Calculating topological $K(X)$ for complex projective manifolds

Question

In the introduction to the book Vector bundles and K-theory

http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html

two approaches to classification of (topological) vector bundles are discussed - the topological $K$-theory and Characteristic classes. Having algebraic geometry in mind, characteristic classes are omni-present when one deals with complex projective manifolds (Grothendick-Riemann-Roch...). One computes them quite often. However I don't seem to remember calculations of the whole ring $K(X)$ in books on algebraic geometry. So I wonder, do people compute (the topological) $K(X)$ for complex projective manifolds? And if they do, in which circumstances and for which purpose?

Answer 1

There is an Atiyah-Hirzebruch spectral sequence $H^*(X;\mathbb{Z}[u,u^{-1}])\Longrightarrow K^*(X)$, in which the differentials are always torsion-valued. Thus, if $H^*(X;\mathbb{Z})$ is torsion-free, then the spectral sequence collapses, and $K^*(X)$ has a natural filtration whose associated graded ring is isomorphic to $H^*(X;\mathbb{Z}[u,u^{-1}])$. There are many complex projective varieties $X$ to which this applies. Moreover, in these cases it is usually not hard to find explicit generators of the $K$-theory. In particular, any complex vector bundle $V$ over $X$ has Chern classes $c^H_i(V)\in H^{2i}(X;\mathbb{Z})$ and also Chern classes $c^K_i(V)\in K^{2i}(X)$. If we have a basis for $H^*(X;\mathbb{Z})$ consisting of monomials in cohomological Chern classes, then the corresponding monomials in $K$-theory Chern classes will give a basis for $K^*(X)$ over $\mathbb{Z}[u,u^{-1}]$.