The higher \(K\)-theory of a complex surface (Q2769560)

The authors investigate the algebraic \(K\)-groups \(K_n(F)\) of the function field of a smooth complex variety \(X\) of dimension at most 2 and prove that the \(K\)-groups of \(F\) are divisible above the dimension of \(X\) and that the \(K\)-groups of \(X\) are divisible-by-finite. NEWLINENEWLINEThe first main result of the paper (Th. 4.8) describe the structure of the abelian group \(K_n(F)\) at least for transcendental degree \(d \leq 2\). The authors prove that \(K_n(F)\) is divisible for all \(n>d\) and determine its torsion subgroup. NEWLINENEWLINEThe other main results of the authors on curves and surfaces (Th. 3.2 \& 6.6) state that for \(n>\) dim(\(X\)) the group \(K_n(X)\) is divisible-by-finite: more precisely that its torsion subgroup is the product of copies of \(\mathbb Q / \mathbb Z\) together with finite summands which are the torsion subgroups of the Betti cohomology groups \(H^*(X,\mathbb Z)\). Moreover this last result holds for all \(n>0\) when \(X\) is proper. As a consequence, if \(X\) is a curve, then \(K_n(X)\) is a divisible group for \(n \geq 2\), and so for \(n=1\) if \(X\) is proper. NEWLINENEWLINEThe paper is organized as follows: Section 1 is devoted to higher Chow groups \(CH^i(F,n)\); Section 2 describes the \(K\)-theory with coefficients and prove the results announced by [\textit{A. Suslin}, Algebraic \(K\)-theory and motivic cohomology, in Proc 94 ICM Zürich, Birkhäuser 342--351 (1995; Zbl 0841.19003)] Section 3 concludes this first part of the article by describing the groups \(K_n(Y)\) for smooth curves \(Y\). The last three sections focus on a smooth surface \(X\) and its function field \(F\): Section 4 introduces Chern classes with value in Deligne-Beilinson cohomology and gives the structure of the groups \(K_n(F)\); then Section 5 applies these results to the \(K\)-cohomology of \(X\); and Section 6 concludes the article by the description of the groups \(K_n(X)\).