Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

Let H ⊂ ℙn be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian G(n + 1, n). Assuming H has a global defining function, we prove H is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension 2n - 2 or dimension 2n - 4. If the singular set is of dimension 2n - 4, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ℙn with a meromorphic (rational of degree 1) first integral. In this case, H is in some sense simply a complex cone over an algebraic curve in ℙ1. Similarly if H has a degenerate singularity, then H is also algebraic. If the dimension of the singular set is 2n - 2 and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ℙ2 of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ℙ2. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.

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HOLOMORPHIC FOLIATIONS AND CHARACTERISTIC NUMBERS

Communications in Contemporary Mathematics ◽

10.1142/s021919970500188x ◽

2005 ◽

Vol 07 (05) ◽

pp. 583-596 ◽

Cited By ~ 1

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].

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On submersions of the complex indicatrix

Filomat ◽

10.2298/fil1716081p ◽

2017 ◽

Vol 31 (16) ◽

pp. 5081-5092

Author(s):

Elena Popovicia

Keyword(s):

Fixed Point ◽

Complex Projective Space ◽

Finsler Space ◽

Hermitian Manifold ◽

Almost Hermitian Manifold ◽

Codimension One ◽

Real Submanifold ◽

Fundamental Equations ◽

Complex Projective ◽

Extrinsic Sphere

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

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

Anais da Academia Brasileira de Ciências ◽

10.1590/s0001-37652001000400001 ◽

2001 ◽

Vol 73 (4) ◽

pp. 475-482 ◽

Cited By ~ 6

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.

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A Big Picard Theorem for Holomorphic Maps into Complex Projective Space

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-018-6 ◽

2009 ◽

Vol 52 (1) ◽

pp. 154-160

Author(s):

Yasheng Ye ◽

Min Ru

Keyword(s):

Projective Space ◽

Complex Manifold ◽

Complex Projective Space ◽

Holomorphic Maps ◽

Analytic Subvariety ◽

Picard Theorem ◽

Complex Projective

AbstractWe prove a big Picard type extension theoremfor holomorphic maps f : X–A → M, where X is a complex manifold, A is an analytic subvariety of X, and M is the complement of the union of a set of hyperplanes in ℙn but is not necessarily hyperbolically imbedded in ℙn.

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Algebraic intersection spaces

Journal of Topology and Analysis ◽

10.1142/s1793525319500778 ◽

2019 ◽

Vol 12 (04) ◽

pp. 1157-1194 ◽

Cited By ~ 1

Author(s):

Christian Geske

Keyword(s):

Tubular Neighborhood ◽

Space Theory ◽

Singular Set ◽

Projective Varieties ◽

Local Duality ◽

Chain Complexes ◽

Singular Sets ◽

Significant Extension ◽

Real Analytic ◽

Complex Projective

We define a variant of intersection space theory that applies to many compact complex and real analytic spaces [Formula: see text], including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze “local duality obstructions,” which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in [Formula: see text] of a tubular neighborhood of the singular set.

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Non-Compact Complex Projective Space as a Phase Space

Physics of Particles and Nuclei Letters ◽

10.1134/s1547477120050210 ◽

2020 ◽

Vol 17 (5) ◽

pp. 744-747

Author(s):

E. Khastyan ◽

H. Shmavonyan

Keyword(s):

Phase Space ◽

Projective Space ◽

Complex Projective Space ◽

Complex Projective

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Sectional curvatures of homogeneous real hypersurfaces of types (A) and (B) in a complex projective space

Journal of Geometry ◽

10.1007/s00022-021-00585-4 ◽

2021 ◽

Vol 112 (2) ◽

Author(s):

Makoto Kimura ◽

Sadahiro Maeda ◽

Hiromasa Tanabe

Keyword(s):

Projective Space ◽

Complex Projective Space ◽

Real Hypersurfaces ◽

Sectional Curvatures ◽

Complex Projective

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From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700040302 ◽

2002 ◽

Vol 66 (3) ◽

pp. 465-475 ◽

Cited By ~ 1

Author(s):

J. Bolton ◽

C. Scharlach ◽

L. Vrancken

Keyword(s):

Projective Space ◽

Minimal Surface ◽

Complex Projective Space ◽

Lagrangian Submanifold ◽

3 Dimensional ◽

Ellipse Of Curvature ◽

Complex Projective

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

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