English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper we study the "holomorphic K-theory" of a projective variety,which is defined in terms of the homotopy type of spaces of holomorphic mapsfrom the variety to Grassmannians and loop groups. This theory was introducedby Lawson, Lima-Filho and Michelsohn, and also by Friedlander and Walker, and arelated theory was considered by Karoubi. Using the Chern character studied bythe authors in a companion paper, we give a rational isomorphism betweenholomorphic $K$-theory and the "morphic cohomology" defined by Lawson andFriedlander. In doing so, we describe a geometric model for rational morphiccohomology groups in terms of algebraic maps from the variety to the"symmetrized loop group" Loops(U(n)/S_n) where the symmetric group S_n acts onU(n) via conjugation. This is equivalent to studying algebraic maps to thequotient of the infinite Grassmannians BU(k) by a similar symmetric groupaction. We then prove a conjecture of Friedlander and Walker stating that ifone localizes holomorphic K-theory by inverting the Bott class, then it isrationally isomorphic to topological K-theory. Finally we produce explicitobstructions to periodicity in holomorphic K-theory, and show that theseobstructions vanish for generalized flag manifolds.

Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups