On \(K_1\) and \(K_2\) of algebraic surfaces (Q1431748)

Let \(X\) be a smooth projective variety over a field \(k\). Bloch's Higher Chow groups \(\text{CH}^p(X,n)\) coincide with the motivic cohomology groups \(H_{\mathcal M}^{2p-n} (X,\mathbb{Z}(p)\) defined by Suslin and Voevodsky and are related to algebraic \(K\)-theory via the isomorphism: \[ K_i(X)^{(j)}\simeq H^{2j-i}\bigl(X,\mathbb{Z}(i) \bigr), \] where \(K_i (X)^{(j)}\) are the eigenspaces for the Adams operations on \(K\)-theory. In particular, when \(X\) is a smooth projective surface, the groups \(\text{CH}^2 (X,1)\) and \(\text{CH}^3(X,2)\) are the most interesting graded pieces respectively of \(K_1(X)\) and \(K_2(X)\). In the case \(k=\mathbb{C}\) several authors (Collino, Müller-Stach, Voisin et al.) have studied the indecomposable part of \(\text{CH}^2 (X,1)\), i.e., the quotient by the image of the map \(\text{CH}^1(X)\otimes \mathbb{C}^* \to \text{CH}^2(X,1)\) modulo torsion. Similarly one can define the indecomposable part of \(\text{CH}^3(X,2)\) to be the quotient of the image of \(\text{Pic}\,X \otimes K_2(C)\) modulo torsion. The aim of the paper is to study these groups in the case \(k= \mathbb{C}\): fix \(Z=\bigcup Z_i\) a simple normal crossing divisor on \(X\) and consider the exact sequence \[ \text{CH}^{r+1} (U,r+1)\to \text{CH}^r(Z,r)\to \text{CH}^{r+1} (X,r). \] Let us define \[ \text{CH}^r(Z,r)^{\text{ind}}= \text{Coker} \bigl(\oplus \text{CH}^r(Z_I,r)\to \text{CH}^r (Z,r)\bigr). \] Then the authors prove the following result, which may be viewed as a generalization of the classical Noether-Lefschetz theorem, which states that a very general surface of degree \(d\geq 4\) in \(\mathbb{P}^3\) has a free Picard group of rank 1, generated by hyperplane section (for \(r=0\) one has \(\text{CH}^1(X,0)= \text{Pic}\,X)\). Theorem 1. Let \(X\subset\mathbb{P}^3\) be a very general hypersurface of degree \(d\) and let \(Z_I=\cup_i Z_i,Z_i\subset X\) be very general hypersurface sections of degree \(e_i\). (i) if \(d\geq 5\) then \(\text{CH}^2(U,2)\to \text{CH}^1(Z,1)^{\text{ind}}\) is surjective; (ii) if \(d\geq 6\) and \((e_i,e_j,e_l)\neq (1,1,2)\) for distinct \(i,j,l\), then \(\text{CH}^3(U,3)\to \text{CH}^2(Z,2)^{\text{ind}}\) is surjective. The paper also contains applications to detect non trivial elements in the indecomposable part of \(\text{CH}^{r+1} (X,r)\) for complete families of surfaces of low degree. A remarkable result in this direction is the following (due to A. Collino). Theorem 2. On a very general quartic \(K3\) surface \(X\) there exists a one-dimensional family \(\alpha_t\) of elements in \(\text{CH}^3(X,2)\), supported on a smooth hyperplane section of \(X\), such that, for \(t\) very general, these elements are indecomposable, modulo the image of \(\text{Pic}\,X\otimes K_2 (\mathbb{C})\).