Self-correspondences of the generic fibre of Lefschetz pencils and the Leray filtration
type: Item
instance of: scholarly article
title: Self-correspondences of the generic fibre of Lefschetz pencils and the Leray filtration
zbMATH Open document ID: 1213.14013
author: José J. Ramón Marí
published in: Advances in Mathematics
publication date: 2010-07-20T00:00:00Z
review text: 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 \…
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 5758026
zbMATH Keywords: standard conjectures
zbMATH Keywords: algebraic cycles
zbMATH Keywords: motives
MaRDI profile type: Publication
full work available at URL: j.aim.2009.07.012
OpenAlex ID: W2019819786
cites work: For an unconditional theory of motives
cites work: Q3941610
cites work: Smoothness, semi-stability and alterations
cites work: La conjecture de Weil. I
cites work: Q4039844
cites work: Q3328651
cites work: Q5598855
cites work: Q3313974
cites work: Motives, numerical equivalence, and semi-simplicity
cites work: Some consequences of the Riemann hypothesis for varieties over finite fields
cites work: Q5595004
cites work: Q4297047
cites work: Q3871006
cites work: Q4297057
DOI: 10.1016/J.AIM.2009.07.012
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description: scientific article; zbMATH DE number 5758026