scholarly journals Generalized transversely projective structure on a transversely holomorphic foliation

2002 ◽  
Vol 30 (3) ◽  
pp. 151-163
Author(s):  
Indranil Biswas
Keyword(s):  
Projective Space ◽  
Normal Bundle ◽  
Holomorphic Section ◽  
Holomorphic Foliation ◽  
Projective Structure ◽  
Transition Functions ◽  
Holomorphic Sections ◽  
Partial Connection

The results of Biswas (2000) are extended to the situation of transversely projective foliations. In particular, it is shown that a transversely holomorphic foliation defined using everywhere locally nondegenerate maps to a projective spaceℂℙn, and whose transition functions are given by automorphisms of the projective space, has a canonical transversely projective structure. Such a foliation is also associated with a transversely holomorphic section ofN⊗−kfor eachk∈[3,n+1], whereNis the normal bundle to the foliation. These transversely holomorphic sections are also flat with respect to the Bott partial connection.

scholarly journals On the geometry of Poincaré's problem for one-dimensional projective foliations

2001 ◽  
Vol 73 (4) ◽  
pp. 475-482 ◽  
Author(s):  
MARCIO G. SOARES
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Projective Variety ◽  
Holomorphic Foliation ◽  
One Dimensional ◽  
Complex Projective Variety ◽  
Geometric Objects ◽  
Complex Projective

We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a one-dimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation.

HOLOMORPHIC FOLIATIONS AND CHARACTERISTIC NUMBERS

2005 ◽  
Vol 07 (05) ◽  
pp. 583-596 ◽  
Author(s):  
MARCIO G. SOARES
Keyword(s):  
Projective Space ◽  
Complex Manifold ◽  
Complex Projective Space ◽  
Holomorphic Foliation ◽  
Compact Complex Manifold ◽  
Holomorphic Foliations ◽  
Singular Set ◽  
One Dimensional ◽  
Characteristic Numbers ◽  
Complex Projective

We relate the characteristic numbers of the normal sheaf of a k-dimensional holomorphic foliation [Formula: see text] of a compact complex manifold Mn, to the characteristic numbers of the normal sheaf of a one-dimensional holomorphic foliation associated to [Formula: see text]. In case M is a complex projective space, we also obtain bounds for the degrees of the components of codimension k - 1 of the singular set of [Formula: see text].

PROJECTIVE STRUCTURES AND -CONNECTIONS

2018 ◽  
Vol 19 (2) ◽  
pp. 571-579
Author(s):  
Radu Pantilie
Keyword(s):  
Projective Space ◽  
Twistor Space ◽  
Projective Structure ◽  
Projective Structures ◽  
Quaternionic Projective Space ◽  
Quaternionic Manifold ◽  
Projective Flatness ◽  
Complex Projective Structure ◽  
Complex Projective

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the $\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

scholarly journals The index of harmonic maps from surfaces to complex projective spaces

2020 ◽  
Vol 31 (09) ◽  
pp. 2050069
Author(s):  
J. Oliver
Keyword(s):  
Projective Space ◽  
Lower Bounds ◽  
Complex Projective Space ◽  
Harmonic Maps ◽  
Projective Spaces ◽  
Line Bundles ◽  
Complex Projective Spaces ◽  
Holomorphic Sections ◽  
Higher Genus ◽  
Complex Projective

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces.

scholarly journals Relativistic Localizing Processes Bespeak an Inevitable Projective Geometry of Spacetime

2017 ◽  
Vol 2017 ◽  
pp. 1-18 ◽  
Author(s):  
Jacques L. Rubin
Keyword(s):  
Projective Space ◽  
Observational Data ◽  
Projective Geometry ◽  
Three Dimensional ◽  
Projective Structure ◽  
Riemannian Structure ◽  
Real Projective Space ◽  
Spacetime Manifold

Surprisingly, the issue of events localization in spacetime is poorly understood and a fortiori realized even in the context of Einstein’s relativity. Accordingly, a comparison between observational data and theoretical expectations might then be strongly compromised. In the present paper, we give the principles of relativistic localizing systems so as to bypass this issue. Such systems will allow locating users endowed with receivers and, in addition, localizing any spacetime event. These localizing systems are made up of relativistic autolocating positioning subsystems supplemented by an extra satellite. They indicate that spacetime must be supplied everywhere with an unexpected local four-dimensional projective structure besides the well-known three-dimensional relativistic projective one. As a result, the spacetime manifold can be seen as a generalized Cartan space modeled on a four-dimensional real projective space, that is, a spacetime with both a local four-dimensional projective structure and a compatible (pseudo-)Riemannian structure. Localization protocols are presented in detail, while possible applications to astrophysics are also considered.

scholarly journals Interpolation for Curves in Projective Space with Bounded Error

2019 ◽  
Author(s):  
Eric Larson
Keyword(s):  
Projective Space ◽  
Normal Bundle ◽  
Hyperplane Section ◽  
Maximal Rank ◽  
Special Linear ◽  
Maximal Rank Conjecture ◽  
Canonical Curve ◽  
Bounded Error ◽  
Rank Conjecture ◽  
Pass Through

Abstract Given $n$ general points $p_1, p_2, \ldots , p_n \in{\mathbb{P}}^r$ it is natural to ask whether there is a curve of given degree $d$ and genus $g$ passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if \begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor.\end{equation*}The case of curves with nonspecial hyperplane section was recently studied in [2], where the above conjecture was shown to hold with exactly three exceptions. In this paper, we prove a “bounded-error analog” for special linear series on general curves; more precisely we show that existence of such a curve subject to the stronger inequality \begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor - 3.\end{equation*}Note that the $-3$ cannot be replaced with $-2$ without introducing exceptions (as a canonical curve in ${\mathbb{P}}^3$ can only pass through nine general points, while a naive dimension count predicts twelve). We also use the same technique to prove that the twist of the normal bundle $N_C(-1)$ satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of general points contained in the hyperplane section of a general curve is at least \begin{equation*}\min\left(d, \frac{(r - 1)^2 d - (r - 2)^2 g - (2r^2 - 5r + 12)}{(r - 2)^2}\right).\end{equation*} As explained in [7], these results play a key role in the author’s proof of the maximal rank conjecture [9].

scholarly journals HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE

2018 ◽  
Vol 238 ◽  
pp. 104-136
Author(s):  
TAKAYUKI KOIKE
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Normal Bundle ◽  
Holomorphic Foliation ◽  
Compact Complex Manifold ◽  
Sufficient Condition ◽  
Existence Problem ◽  
Hermitian Metric ◽  
Flat Normal Bundle ◽  
Compact Submanifold

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

Sign in / Sign up

Export Citation Format

Share Document