Complex Projective Space Pn Research Articles

Non-normal surfaces of degree 5 and 6 in P n

The purpose of this paper is to describe the non-normal surfaces of degree 5 and 6 embedded in the complex projective space P n , with n⩾4. The idea is to study the normalization of such a surface, and then to find how a non-normal surface can be obtained from its normalization.

On the dual and hessian mappings of projective hypersurfaces

We investigate the first-order Thom–Boardman singularity sets of the dual mapping for an arbitrary (and then for a generic) smooth hypersurface in the complex projective space ℙn. Our results focus on nonemptiness, connectedness, regular stratifications and numerical invariants for these sets.

Nonexistence of Almost-Quaternion Substructures on the Complex Projective Space

AbstractIt is shown that there are no almost-quaternion substructures on the complex projective space Pn(ℂ).

On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces

The purpose of this paper is to establish the following isomorphism of Lie algebras.Main Theorem. Let X be the complement of a hypersurface S in the complex projective space PN. Then the tower of nilpotent complex Lie algebras associated with the fundamental group π1(X, *) and the holonomy Lie algebraattached to S are isomorphic.

The duals of generic space curves and complete intersections

In a previous paper we discussed the duals of generic hypersurfaces: both smooth hypersurfaces in ℝn and algebraic hypersurfaces in real or complex projective space ℙn. In this note we show how to extend the methods of [1] to cover the case of complete intersections in ℙn and preface this with a brief discussion on the contact of space curves in ℝn with planes. We shall use the notation of [1].

Complex 2-plane bundles over complex projective space

In recent years there has been some interest in rank2 algebraic (or holomorphic) vector bundles over the complex projective space pn. One of the main reasons for this interest is the connection of such bundles with projective varieties of codimension 2 and in particular with the question whether such varieties are necessarily complete intersections. In [2] Hartshorne formulates the conjecture that every rank2 algebraic bundle over pn splits as the sum of 2 line bundles at least if n > 7. Rank 2 bundles over pa are classified topologically by their Chern classes cl, c 2. In [1] Atiyah and Rees carry out the topological classification for p3 and show the existence of "sufficiently many" rank 2 topological bundles over p4. On p2 and p3 every topological 2-plane bundle has an algebraic structure. It is not known whether this is true for n > 4. In [6] and [-7] non-trivial 2-plane bundles are constructed over P", n>5 , with c a = c 2 = 0 . These are possible candidates for topological bundles without algebraic structure. In this paper we carry the classification of topological 2-plane bundles over P~ a few steps further. Although this by no means solves the classification problem for algebraic bundles, we hope the results may be of some use to those interested in algebraic bundles and projective varieties. In the following we shall not distinguish between a map f:X~BU(2), its homotopy:class and the induced isomorphism class of rank 2 bundles. Thus we shall often write r/$X or q :X~BU(2) . As H*(P";2~) is a truncated polynomial algebra on a 2-dimensional generator x, the Chern classes cl, c 2 are integer multiples of x resp. x 2. Therefore we shall regard them as integers. Let us write

Remarks to the uniqueness problem of meromorphic maps into PN(C), III

In the previous papers [3], [4], [5] the author gave some results on the uniqueness of meromorphic maps of Cn into the N-dimensional complex projective space PN(C) which have the pre-assigned inverse images for some hyperplanes in PN(C). Relating to these results, we attempt in this paper to generalize the following Cartan-Nevanlinna’s theorem to the case of meromorphic maps into PN(C).

C3-Actions and algebraic threefolds with ample tangent bundle

One of the most challenging problems in complex differential geometry is the following conjecture of Frankel [3].(F-n) A compact Kaehler manifold M of dimension n with positive sectional (or more generally, positive holomorphic bisectional) curvature is biholomorphic to the complex projective space Pn(C).

A uniqueness theorem of algebraically non-degenerate meromorphic maps into PN(C)

In the previous paper [3], the author generalized the uniqueness theorems of meromorphic functions given by G. Pólya in [5] and R. Nevanlinna in [4] to the case of meromorphic maps of Cn into the N- dimensional complex projective space PN(C).

A relation between order and defects of meromorphic mappings of Cn into Pn(C)

Let f be a meromorphic mapping of the n-dimensional complex plane Cn into the N-dimensional complex projective space PN(C). We denote by T(r,f) the characteristic function of f and by N(r,f*H) the counting function for a hyperplane H ⊂ PN(C). The purpose of this paper is to establish the following results.

The uniqueness problem of meromorphic maps into the complex projective space

In 1921, G. Pólya showed that non-constant meromorphic functions ϕ and ψ of finite genera on the complex plane C are necessarily equal if there are distinct five values ai(1 ≦ i ≦ 5) such that ϕ(z) — ai and ψ(z) — ai have the same zeros of the same multiplicities for each i ([8]). Afterwards, R. Nevanlinna obtained the same conclusion for arbitrary ϕp and ψ satisfying ϕ— 1(ai) = ψ— 1(1 ≦ i ≦ 5) regardless of multiplicities. And, some other results relating to this were given by H. Cartan ([2], [3]), E. M. Schmid ([9]) and others. The purpose of this paper is to give some types of generalizations of these results to the case of meromorphic maps into the N-dimensional complex projective space PN(C).

Holomorphic Vector Fields on Algebraic Manifolds

Then M does not admit any non-trivial holomorphic p-form for p > m. We recall that a Hodge (or projective algebraic) manifold is a compact complex manifold which admits a holomorphic imbedding into complex projective space pN for some N. A holomorphic vector field X is one which can be represented locally as X = EXi3/azi, where each Xi is a holomorphic function. The set zero (X) = {x C M: X= O} is an analytic subspace of M, and by dim zero (X) we mean the maximum dimension of its components. Our hypothesis on the dimension includes the assumption that zero (X) is non-empty. Finally, a non-trivial form is one which does not vanish identically. As a special case of our theorem, we obtain:

A Theorem on Analytic Mappings of Complex Manifolds

We prove in this paper a theorem on analytic mappings of the complex space Cn into the complex projective space Pn. The theorem is closely related to that of S. S. Chern in [1], and the main idea of the proof is the same with the latter, though the calculations are rather different. The background of our calculation is the normal contact metric structure which was found by S. Sasaki and Y. Hatakeyama [4].