Constructing indecomposable motivic cohomology classes on algebraic surfaces.

*(English)*Zbl 0910.14017Let \(X\) be a smooth algebraic surface over \(\mathbb{C}\). It is well known that \(K_0(X) \otimes \mathbb{Q} \cong \bigoplus\text{CH}^p (X)\otimes \mathbb{Q}\) as a consequence of Grothendieck’s Riemann-Roch theorem. S. Bloch has generalized this to higher algebraic \(K\)-theory [S. Bloch Adv. Math. 61, 267-304 (1986; Zbl 0608.14004); see also M. Levine, Invent. Math. 91, No. 3, 423-464 (1988; Zbl 0646.14010)]. For example one obtains
\[
K_1(X) \otimes \mathbb{Q}\cong \bigoplus_p\text{CH}^p(X,1) \otimes \mathbb{Q}.
\]
The groups CH\(^p(X,n)\) are called higher Chow groups. The purpose of this paper is to give an explicit way for constructing classes in CH\(^2(X,1)\) that are nontrivial modulo the image of the natural map
\[
\gamma: \text{Pic} (X)\otimes \mathbb{C}^* \to\text{CH}^2(X,1)
\]
and modulo torsion. We call such classes indecomposable. Cycles in CH\(^2(X,1)\) can be constructed by finding curves \(Z_i\) in \(X\) with rational functions \(f_i\) on them that satisfy \(\sum \text{div} (f_i)=0\) on \(X\). By general conjectures (see section 7), the cokernel of \(\gamma\) is expected to be a countable group on any smooth surface. We will construct indecomposable elements in \(CH^2(X,1)\) on general quartic K3-surfaces that contain a line and on some special quintic surfaces of general type. Other examples over the complex numbers have been constructed by M. Nori on abelian surfaces, by A. Collino on Jacobian varieties, and by C. Voisin and C. Oliva on K3 surfaces. Our method consists of deforming the complex structure of the pair \((X,Z)\) and studying the variation of mixed Hodge structures associated to the open complements. The ideas of this technique in the case of ordinary Chow groups are similar to those developed earlier by F. Bardelli and S. Müller-Stach (1991; unpublished) and by C. Voisin [Ann. Sci. Éc. Norm. Supér., IV. Sér. 27, No. 2, 209-226 (1994; Zbl 0808.14030)], but eventually go back to the fundamental idea of Griffiths to show the non-triviality of cohomology classes on the general member of a family of varieties by showing that their derivatives are non-zero. The advantage of our method is that it is not restricted to surfaces with trivial canonical bundle and that it is very simple to apply in situations where some geometry is known, in particular if an explicit family or a degeneration to a singular configuration can be written down.