We embed this torus into the complex projective plane c geometry. A theorem of ilyashenko states that except maybe for a residual set, all foliations which are topologically conjugate are in fact analytically thus homographically conjugate, that is, the generic. The theory of line arrangements is a classical and rich subject of studies with many deep results with applications and impact in numerous branches of mathematics. The fubinistudy metric gon cp2 is k ahler with the k ahler form k i 2 g a b dz adz b. The trajectories of in the real plane are then the intersection of these complex curves with the plane im ximy0. Graeme segal, the stable homotopy of complex of projective space, the quarterly. Example of a topologically nonrigid foliation of the complex. The projective plane p2 is the set of lines through an observation point oin three dimensional space. A quadrangle is a set of four points, no three of which are collinear. Suppose the sphere represents kfi, where p g h2m is a generator. A classical result asserts that the complex projective plane modulo complex conjugation is the 4dimensional sphere. Group actions on the complex projective plane 709 proof.
Conicline arrangements in the complex projective plane. In mathematics, complex projective space is the projective space with respect to the field of complex numbers. Fora systematic treatment of projective geometry, we recommend berger 3, 4, samuel. We have presented this formula in a more precise form in a, showing an explicit relation with the infinitely near points.
We denote the rst chern class of the dual of the hyperplane bundle overr cp2 by h k. The original proof of existence, due to kuhnel, as well as the original proof of uniqueness, due to kuhnel and lassmann, were based on. The complex projective line cp1 for purposes of complex analysis, a better description of a onepoint compacti cation of c is an instance of the complex projective space cpn, a. The projective space associated to r3 is called the projective plane p2. Surfaces in the complex projective plane and their mapping. We study a special family of such critical points, p k. More generally, if a line and all its points are removed from a. When v r2, the projective space is the projective line p1 r. Axiom ap2 for the real plane is an equivalent form of euclids parallel postulate. Any two lines l, m intersect in at least one point, denoted lm. We find obstructions to the existence of topologically locally flat spheres realizing a.
Request pdf surveying points in the complex projective plane we classify sicpovms of rank one in cp2, or equivalently sets of nine equallyspaced points in cp2, without the assumption of. The points at in nity in this case are lines in c2. Table of contents introduction 1 the projective plane. To see why this space has some interesting properties as an abstract manifold, we start by examining the real projective plane, rp2. Riemannian metric in the projective space mathematics. Dynamics of singular holomorphic foliations on the complex. In this article we describe such a triangulation for one of the most significant objects in topology, the complex projective plane. From the perspective of real numbers already the complex plane c is a two. It is the study of geometric properties that are invariant with respect to projective transformations. It is known that any nonsingular rational variety is obtained from the plane by a sequence of blowing up transformations and their inverses blowing down of curves, which must be of a very particular type. Complex projective space the complex projective space cpn is the most important compact complex manifold. When k r, our intuition is that the real projective line p2r is an ordinary line with a point at in nity identifying its opposite directions, and the projective plane is an ordinary plane surrounded by a circle at in nity identifying its opposite directions.
Note that for any given line in c3 passing through the origin, we can identify a point x 1. In mathematics, the complex projective plane, usually denoted p 2 c, is the twodimensional complex projective space. Projective geometry in a plane fundamental concepts undefined concepts. In the early days of topology, most of the objects of gether. Pdf conicline arrangements in the complex projective plane. One can easily see that this foliation extends to the complex projective plane cp2, which is obtained by adding a line at in nity to the plane c2. The points at in nity in this case are lines in c2 that are parallel to reference plane. Algebra and geometry through projective spaces department of. A projective line lis a plane passing through o, and a projective point p is a line passing through o. Master mosig introduction to projective geometry a b c a b c r r r figure 2.
S2, and that the real projective space rp3 is homeomorphic to the group of. Such critical points arise in the calculation of a metric invariant called the filling radius, and are akin to the critical points of the distance function. Example of a topologically nonrigid foliation of the. Riemann sphere, projective space november 22, 2014 2.
The integer q is called the order of the projective plane. F we usually call this extra point 0,1 the point when f c, the complex numbers, the projective line is what is called in. In this paper we are going to apply these results to the theory of line arrangements in the complex projective. For any field f, the projective plane p2f is the set of equivalence classes of nonzero points in f3. There exists a projective plane of order n for some positive integer n. The space of all such foliations with p and q of a fixed degree d is a complex projective space of finite dimension, endowed with the natural topology. And lines on f meeting on m will be mapped onto parallel lines on c. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts.
Complex projective varieties where the corrections concerned the wiping out of some misprints, inconsistent notations, and other slight inaccuracies. Riemannian metric in the projective space mathematics stack. It is denoted or alternatively, it can be viewed as the quotient of the space under the action of by multiplication. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. We explain what it means for polynomials to be \a ne equivalent. In 1976, the author published the first volume under the title lgebraic geometry.
The structure jacobi operator for real hypersurfaces in the complex projective plane and the complex hyperbolic plane kurihara, hiroyuki, tsukuba journal of mathematics, 2011 real hypersurfaces some of whose geodesics are plane curves in nonflat complex space forms adachi, toshiaki, kimura, makoto, and maeda, sadahiro, tohoku mathematical. The projection of v r3 is the projective plane p2 r. We present an elementary combinatorial proof of the existence and uniqueness of the 9vertex triangulation of. The latter result is a direct consequence of the uniformization. By analogy, whereas the points of a real projective space label the lines through the origin of a real euclidean space, the points of a complex projective space label the complex lines through the origin of a complex euclidean space see below for an intuitive account. The complex projective plane is the complex projective space of complex dimension 2. In the case k r we can also think of it as a compacti. By passing to a power of g, if necessary, we can assume that n is a prime number.
Instead, let p2cbe one of the rami cation points, and suppose. When n 1, the complex projective space cp 1 is the riemann sphere, and when n 2, cp 2 is the complex projective plane see there for a more. Surveying points in the complex projective plane request pdf. The basic intuitions are that projective space has more points than euclidean space. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the. The 9vertex complex projective plane brown university. Pdf almost complex rigidity of the complex projective plane. Pyramids in the complex projective plane springerlink. Pdf on k hnels 9vertex complex projective plane basudeb. The closure in p2 wont serve the purpose, since it is singular for g 2. Pdf on kuhnels 9vertex complex projective plane basudeb. To rst approximation, a projective variety is the locus of zeroes of a system of homogeneous polynomials. Pdf projective planes, severi varieties and spheres.
Similarly, consider the projective plane p2 and let x 0 1 be a complex reference plane not passing through the origin. One may observe that in a real picture the horizon bisects the canvas, and projective plane. Alternatively, it can be viewed as the quotient of the space under the action of by multiplication. The trajectories of in the real plane are then the intersection of these complex curves with the. Let y be a smooth compact complex surface given by a. M on f given by the intersection with a plane through o parallel to c, will have no image on c. It is also, of course, the unique steiner triple system of order 7. Mosher, some stable homotopy of complex projective space, topology 7 1968, 179193. That is, these are homogeneous coordinates in the traditional. The projective space pv is the set of lines passing through the origin of a vector space v. The smallest projective plane has order 2 see figure 1. For example the hirzebruch inequality 2, 11, 12 appreciated very much in combinatorics is motivated by problems in algebraic geometry. We study the critical points of the diameter functional. A theorem of hacking and prokhorov 9, 10 asserts that if x is a projective algebraic surface with quotient singularities which admits a qgorenstein smoothing to the complex projective plane cp2, then x is obtained by partially smoothing a weighted projective plane cpp2 1,p 2 2,p 2 3 for a markov triple p1,p2,p3.
It is a complex manifold of complex dimension 2, described by three complex coordinates. In birational geometry, a complex rational surface is any algebraic surface birationally equivalent to the complex projective plane. It is called playfairs axiom, although it was stated explicitly by proclus. We can understand projective planes based on equivalence classes and homogeneous coordinates.
The complex projective line cp1 for purposes of complex analysis, a better description of a onepoint compacti cation of c is an instance of the complex projective space cpn, a compact space containing cn, described as follows. Description of cells and attaching maps there is one cell in dimension. This article discusses a common choice of cw structure for real projective space, i. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes. It is the goal of this chapter to study this process in detail, leading to plane curves that are compacti. Complex ball quotients and line arrangements in the. Pdf the main goal of the paper is to begin a systematic study on conicline arrangements in the complex projective plane. We introduce the general projective space rpn, but focus almost exclusively on rp2.
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