A result on cycles algebraically equivalent to zero (Q678255)

For a complex projective algebraic manifold \(X\), let \(\text{CH}_k(X)_{\text{alg}}\) be the Chow group of algebraic cycles, algebraically equivalent to zero modulo rational equivalence. Also, if \(H=H_\mathbb{Q}\) is a finite-dimensional Hodge structure with Hodge decomposition \(H_\mathbb{C}= \bigoplus_{p,q} H^{p,q}\), we define \(\text{Level}(H)= \max\{p-q\mid H^{p,q}\neq 0\}\) if \(H\neq 0\) and \(-\infty\) if \(H=0\). To describe the main result, we introduce the following setting (all varieties are complex): (i) Let \(\{E_c\}_{c\in\Omega}\) be a flat family of \(k\)-dimensional (irreducible) subvarieties in some \(\mathbb{P}^N\). (ii) Let \(\{X_t\}_{t\in W}\) be a flat family of subvarieties in \(\mathbb{P}^N\), with generic member smooth. (iii) \(P=\{(c,t)\in \Omega\times W\mid E_c\subset X_t\}\), with projections \(\rho:P\to \Omega\), \(\pi:P\to W\). (iv) Assume \(W,\Omega,P\) are smooth varieties, \(\pi,\rho\) are surjective with connected fibers, and that \(\rho\) is a smooth morphism. Also, we will set \(\Omega_{X_t}= \rho(\pi^{-1}(t))\), and let \(\delta= \dim \Omega_{X_t}\) for general \(t\in W\). (v) Fix a closed point \(t_0\in W\), and an integer \(\ell\geq 2\). Assume that there is an irreducible component \(\Omega_{t_0} \subset \Omega_{X_{t_0}}\) of dimension \(m\geq\ell\), with desingularization \(\widetilde{\Omega}_{t_0}\), such that the corresponding cylinder homomorphism \(H_\ell (\widetilde{\Omega}_{t_0},\mathbb{Q})\to W_{-2k-\ell} (X_{t_0},\mathbb{Q})\) has image Hodge level \(\ell\). Finally, assume \(\delta\geq (m-\ell)+1\). Our main result is: Theorem. Assume given the above setting. Then for general \(t\in W\), there are (an uncountable number of) non-torsion classes in \(\text{CH}_k(X_t)_{\text{alg}}\). As an application of the theorem, we are able to deduce the following: Corollary. Let \(X\subset \mathbb{P}^{n+1}\) be a general hypersurface of degree \(d\geq 3\). Assume given positive integers \(d_0,\ell,k\) satisfying: \[ \begin{alignedat}{4} &\text{(i)} &&k=\biggl[{{n+1}\over{d_0}}\biggr] &&\text{(ii)} &&n-2k>2,\\ &\text{(iii)} \quad &&k(n+2-k)+1- {{d_0+k}\choose k}\geq 0, \qquad &&\text{(iv)}\quad &&0\leq {{d+k}\choose k}- {{d_0+k}\choose k}\leq n-2k-1.\end{alignedat} \] Then \(\text{CH}_k(X)_{\text{alg}}\) is uncountable. In particular \(\text{CH}_k(X)_{\text{alg}}\) contains non-torsion classes.