26. CMB 2006 (vol 49 pp. 628)
 Zeron, E. S.

Approximation and the Topology of Rationally Convex Sets
Considering a mapping $g$ holomorphic on a neighbourhood of a rationally
convex set $K\subset\cc^n$, and range into the complex projective space
$\cc\pp^m$, the main objective of this paper is to show that we can
uniformly approximate $g$ on $K$ by rational mappings defined from
$\cc^n$ into $\cc\pp^m$. We only need to ask that the second \v{C}ech
cohomology group $\check{H}^2(K,\zz)$ vanishes.
Keywords:Rationally convex, cohomology, homotopy Categories:32E30, 32Q55 

27. CMB 2006 (vol 49 pp. 72)
 Dwilewicz, Roman J.

Additive RiemannHilbert Problem in Line Bundles Over $\mathbb{CP}^1$
In this note we consider $\overline\partial$problem in
line bundles over complex projective space $\mathbb{CP}^1$
and prove that the
equation can be solved for $(0,1)$ forms with compact support. As a
consequence, any CauchyRiemann function on a compact real hypersurface in
such line bundles is a jump of two holomorphic functions defined on the
sides of the hypersurface. In particular, the results can be applied to
$\mathbb{CP}^2$ since by removing a point from it we get a line bundle over
$\mathbb{CP}^1$.
Keywords:$\overline\partial$problem, cohomology groups, line bundles Categories:32F20, 14F05, 32C16 

28. CMB 2005 (vol 48 pp. 414)
 Kaveh, Kiumars

Vector Fields and the Cohomology Ring of Toric Varieties
Let $X$ be a smooth complex
projective variety with a holomorphic vector field with isolated
zero set $Z$. From the results of Carrell and Lieberman
there exists a filtration
$F_0 \subset F_1 \subset \cdots$ of $A(Z)$, the ring of
$\c$valued functions on $Z$, such that $\Gr A(Z) \cong H^*(X,
\c)$ as graded algebras. In this note, for a smooth projective
toric variety and a vector field generated by the action of a
$1$parameter subgroup of the torus, we work out this filtration.
Our main result is an explicit connection between this filtration
and the polytope algebra of $X$.
Keywords:Toric variety, torus action, cohomology ring, simple polytope,, polytope algebra Categories:14M25, 52B20 

29. CMB 2005 (vol 48 pp. 473)
30. CMB 2003 (vol 46 pp. 617)
 Pak, Hong Kyung

On Harmonic Theory in Flows
Recently [8], a harmonic theory was developed for a compact
contact manifold from the viewpoint of the transversal geometry of
contact flow. A contact flow is a typical example of geodesible
flow. As a natural generalization of the contact flow, the present
paper develops a harmonic theory for various flows on compact
manifolds. We introduce the notions of $H$harmonic and
$H^*$harmonic spaces associated to a H\"ormander flow. We also
introduce the notions of basic harmonic spaces associated to a weak
basic flow. One of our main results is to show that in the special
case of isometric flow these harmonic spaces are isomorphic to the
cohomology spaces of certain complexes. Moreover, we find an
obstruction for a geodesible flow to be isometric.
Keywords:contact structure, geodesible flow, isometric flow, basic cohomology Categories:53C20, 57R30 

31. CMB 2003 (vol 46 pp. 268)
 Puls, Michael J.

Group Cohomology and $L^p$Cohomology of Finitely Generated Groups
Let $G$ be a finitely generated, infinite group, let $p>1$, and let
$L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in
G} a_x ^p < \infty \}$. In this paper we will study the first
cohomology group of $G$ with coefficients in $L^p(G)$, and the first
reduced $L^p$cohomology space of $G$. Most of our results will be for a
class of groups that contains all finitely generated, infinite nilpotent
groups.
Keywords:group cohomology, $L^p$cohomology, central element of infinite order, harmonic function, continuous linear functional Categories:43A15, 20F65, 20F18 

32. CMB 2000 (vol 43 pp. 3)
 Adin, Ron; Blanc, David

Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and
free Lie algebras in the category of nonassociative algebras. These
resolutions derive in both cases from geometric objects, which in turn
reflect the combinatorics of suitable collections of leaflabeled
trees.
Keywords:resolutions, homology, Lie algebras, associative algebras, nonassociative algebras, Jacobi identity, leaflabeled trees, associahedron Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50 

33. CMB 1999 (vol 42 pp. 129)
 Baker, Andrew

Hecke Operations and the Adams $E_2$Term Based on Elliptic Cohomology
Hecke operators are used to investigate part of the $\E_2$term of
the Adams spectral sequence based on elliptic homology. The main
result is a derivation of $\Ext^1$ which combines use of classical
Hecke operators and $p$adic Hecke operators due to Serre.
Keywords:Adams spectral sequence, elliptic cohomology, Hecke operators Categories:55N20, 55N22, 55T15, 11F11, 11F25 

34. CMB 1997 (vol 40 pp. 54)
 Kechagias, Nondas E.

A note on $U_n\times U_m$ modular invariants
We consider the rings of invariants $R^G$, where $R$ is the symmetric
algebra of a tensor product between two vector spaces over the field $F_p$
and $G=U_n\times U_m$. A polynomial algebra is constructed and these
invariants provide Chern classes for the modular cohomology of $U_{n+m}$.
Keywords:Invariant theory, cohomology of the unipotent group Category:13F20 
