Dimensional Projective Space Research Articles

Real Forms of the Radon–Penrose Transform

The complex Radon correspondence relates an n -dimensional projective space with the Grassmarm manifold of its p -dimensional planes. This is the geometric background of the Radon–Penrose transform, which intertwines cohomology classes of homogeneous line bundles with holomorphic solutions to the generalized massless field equations. A good framework to deal with such problems is provided by the recently developed theory of integral transforms for sheaves and D -modules. In particular, an adjunction formula describes the range of transforms acting on general function spaces, associated with constructible sheaves. The linear group SL ( n + 1,C) naturally acts on the Radon correspondence. A distinguished family of function spaces is then the one associated with locally constant sheaves along the closed orbits of the real forms of SL ( n + 1,C). In this paper, we systematically apply the above-mentioned adjunction formula to such function spaces. We thus obtain in a unified manner several results concerning the complex, conformal, or real Radon transforms.

Approximations algébriques des points dans les espaces projectifs I

We establish approximation properties by algebraic points, of points in projective spaces of dimension ⩽3. We introduced this kind of properties in a previous text where it was shown how they can be used to prove algebraic independence results. In dimension one, a finer approximation property has been developped by D. Roy and M. Waldschmidt then M. Laurent and D. Roy, for similar purposes. The essential tool which enables us to reach dimension three is a refined effective lower bound for the Hilbert function of a prime ideal.

Geometry of distributions of flags on Grassmann manifolds of a projective space. I

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space.

Envelopes of meromorphy of domains over infinite dimensional complex projective spaces

Let E be a separable complex Fréchet space with bounded approximation property or a complex DFN-spaceP(E) be the complex projective space induced from E and (Ω, ϕ) be a Riemann domain over P(E). We prove the coincidence of the envelope of meromorphy of the domain (Ω, ϕ) with the envelope of holomorphy of the domain (Ω, ϕ).

Gherardelli linkage and complete intersections.

Our main theorem characterizes the complete intersections of codimension 2 in a projective space of dimension 3 or more over an algebraically closed field of characteristic 0 as the subcanonical and self-linked subschemes. In order to prove this theorem, we'll prove the Gherardelli linkage theorem, which asserts that a partial intersection of two hypersurfaces is subcanonical if and only if its residual intersection is, scheme-theoretically, the intersection of the two hypersurfaces with a third.

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.

Transitive and co-transitive caps

Let PG(r, q) be the projective space of dimension r over GF (q). A k–cap K in PG(r, q) is a set of k points, no three of which are collinear [10], and a k–cap is said to be complete if it is maximal with respect to set–theoretic inclusion. The maximum value of k for which there is known to exist a k–cap in PG(r, q) is denoted by m2(r, q). Some known bounds for m2(r, q) are given below. Suppose that K is a cap in PG(r, q) with automorphism group Ḡ0 ≤ PΓL(r + 1, q). Then K is said to be transitive if Ḡ0 acts transitively on K, and co-transitive if Ḡ0 acts transitively on PG(r, q)− K. Our main result is the following theorem.

Adjacency preserving mappings of invariant subspaces of a null system

In the space I r I_r of invariant r r -dimensional subspaces of a null system in ( 2 r + 1 ) (2r+1) -dimensional projective space, W.L. Chow characterized the basic group of transformations as all the bijections φ : I r → I r \varphi :I_r\to I_r , for which both φ \varphi and φ − 1 \varphi ^{-1} preserve adjacency. In the present paper we show that the two conditions φ : I r → I r \varphi :I_r\to I_r is a surjection and φ \varphi preserves adjacency are sufficient to characterize the basic group. At the end of this paper we give an application to Lie geometry.

On Modified Mabuchi Functional and Mabuchi Moduli Space of Kähler Metrics on Toric Bundles

Mabuchi introduced the Mabuchi functional in [Mb1], and it turns out that it is very useful for dealing with Kahler metrics with constant scalar curvatures on compact manifolds (see [BM], etc.). One can also expect that the existence of Kahler metric with constant scalar curvature is almost equivalent to the existence of a lower bound of the Mabuchi functional (see, e.g., [Ti]). But for the case of extremal metrics, the Mabuchi functional is not applicable. Therefore, we need a new (or a modified) functional for metrics which are invariant under a maximal compact connected subgroup K of Aut(M). We did not obtain this functional until the appearing of [FM] (while we were reviewing [FM] in 1995). Mabuchi also found this functional independently [Mb3] (see also [Sm]). A definition of this functional was given in [GC]. We shall give some results and applications in this paper. It turns out that our modified Mabuchi functional M(ω1, ω2) has the property that M(ω1, g∗ω2) = M(ω1, ω2), for any g ∈ CKC(K), where CKC(K) is the centralizer of K in the complexification K of K. Moreover, the extremal metrics are exactly the local minimal points of this functional. Therefore, we expect that the existence of an extremal metric is almost equivalent to the existence of a lower bound of this functional. Surprising enough that the first application of this functional is not the existence but the uniqueness of extremal metrics on smooth toric varieties, i.e., smooth Kahler manifolds with an open (C∗)n-orbit. Therefore, there is for example at most one extremal metric in any Kahler class of the manifold obtained by blowing up two points or three points of a two dimensional complex projective space. To have the uniqueness, we consider the Mabuchi moduli space of the Kahler metrics on the toric varieties (see [Mb2], which was rediscovered by Semmes [Se1] and Donaldson [Ch]). It turns out that the moduli space is flat in this situation (see also [Se1,2]). Moreover, for any two Kahler metrics there is a unique geodesic

A remark on the geometry of elliptic scrolls and bielliptic surfaces

The union of two quintic elliptic scrolls in P^4 intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface. In the appendix to the paper [W. Decker et al.: Syzygies of abelian and bielliptic surfaces in P^4, alg-geom/9606013] where the equations of this singular surface were computed, we proved that Z defines a smooth point in the appropriate Hilbert scheme and that Z cannot be smoothed in P^4. Here we consider the analogous situation in higher dimensional projective spaces P^{n-1}, where, to our surprise, the answer depends on the dimension n-1. If n is odd the union of two scrolls cannot be smoothed, whereas it can be smoothed if n is even. We construct an explicit smoothing.

Subregular Spreads of[formula omitted](2n+1,q)

In this paper, we develop some of the theory of spreads of projective spaces with an eye towards generalizing the results of R. H. Bruck (1969,in“Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514, Univ. of North Carolina Press, Chapel Hill). In particular, we wish to generalize the notion of asubregularspread to the higher dimensional case. Most of the theory here was anticipated by Bruck in later papers; however, he never provided a detailed formulation. We fill this gap here by developing the connections between a regular spread of (2n+1)-dimensional projective space and ann-dimensional circle geometry, which is the appropriate generalization of the Miquelian inversive plane. After developing this theory, we provide a fairly general method for constructing subregular spreads ofPG(5,q). Finally, we explore a special case of this construction, which yields several examples of three-dimensional subregular translation planes which are not André planes.

Differential topology of numerical range

The numerical range of an n × n matrix, also known as its field of values, is reformulated as the image of a smooth quadratic mapping from the n − 1 dimensional complex projective space to the complex plane. This paper investigates the numerical range from the perspective of differential topology (Morse theory). More specifically, the boundary of the range is interpreted as a rank 1 critical value curve and its sharp points are interpreted as rank 0 critical values. More importantly, the map is shown to have additional critical value curves in the interior of the numerical range. These additional curves are shown to have such singularity phenomena as cusps and swallow tails, to be the caustic envelopes of families of lines, and to exhibit the so-called “normal bifurcation” when an eigenvalue becomes unitarily decoupled.

The rigidity for real hypersurfaces in a complex projective space

The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.

On the Non-Existence of Certain Cameron-Liebler Line Classesin PG(3, q)

Our main result is a non-existence theorem for certain families of lines in three dimensional projective space PG(3, q) over a finite field GF(q). Specifically, a Cameron-Liebler line class in PG(3, q) is a set of lines which intersects every spread of PG(3, q) in the same number x of lines (this number is called its parameter). These sets arose in connection with an attempt by Cameron and Liebler to determine the subgroups of PGL(n+1, q) which have the same number of orbits on points (of PG(n, q)) as on lines; they satisfy several equivalent properties. Here we prove that for 2 < x ≤ √q, no Cameron-Liebler line class of parameter x exists in PG(3, q). A relevant general question on incidence matrices is described.

The Dolbeault complex in infinite dimensions I

In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy–Riemann equations on affine spaces.

Flag-transitive extensions of dual projective spaces

AbstractWe classify the ag{transitive circular extensions of line{point systems of nite projective geometries. 1 Introduction We consider geometries belonging to the following diagram of rank 3, where 0 ; 1 ; 2are the types, q;s are nite orders with q> 1and s +1=( q n 1) = ( q 1) for someinteger n> 1, the label c denotes the class of circular spaces and PG stands for theclass of dual projective spaces, namely geometries of lines and points of a projectivegeometry.(c : PG)   1 sq 01 2 c PG We call these geometries c : PG{ geometries .Givenac : PG{geometry with orders1 ;s;q as above, we call q the order of . As s +1=( q n 1) = ( q 1), the residuesof the elements of of type 0 are dual n {dimensional projective spaces of order q .We call n the residual dimension of .Ac : PG{geometry of residual dimension2 is a nite extendedprojectiveplane. Itis well{known that just two nite extended projective planes exist, namely AG(3 ; 2) Charge de Recherches du Fonds National Belge de la Recherche Scienti quey

Moduli Spaces of Curves with Quasi-Symmetric Weierstrass GAp Seqnences

In this paper we give an explicit construction of the moduli space of the pointed complete Gorenstein curves of arithmetic genus g with a given quasi-symmetric Weierstrass semigroup, that is, a Weierstrass semigroup whose last gap is equal to 2g − 2. We identify such a curve with its image under the canonical embedding in the (g − 1)-dimensional projective space. By normalizing the coefficients of the quadratic relations and by constructing Grobner bases of the canonical ideal, we obtain the equations of the moduli space in terms of Buchberger's criterion. Moreover, by analyzing syzygies of the canonical ideal we establish criteria that make the computations less expensive.

Frobenius collineations in finite projective planes

of order n on V . It follows that R induces a projective collineation φ on the (n−1)dimensional projective space PG(n−1, q). We call φ and any projective collineation conjugate to φ a Frobenius collineation. In the present paper we shall study the case n = 3, that is, the Frobenius collineations of the projective plane PG(2, q). Let P = PG(2, q). Then every Singer cycle σ (see Section 3) of P defines a partition P(σ) of the point set of P into pairwise disjoint Baer subplanes. These partitions are called linear Baer partitions or, equivalently, Singer Baer partitions [17]. If % is a Frobenius collineation of P , then we define E% to be the set of Baer subplanes of P fixed by %. It turns out that for q ≡ 2 mod 3 we have |P(σ)∩E%| ∈ {0, 1, 3} with |P(σ) ∩ E%| = 3 if and only if % ∈ NG( ), where G = PGL3(q) (see 3.5).

Arguesian Identities in Invariant Theory

Having been motivated by an example of Doubilet, Rota, and Stein [Stud. Appl. Math.56(1976), 185–216], we present a technique for constructing geometric identities in a Grassmann–Cayley algebra. Each identity represents a projective invariant closely related to the Theorem of Desargues in the plane and its generalizations to higher dimensional projective space. The construction employs certain combinatorial properties of matchings in bipartite graphs. We also prove a dimension independence result for Arguesian identities, thereby connecting the identities with lattice theory.

Compact Quotient Manifolds of Domains in a Complex 3-Dimensional Projective Space and the Lebesgue Measure of Limit Sets

We shall consider compact complex manifolds of dimension three which have subdomains in a three dimensional projective space as their unramified even coverings. Assume that the subdomains contain projective lines. Then any pair of such manifolds can be connected together complex analytically by an analogue of Klein combinations [K1]. In this paper, we shall prove a (weak) analogue of two results of B. Maskit [M] on the Lebesgue measures of the limit set of Kleinian groups (Theorems A and B). In section 1, we shall give definitions of terms and precise statement of our result. In section 2, we shall analyze the limit set by the same method as that of Maskit. Section 3 is devoted to proving Theorem 2.1. As a corollary to Theorem 2.1, we obtain Theorem A, which is an analogue of Combination Theorem I of [M]. In section 4, we shall prove Theorem B, which is an analogue of Combination Theorem II of [M].