Self-correspondences of the generic fibre of Lefschetz pencils and the Leray filtration
Let \(X\) be a smooth projective variety of dimension \(n\) over a field \(k\). Let \(L\) be the Lefschetz operator on \(X\), given by a smooth hyperplane section. The Hard-Lefschetz theorem states that the maps \[ L^{n-i}:H^i(X)\to H^{2n-1} \] are isomorphisms, where \(H^*\) are the etale cohomology groups with \(\mathbb Z_l\) coefficients, with \(l\neq\operatorname{char}k\). Let \(\Lambda\) be the Lefschetz operator, i.e., \[ \Lambda x= \sum_{j\geq\max(i-n,1)} L^{j-1}x_J \] where, for \(x\in H^i(X)\), \[ x= \sum_{j\geq(i-n,0)} L^jx_j \] is its primitive decomposition. For \(x=\sum_L^j x_{i-2j}\in H^i(X)\) let \(p_X^kx= \delta_{i,k}x_k\), when \(i\leq n\) and \(p_X^kx= \delta_{i,k}x_{2n-k}\) for \(k>n\). Then \(p_X^i\) is a projector for \(i\leq n\). The Grothendieck's standard conjecture \(B(X)\) states that the operator \(\Lambda\) is algebraic. The standard conjecture \(B(X)\) implies the standard conjecture \(C(X)\), which states the algebraicity of the projectors \(\pi^i(X)\), giving the Künneth decomposition of the diagonal \(\Delta_X\) in \(H^{2n}(X\times X)\). Therefore, under \(C(X)\), \(\pi^i(X)\in CH^n(X\times X)\). \(B(X)\) is known for curves, surfaces, abelian varieties and it is stable under products and hyperplane sections. Under \(C(X)\) the Künneth decomposition of the diagonal yields a Chow-Künneth decomposition of the motive \(h_{\text{rat}}(X)\) in the category \({\mathcal M}_{\text{rat}}(k)\) of Chow motives over \(k\) \[ h_{\text{rat}}(X)= \sum_{0\leq i\leq 2n} h^i(X) \] where \(h_i(X)= (X, \pi^i(X))\). Let \(k=\bar k\) be algebraically closed and let \(X\subset\mathbb P^N\) be a suitable projective embedding, given by a very ample line bundle \({\mathcal L}_X\) Then there exists a line \(L\subset(\mathbb P^N)^\vee\) cutting the dual variety \((X)^\vee\) of \(X\subset\mathbb P^N\) transversally. Let \(\Delta\) be the base locus of \(L\) in \(X\); \(\Delta\) is smooth of dimension \(n-2\). If \(\widetilde{X}\) denotes the blow-up of \(X\) at \(\Delta\) then the projection induces a map \(X-\Delta\to\mathbb P^1\simeq L\) which induces a Lefschetz pencil \[ \rho:\widetilde{X}\to\mathbb P^1. \] The main result of this paper is the following Theorem 1. Let \(X\) be a smooth projective variety of dimension \(n\geq 3\). Assume the conjecture \(B({\mathcal Y})\) for the generic fibre \({\mathcal Y}\) of a Lefschetz pencil of \(X\). Then the operator \(\lambda_X-p_X^{n+1}\) is algebraic.
scientific article; zbMATH DE number 5758026
scientific article