Sheaf Of Germs Research Articles

Cohomology of locally q-complete sets in Stein manifolds

We prove that if X is a Stein complex manifold of dimension n and Ω ⊂ X a locally q-complete open set in X with q ≤ n−2, then the cohomology groups H p (Ω , OΩ) vanish if p ≥ q and OΩ is the sheaf of germs of holomorphic functions on Ω.

A universal complex structure on complete metrizable topological spaces

Let X be a topological space whose topology may be defined by a complete metric d. Taking all such metrics d we define a universal complex structure on X. For this complex structure the sheaf of germs of holomorphic functions on X coincides with the sheaf of germs of continuous functions on X, and hence the theories of topological and holomorphic vector bundles on X are the same.

On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons

AbstractFor an integer n ≥ 3, let Mn be the moduli space of spatial polygons with n edges. We consider the case of odd n. Then Mn is a Fano manifold of complex dimension n − 3. Let ΘMn be the sheaf of germs of holomorphic sections of the tangent bundle TMn. In this paper, we prove Hq(Mn, ΘMn) = 0 for all q ≥ 0 and all odd n. In particular, we see that the moduli space of deformations of the complex structure on Mn consists of a point. Thus the complex structure on Mn is locally rigid.

Vanishing theorems on manifolds with piecewise strictly pseudodonvex boundaris

Let ω be a relatively compact subdomain of a complex manifold X with boundary ∂ ω strictly pseudoconvex and let there be a C2 strictly plurisubhannonic function defined in a neighborhood of the closure of ω in X. It is shown that the cohomology groups Hq (ωDp ) vanish for q ≥ 1, p ≥ 0, where Dp is the sheaf of germs of holomorphic p-forms on ω. This settles, in the negative, the question raised by I. Lieb concerning whether the vanishing of Hq (ωDp ) for q ≥ 1, p ≥. 0 implies that ω is Stein or not. When Ω has only a piecewise strictly pseudoconvex boundary, it also shows that Hq (ωVn) vanishes for q ≥ 1. This is applied to the Gleason problem, the weak corona problem and an approximation problem.

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of ‘minimal herissons’ inℝ3 is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC3 are described by a single map from the etale space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ℙ1 to C3, which islinear on stalks. There is an analogous construction for null curves inC4. This gives a similar class of minimal surfaces in ℝ4.

Cohomology on a Riemannian foliated manifold with coefficients in the sheaf of germs of foliated currents

Foliated differential forms were introduced in [7], [9], to study the cohomology on a RIEMANNian foliated manifold with coefficients in the sheaf of germs of foliated differential forms. In this paper the notion of DE RHAM like current of the type (p, q) is defined for a RIEMANNian foliated manifold and some properties of various differential operators acting on the spaces of currents are given. In particular, special DE RHAM like currents are considered namely the foliated ones. It turns out that the space of foliated p-forms is dense in the space of foliated p-currents with the usual topology. We get certain results concerning the cohomology on a RIEMANNian foliated manifold with coefficients in the sheaf of germs of foliated currents.

Deformations of holomorphic Seifert fiber spaces

Let M be a holomorphic Seifert fiber space as constructed in Conner and Raymond [3] and let O M be the sheaf of germs of holomorphic vector fields on M. The purpose of this paper is to study the cohomology group H I(M, OM) and to describe the small deformations of M. Since M is given as a quotient of a principal torus bundle, it would be natural to use the cohomology of groups with coefficients in sheaves with operators to describe M and objects associated to it. These groups were introduced in Grothendieck [4] and also in Kodaira [7], Conner [2] and Conner and Raymond [3] by different approaches. In Sect. 1, we review these groups. Using the equivariant version of the Leray spectral sequence obtained in Sect. 1, we give a decomposition of H~(M, OM) into the direct sum of complex vector groups C, E, F and H in Sect. 2. In Sect's. 3 through 6, we study the geometric significance of the groups C, E, F and H. In Sect. 7, we compute these vector groups when the dimension of the base space is one and give some examples. It is possible to deal with r fibrations in a similar manner. It will be done in a forthcoming paper [14]. I would like to thank F. Raymond for many valuable conversations.

Chern classes of the Grassmannians and Schubert calculus

IN THIS note we continue the theme of [3,4] by using the zeros of a holomorphic vector field to give a geometric demonstration that the ring structure of the cohomology of the Grassman manifolds (the multiplicative aspect of Schubert calculus) is a consequence of general properties of symmetric functions. More precisely, this amounts to showing that certain properties of Schubert cycles such as independence; multiplicative structure etc. are formally deducible from similar properties of a class of symmetric functions called Schur functions provided certain so called meaningless Schubert conditions are ignoied. This principle seems to have originated with Giambelli and was formulated precisely by Lesieur[9]. More recently, Horrocks[6], Lascoux[8], Porteous [ 1 l] and Stanley [ 131 have discussed various aspects of the principle. Recall that if, X is a compact complex manifold and V a holomorphic vector field on X with zeros 2, one may view 2 as the variety defined by the sheaf of ideals i(V)R’ C 6 where i(V):W+fP-’ is the contraction from the sheaf of germs of holomorphic p-forms to (p l)-forms and 6 = R”. The structure sheaf fY2 = Q/i( V)Q’ of 2 is a coherent possibly unreduced sheaf of rings on Z. The main result of [4] is that if X is in addition compact Kaehler and Z finite but nontrivial, then H”(X, I!?~), the ring of functions on Z, has a decreasing filtration F having the property that fiFj C F;+j such that there is an isomorphism of graded rings

Criteria for the injectivity of analytic sheaves

It is shown that a moduleL over the sheafO of germs of holomorphic functions on a domain G of Cn is injective if and only if the following conditions are satisfied; a)L is flabby; b) for every closed set S ⊂G and every point z λ G, the stalk se z of the sheafS L;U1→Γ S (U:L) is an injectiveO z -module. It follows in particular that the sheaf of germs of hyperfunctions is injective over the sheaf of germs of analytic functions.

Algebras of Holomorphic Functions in Ringed Spaces, I

A pair () is a ringed space if it is a subsheaf of rings with 1 of the sheaf of germs of continuous functions on X. If U is an open subset of X, we denote the set of sections over U relative to by . If , then implies that there exists some open neighbourhood V of u, V ⊂ U, and some g continuous on V such that the germ of g at u, ug is ϕ(u). Now we define ϕ(u) (u) to be g(u) and in this way we obtain, in a unique fashion, a continuous complex-valued function on U. The collection of all such functions for a given set is denoted by and is called the -holomorphic functions on U.THEOREM. Let X be a locally connected Hausdorff space and () a ringed space.

Sheaves Defined by Differential Equations and Application to Deformation Theory of Pseudo-Group Structures

For a vector bundle E over a manifold V., denote by S (E) the sheaf of germs of local cross-sections of E. By using local coordinate we may consider linear differential equations on E. When V = Rn and E = V X Rm, the linear differential equations on E are linear differential equations on m unknown functions in n variables. The germs of solutions of a linear differential equation on E forms a subsheaf, say G, of S (E). The importance of such subsheaves G can be seen, for instance, in the recent works of Kodaira and Spencer in the deformation theory of pseudo-group structures. In the first half of the present paper, we study some of properties of such sheaves. Since it is very difficult to treat the general case, we restrict ourselves to the category of real analyticity and to the case where inconvenient degeneracy does not occur in the differential equations. In our argument these restrictions are necessary because in the last analysis we depend on Cartan's theorem of solvability of involutive real analytic differential equations. G together with its defining linear differential equation which is nondegenerate, is called a (S)-sheaf. Our main result is the existence of quotient (S)-sheaves of (S)sheaves by (S) -subsheaves. In the second half the writer gives concise description of how the above results can be applied to deformation theory of transitive continuous pseudo-group compact structures. Since our argument follows the more or less well known line developed in [3], details are omitted. It is noted that another approach, which is still within the same framework as ours but which is more differential geometric, is published in [5] by D. C. Spencer. In the present paper every notions are assumed to be in the category of real analyticity. So when we say manifolds or mappings for example, we mean always manifolds or mappings which are real analytic.