C ⊂ is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Let be the Hartshorne-Rao module of finite length (cf. [R]). Here Z is the set of integers and ℐc the ideal sheaf of C. In [GMV] it is shown that , where is the homogeneous ideal of C, is the first local cohomology module of the R-module M with respect to . Thus there exists a smallest nonnegative integer k ∊ N such that , (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown that k = 0 if and only if C is arithmetically Cohen-Macaulay and C is arithmetically Buchsbaum if and only if k ≤ 1. We therefore have the following natural definition.

All rings are commutative with unit and modules are unital. With one exception all rings are Noetherian. We consider only rings of positive characteristic p. Section 0 contains background, definitions and terminology.

A real hypersurface M in a complex manifold X is said to be Levi flat if it separates X locally into two Stein pieces. It is proved that there exist no real analytic Levi flat hypersurfaces in ℙ2.

In this paper, we improve an inequality given in [3]. Let Lp (1 ≦ p ≦ ∞) and BMO denote the Lp space and the BMO space on the real line. Their norms are denoted by ‖ · ‖p and ‖ · ‖BMO, respectively.