scholarly journals Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Theocharis Theofanidis
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Additional Restriction ◽  
Real Hypersurfaces ◽  
Specific Function ◽  
Complex Projective Plane ◽  
Jacobi Structure ◽  
Complex Projective

Real hypersurfaces satisfying the conditionϕl=lϕ(l=R(·,ξ)ξ)have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective planeCP2satisfying a generalization ofϕl=lϕunder an additional restriction on a specific function.

Examples of real hypersurfaces with two principal curvatures in complex projective planes

2015 ◽  
Vol 12 (08) ◽  
pp. 1560013 ◽  
Author(s):  
J. Carlos Díaz Ramos ◽  
Cristina Vidal-Castiñeira
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Real Hypersurfaces ◽  
Complex Projective Plane ◽  
Principal Curvatures ◽  
Complex Projective

We give examples of real hypersurfaces with two distinct principal curvatures in the complex projective plane ℂP2.

LEVI-UMBILICAL REAL HYPERSURFACES IN A COMPLEX SPACE FORM

2016 ◽  
Vol 229 ◽  
pp. 99-112 ◽  
Author(s):  
JONG TAEK CHO ◽  
MAKOTO KIMURA
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Levi Form ◽  
Complex Space Form ◽  
Real Hypersurfaces ◽  
Complex Projective Plane ◽  
Nonzero Constant ◽  
Local Construction ◽  
Complex Projective

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$, $n\geqslant 3$, whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$, we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$.

On the Hyperplane Sections of Blow-Ups of Complex Projective Plane

1989 ◽  
Vol 41 (6) ◽  
pp. 1005-1020 ◽  
Author(s):  
Aldo Biancofiore
Keyword(s):  
Line Bundle ◽  
Projective Plane ◽  
Algebraic Surfaces ◽  
Projective Surface ◽  
Complex Projective Plane ◽  
Complex Projective ◽  
Hyperplane Sections

Let L be a line bundle on a connected, smooth, algebraic, projective surface X. In this paper we have studied the following questions:1) Under which conditions is L spanned by global sections? I.e., if ɸL : X →PN denotes the map associated to the space Г(L) of the sections of L, when is ɸL a morphism?2) Under which conditions is L very ample? I.e., when does ɸL give an embedding?These problems arise naturally in the study, and in particular in the classification, of algebraic surfaces (see [8], [3], [5]).

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

Spectral Sequences

2020 ◽  
pp. 45-56
Author(s):  
Loring W. Tu
Keyword(s):  
Projective Plane ◽  
Spectral Sequence ◽  
Fiber Bundles ◽  
Spectral Sequences ◽  
Complex Projective Plane ◽  
Computational Tool ◽  
Short Introduction ◽  
Complex Projective

This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined by Jean-Louis Koszul, Henri Cartan, Jean-Pierre Serre, and many others. The chapter provides a short introduction, without proofs, to spectral sequences. As an example, it computes the cohomology of the complex projective plane. The chapter then details Leray's theorem. A spectral sequence is like a book with many pages. Each time one turns a page, one obtains a new page that is the cohomology of the previous page.

scholarly journals SMOOTH, NONSYMPLECTIC EMBEDDINGS OF RATIONAL BALLS IN THE COMPLEX PROJECTIVE PLANE

2020 ◽  
Vol 71 (3) ◽  
pp. 997-1007
Author(s):  
Brendan Owens
Keyword(s):  
Projective Plane ◽  
Infinite Family ◽  
Complex Projective Plane ◽  
Rational Homology ◽  
Complex Projective

Abstract We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

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