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RATIONAL ISOMORPHISMS BETWEEN KTHEORIES AND COHOMOLOGY THEORIES
"... The well known isomorphism relating the rational algebraic Ktheory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the “Segre map”) of infinite loop spaces. Moreover, the associated Chern character map on rati ..."
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The well known isomorphism relating the rational algebraic Ktheory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the “Segre map”) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced which establishes a useful general criterion for a natural transformation of functors on quasiprojective complex varieties to induce a homotopy equivalence of semitopological singular complexes. Since semitopological Ktheory and morphic cohomology can be formulated as the semitopological singular complexes associated to Ktheory and motivic cohomology, this criterion provides a rational isomorphism between the semitopological Ktheory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a RiemannRoch theorem for the Chern character on semitopological Ktheory and an interpretation of the “topological filtration” on singular cohomology groups in Ktheoretic terms.
SEMITOPOLOGICAL KTHEORY OF REAL VARIETIES
"... The semitopological Ktheory of real varieties, KRsemi (−), is an oriented multiplicative (generalized) cohomology theory which extends the authors ’ earlier theory, Ksemi (−), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpene ..."
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The semitopological Ktheory of real varieties, KRsemi (−), is an oriented multiplicative (generalized) cohomology theory which extends the authors ’ earlier theory, Ksemi (−), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpened to real semitopological equivalence), consideration of Z/2equivariant mapping spaces of morphisms of algebraic varieties to Grassmannian varieties, and consideration of the algebraic Ktheory of real varieties. The authors verify that the semitopological Ktheory of a real variety X interpolates between the algebraic Ktheory of X and Atiyah’s Real Ktheory of the associated Real space of complex points, XR(C). The resulting natural maps of spectra K alg (X) → KR semi (X) → KRtop(XR(C)) satisfy numerous good properties: the first map is a modn equivalence for any projective real variety and any n> 0; the second map is an equivalence for smooth projective curves and flag varieties; the triple fits in a commutative diagram of spectra mapping via total Segre classes to a triple of cohomology theories. The authors also establish results for the semitopological Ktheory of real varieties, such as Nisnevich excision and a type of localization result, which were previously unknown even for complex varieties.
HOLOMORPHIC K THEORY, ALGEBRAIC COCYCLES, AND LOOP GROUPS
"... Abstract. In this paper we study the “holomorphic Ktheory ” of a projective variety. This K theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a re ..."
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Abstract. In this paper we study the “holomorphic Ktheory ” of a projective variety. This K theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a related theory was considered in [11]. This theory is built out of studying algebraic bundles over a variety up to “algebraic equivalence”. In this paper we will give calculations of this theory for “flag like varieties ” which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors in [6], we will show that there is a rational isomorphism of graded rings between holomorphic K theory and the appropriate “morphic cohomology ” groups, defined in [7] in terms of algebraic cocycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the “symmetrized loop group ” ΩU(n)/Σn where the symmetric group Σn acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K theory by inverting the Bott class, then rationally this is isomorphic to topological K theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K theory, and show that these obstructions vanish for generalized flag manifolds.