Some boundedness results for zero-cycles on surfaces (Q914765)

Let X be a smooth complex projective algebraic surface, and let \(A_ 0(X)\) be the Chow group of zero-cycles of degree 0 on X (modulo rational equivalence). Also, denote by T(X), the kernel of the Albanese map \(\Phi:\;A_ 0(X)\to Alb(X)\) and \(Pic^ 0(X)\) the Picard variety of X (with Lie algebra \(H^{0,1}(X))\). Consider the following schema setting: \[ \begin{alignedat}{4} H^{0,1}(X) & \otimes_{\mathbb{C}}H^{0,1}(X)\overset\wedge\rightarrow &&H^{0,2}(X)\rightarrow && V^{0,2}\rightarrow && 0\\ & \updownarrow && \updownarrow && \updownarrow \\ Pic^ 0(X) & \oplus_{\mathbb{Z}}Pic^ 0(X)\overset\cap\rightarrow && T(X)\rightarrow && Coker\rightarrow && 0 \end{alignedat} \] where \(V^{0,2}\) and Coker are the respective cokernels. We prove: Theorem: (1) Suppose \(V^{0,2}\neq 0\). Then Coker is ``infinite'' dimensional. Conversely, (2) if Kodaira dimension \(\kappa\) (X)\(\leq 1\), then \(Co\ker =0\) if \(V^{0,2}=0.\) One expects the following to hold: Conjecture: For all smooth surfaces X, \(V^{0,2}=0\) iff \(Co\ker =0\).