The space v may be a euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings. This paper will presume a fair understanding of the basic ideas of algebraic topology, including homotopy theory and co. Thus in the axiomatic development of the theory, the notions of distance and angle will play no part. An algebraic k3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. Grothendieck group of finitely generated projective rmodules for a com mutative ring r if. Background information and notation are given in chapter1. Topological ktheory of complex projective spaces uc davis. Analytic projective geometry mathematical association of. The most fundamental result concerns the case of projective space and states that there is an isomorphism k. We will refer to k r as the algebraic k theory space of r. Then the singular simplicial set singx is a kan complex, and. Projective geometry is a branch of mathematics which deals with the properties and invariants of geometric. By combining the atiyahhirzebruch spectral sequence and k or ko theory, we get a proof of a property of projective spaces called james.
If x is a smooth complex projective variety, then i k x. Equivariant quantum differential equation, stokes bases, and k theory for a projective space. One can define a projective space axiomatically in terms of an incidence structure a set of points p, lines l, and an incidence relation i specifying which points lie on which lines satisfying certain axioms an automorphism of a projective space thus defined then being an automorphism f of the set of points and an automorphism g of the. For a complex projective variety x, the group zpx has additional natural structure. The algebraic ktheory of spaces lecture 21 october 22, 2014 let x be a topological space. More restrictively, a compact complex manifold is called projective if it embeds holomorphically as a subvariety of complex projective space. K theory of perfect complexes contravariant functoriality theorem poincar. To see why this space has some interesting properties as an abstract manifold, we start by examining the real projective plane, rp2.
H spaces, division algebras, and tangent bundles of spheres. The k theory, in the sense of c algebras, of the space of continuous sections of this bundle, written k 0 x. Beltrametti and others published introduction to the theory of weighted projective spaces find, read and cite all the research you need on researchgate. The bordism ring mutcpa is central to the theory of formal groups as applied by d. According to the chow theorem, there is a canonical way, by means of chow forms, to give cp.
Pdf riemannroch and topological k theory for singular. Citeseerx equivariant ktheory of real vector spaces and. Let us come back now to the general theory k x of grothendieck where, for simplicity, we assume x to be a complex projective variety. A minimal, smooth, and projective surface over complex numbers satis. One purpose of this paper is to fill this gap, at least for real vector spaces considered as vector. Equivariant algebraic ktheory northeastern university. Like for example, just being able to write down a resolution by powers of twisting sheaves. Equivariant quantum differential equation, stokes bases. Using k theory, adams and atiyah were able to give a simple proof that the only spheres which can be provided with h space structures are s1, s3 and s7. Let v be a nitedimensional vector space over a eld kand denote the dual space homv. In 1988 i wrote out a brief outline, following quillens paper higher algebraic ktheory i. Gx, the category of gequivariant vector bundles on a topological space x. Pdf introduction to the theory of weighted projective spaces.
We will now investigate these additional points in detail. Bennett is professor of mathematics at the university of massachusetts, amherst, where she earned her phd in 1966. Examples lines are hyperplanes of p2 and they form a projective space of dimension 2. Citeseerx document details isaac councill, lee giles, pradeep teregowda. In mathematics, a complex analytic k3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. The perspective camera model uses the algebra of the projective space to describe the way in which 3d space points are mapped into an image plane. Hirzebruch, in 1959 studied k0cwhere c is the category vectc xof finite dimensional complex vector bundles over a compact space x yielding what became known as topological k theory. In the 1960s and 70s, there was some transatlantic controversy over whether the projective space associated. The aim of the present paper is to generalise this splitting result to arbitrary projective toric. We recall that the baumfultonmacpherson class of a complex projective variety v is the k. Some of the theorems which we shall find, have of course been known for a.
Since equivariant k theory on a space with free gaction is the same as the k theory of the quotient, we have the following consequence of the completion theorem. Then the singular simplicial set sing x is a kan complex, and in particular an 1category. The remarkable fact, proved by rost, is that for smooth varieties over a eld, the obvious relations are enough, i. However, prior to the development of etale cohomology. The projective space pn thus contains more points than the a. Ktheory of the infinite dimensional projective space. Well known restrictions stem from hodge theory, see 2 for various results. Unfortunately, this is caused by preculiarities in the cohomology of real projective spaces and not a generic feature. Informally, \ k \ theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebraic and geometric questions. Chapter3concerns the equivariant quantum k theory of projective space pn and the demonstrations of theoremsd,e,f. Some formulas in operator theory, involving determinants, are best understood in terms of algebraic ktheory.
A vector space can be viewed, according to ones predilections, either as a module. We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. In geometry, a hyperplane of an ndimensional space v is a subspace of dimension n. Exact sequence spectral sequence cohomology theory chern character real projective space. These two theorems give nontrivial constraints on the possible homotopy type of compact k ahler manifolds and smooth complex projective varieties. For simplicity, we will restrict the category of topological spaces considered to. Some basics of algebraic ktheory in this paper, we.
Notes for the masterclass, strasbourg, february 2015. Introduction a basic result in the k theory of algebraic varieties is the computation of the k groups for projective space bundles 2, 5, 11. Therefore, by vector space, we shall always understand complex vector space unless otherwise specified. C cohx, the category of coherent sheaves on an algebraic variety x. The set of subspaces of pn with the same dimension is also a projective space. The equivariant cohomology ring of weighted projective space. Because k0cclassifies the isomorphism classes in c and he wanted the name of the theory to reflect class, he used the first letter k in. Algebraic ktheory plays an important role in many subjects, especially number theory, algebraic topology and algebraic geometry. The nonabelian hodge theory due to simpson says \the space of local systems is equal to the space of higgs bundles. As an application, we develop an intersection theory for quantum ruled surfaces. It is the main purpose of this paper to bring up the general projective theory of curves, based upon the theory of invariants, to the same level of perfection as the corresponding theory of ruled surfaces.
By using homogeneous coordinates the geometry of image formation is simply defined by the projection of a 3d point into the plane by one special type of homography known as a projection. Fora systematic treatment of projective geometry, we recommend berger 3, 4, samuel. In the enriqueskodaira classification of surfaces, k3 surfaces form one of the four classes of. In this paper we study the \holomorphic k theory of a projective varie. She was a john wesley young postdoctoral research fellow at dartmouth college, has authored numerous research articles on lattice theory, geometry, and quantum logics and has lectured on her work around the globe.
K vect gx is known as equivariant topological k theory. Some results publications and preprints \a formula for quantum k theory of projective space in preparation expected submission. A geometric introduction to ktheory university of oregon. Both methods have their importance, but thesecond is more natural. Pdf on the bordism ring of complex projective space. X ii a finite dimensional vector space structure on eac. It is known that k theory of homogeneous projective varieties could be expressed by means of k theory of central simple algebras. Topological ktheory has become an important tool in topology. A family of vector spaces over x is a topological space e, together with. If kq g x is nitely generated as a module over rg, then k \ gx. A homology and cohomology theory for real projective varieties. Much is known in this direction, for example the topological and holomorphic kgroups of all complex projective spaces are isomorphic. I talked to hy bass, the author of the classic book algebraic ktheory, about what would be involved in writing such a book. Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry.
When the ground eld k c, bezouts theorem can be proved using integral cohomology. We compute the k theory of complex projective spaces. We deduce that the ring is a perfect invariant, and prove a chern class formula for weighted projective bundles. Chpx on which the intersection theory of algebraic varieties can be built 9. Another natural example is the topological k theory of a compact space. V of the thom space of a real vector bundle has been done successfully only under some spinoriality conditions 1, thanks to a clever use of the atiyahsinger index theorem even if g is a finite group. Adams operations 17 acknowledgments 19 references 19 1. Grotherdieck who first studied k0c then written k c where for a scheme x, c is the category px of locally free sheaves of oxmodules. A subscheme of projective space is determined by its equations.
Wolfgang boehm, hartmut prautzsch, in handbook of computer aided geometric design, 2002. This is called algebraic k theory if we wish to generalize this. Examples projective space bundle formula chern classes and higher chern classes 9. For instance, the class group of a number eld kis essentially k0ok.
Lectures on ktheory international nuclear information system. One source for projective geometry was indeed the theory. Ill give an application at the very end of the post which relates the k theory of projective space to the g theory of projective space. Introducing points at infinity leads to the projective space and allows a unified and most elegant treatment of geometry 4 2. Some of the theorems which we shall find, have of course been known for a long time. This is called algebraic k theory if we wish to generalize this last example to the equivariant setting, we have to be. Conformal field theory on ddimensional real projective space. We compute the ktheory of complex projective spaces. M bx is nite union of torsion translates of subtori, for any i.
The atiyahhirzebruch spectral sequence in ktheory springerlink. Moreover, it is possible to derive a substantial part of stable homotopy theory from ktheory. Given a compact hausdorff space consider the set of isomorphism classes of finitedimensional vector bundles over, denoted and let the isomorphism class of a vector bundle. A projective manifold is also a kahler manifold, and its fundamental group is called a projective kahler group. Since cohomology is known for many spaces, with this. Sep 24, 2017 for a long time this was really confusing to me but, it turns out it really didnt need to be. However, one of the most important examples of the theory is the real pro. In the isolated case, i am pretty sure that the map from algebraic k theory to topological k theory is an isomorphism integrally but i havent checked the details of this. Equivariant quantum differential equation, stokes bases, and.
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