FIRST BETTI NUMBERS OF ABELIAN COVERINGS OF THE COMPLEX PROJECTIVE PLANE BRANCHED OVER LINE CONFIGURATIONS

2000 ◽  
Vol 09 (02) ◽  
pp. 271-284 ◽  
Author(s):  
IKUO TAYAMA
Keyword(s):  
Projective Plane ◽  
Real Line ◽  
General Position ◽  
Betti Numbers ◽  
Complex Projective Plane ◽  
The Real ◽  
Complex Projective

We prove the following results concerning the first Betti numbers of abelian coverings of CP2 branched over line configurations: (1) An estimate of the first Betti numbers. (2) A characterization of central and general position line configurations in terms of the first Betti numbers of abelian coverings. (3) The first Betti numbers of the abelian coverings of the real line configurations up to 7 components.

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.

General position properties of ANR's

1982 ◽  
Vol 92 (3) ◽  
pp. 451-466 ◽  
Author(s):  
W. J. R. Mitchell
Keyword(s):  
Real Line ◽  
General Position ◽  
Cartesian Product ◽  
Vital Role ◽  
The Real ◽  
Homology Groups ◽  
Local Homology

This paper investigates the ‘general position’ properties which ANR's may possess. The most important of these is the disjoint discs property of Cannon (5), which plays a vital role in recent striking characterizations of manifolds (5, 9, 12, 18, 19, 22). Also considered are the property Δ of Borsuk(2) (which ensures an abundance of dimension-preserving maps), and the vanishing of local homology groups up to a given dimension (cf. (9)). Our main results give relations between these properties, and clarify their behaviour under the stabilization operation of taking cartesian product with the real line. In the last section these results are applied to give partial solutions to questions about homogeneous ANR's.

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.

The Space of Harmonic Maps from the 2-Sphere to the Complex Projective Plane

1997 ◽  
Vol 40 (3) ◽  
pp. 285-295 ◽  
Author(s):  
T. Arleigh Crawford
Keyword(s):  
Projective Plane ◽  
Harmonic Maps ◽  
Complex Manifolds ◽  
Complex Projective Plane ◽  
Fixed Degree ◽  
Complex Projective ◽  
Path Connected

AbstractIn this paper we study the topology of the space of harmonic maps from S2 to ℂℙ2.We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to ℂℙn for n ≥ 2. We show that the components of maps to ℂℙ2 are complex manifolds.

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.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

2013 ◽  
Vol 24 (02) ◽  
pp. 1350017
Author(s):  
A. MUHAMMED ULUDAĞ ◽  
CELAL CEM SARIOĞLU
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Free Product ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Galois Covering ◽  
Cyclic Groups ◽  
Galois Coverings ◽  
Complex Projective ◽  
Branching Index

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

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