THE JUMPING PHENOMENON OF THE DIMENSIONS OF COHOMOLOGY GROUPS OF TANGENT SHEAF

Let X be a compact complex manifold.Consider a small deformation π :X→B of X,the dimensions of the cohomology groups of tangent sheaf H q(X_t,T_X_t) may vary under this deformation.This article studies such phenomena by studying the obstructions to deform a class in H q(X,T X) with parameter t and gets a formula for the obstructions.     

关于非主Hopf曲面的模空间

构造了Ｓ＾１上Ｓ＾３｜Ｈ－丛的所有复结构的模空间,其中丛的转换函数ｕ：ｓ＾３／Ｈ→Ｓ＾３／Ｈ是Ｓ＾３／Ｈ的一个对合,ＨＵ（２）,Ｈ为有限群且在Ｓ＾３／Ｈ上的作用是自由的。真不连续的。     

The second bounded cohomology of an amalgamated free product of groups

We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we show that a group with infinitely many ends has infinite dimensional second bounded cohomology.

     

Minus class groups of the fields of the -th roots of unity

We show that for any prime number the minus class group of the field of the -th roots of unity admits a finite free resolution of length 1 as a module over the ring . Here denotes complex conjugation in . Moreover, for the primes we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.

     

The integral cohomology of the Bianchi groups

We calculate the integral cohomology ring structure for various members of the Bianchi group family. The main tools we use are the Bockstein spectral sequence and a long exact sequence derived from Bass-Serre theory.

     

Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves

Let E be an elliptic curve over Q and ? be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ?. Under certain assumptions, we prove the vanishing of the Galois cohomology group H¹(Gal(K(E[?ⁱ])/K),E[?ⁱ]) for all i?1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ?-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture.     

Approximately vanishing of topological cohomology groups

In this paper, we establish the pexiderized stability of coboundaries and cocycles and use them to investigate the Hyers-Ulam stability of some functional equations. We prove that for each Banach algebra A, Banach A-bimodule X and positive integer n,Hn(A,X)=0 if and only if the nth cohomology group approximately vanishes.     

Computation of the cohomology rings of Kac-Moody groups,their flag manifolds and classifying spaces

In this paper we introduce the history and present situation of the computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces, and give some problems and conjectures that deserve further study.     

On the rigidity of super-Grassmannians

The rigidity of the complex super-Grassmannian Gr _m|n,k|l with 0<k<m, 0<l<n, supposing that (k, l) (1,n–1), (m–1, 1), (1,n–2), (m–2, 1), (2,n–1), (m–1, 2), is proved.Partially supported by Erwin Schrödinger International Institute for Mathematical Physics (Vienna, Austria)     

The Hochschild cohomology ring of a cyclic block

Suppose is a block of a group algebra with cyclic defect group. We calculate the Hochschild cohomology ring of , giving a complete set of generators and relations. We then show that if is the principal block, the canonical map from to the Hochschild cohomology ring of induces an isomorphism modulo radicals.     

The cohomology of the elliptic tangent bundle

In this note we compute the cohomology of the elliptic tangent bundle, a Lie algebroid introduced in Cavalcanti and Gualtieri (2018), Cavalcanti et al. (2020) used to describe singular symplectic forms arising from generalised complex geometry.     

On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior

This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53-64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119-171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173-177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91-117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have Q-forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17-22, 2001, TIFR, Mumbai, 2001, pp. 469-490] of D. Morris (Witte) on a completely different topic. We will answer that question too.     

Sub-bundles of the complexified tangent bundle

We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold. The aim is to understand implications of properties of interest in partial differential equations.

     

On loop extensions satisfying one single identity and cohomology of loops

In this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities and in two variables for the case of commutativity. It is considered a large amount of identities. These groups generalize those defined in Nishigori [3 Nishigori, N. (1963). On loop extensions of groups and M-cohomology groups I. J. Sci. Hiroshima Univ. Ser. A-I 27:151–165. [Google Scholar]] and of Kenneth and Leedham-Green [2 Kenneth, W. J., Leedham-Green, Ch. R. (1990). Loop cohomology. Czec. Math. J. 40(2):182–194.[Web of Science ®] , [Google Scholar]]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.     

Computation of cohomology groups of the Lie algebras of type Bn and Cn

We adduce the results of the numerical experiment in calculating the spaces of local deformations of classical Lie algebras of type B _n and C _n in characteristic p = 2. Original Russian Text ? D.V. Reshetnikov, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 8, pp. 71–72.     

The Hochschild cohomology of the quasi-entwining structure

We give a concept of quasi-entwining structure,and investigate the Hochschild cohomology of the quasi-entwining structure.We obtain the equivalent theorems on the Hochschild cohomology of the quasientwining structure.In particular,we get the isomorphism theorem between the Hochschild cohomology of coalgebra structures and the Hochschild cohomology of the dual algebra structures for the quasi-entwining structures of finite-dimensional algebras and coalgebras.