Remarks on Grothendieck's standard conjectures (Q2906513)

The author shows that Grothendieck's standard conjectures over a field of characteristic zero follow from either of two other motivic conjectures.NEWLINENEWLINEThe first conjecture is the one on the existence of a motivic \(t\)-structure. Let \(k\) be any field, and \(DM_k\) be the triangulated category of geometric motives with \({\mathbb Q}\)-coefficients over \(k\). It is an idempotently complete triangulated rigid tensor \({\mathbb Q}\)-category. Denote by \(\mathrm{Vec}_{{\mathbb Q}}\) and \(\mathrm{Vec}_{{\mathbb Q}_{\ell}}\) the categories of finite-dimensional \({\mathbb Q}\)- and \({\mathbb Q}_{\ell}\)-vector spaces, respectively. For \(\ell\) a prime different from the characteristic of \(k\), there is the \(\ell\)-adic realization functor \(r_{\ell}: DM_k \to D^b(\mathrm{Vec}_{{\mathbb Q}_{\ell}})\). For \(k\) of characteristic zero, each embedding \(\iota:k \hookrightarrow {\mathbb C}\) yields the Betti realization functor \(r_{\iota}:DM_k \to D^b(\mathrm{Vec}_{\mathbb Q})\). Let \(r\) be either of these realization functors. For a \(t\)-structure \(\mu\) on \(DM_k\), let \(\mathcal{M}_k\) be its heart and \(^{\mu}H: DM_k \to \mathcal{M}_k\) be the cohomology functor. The \(t\)-structure \(\mu\) is called \textit{motivic} if it is non-degenerate and compatible with \(\otimes\) and \(r\). It is a conjecture that a motivic \(t\)-structure exists. The author shows that the existence of a motivic \(\mu\) would imply the Lefschetz and Künneth type standard conjectures.NEWLINENEWLINEThe second conjecture is a weak version of Suslin's homology conjecture. For a smooth projective complex variety \(X\), let \(C_r(X)\) be the topological monoid of effective \(r\)-cycles on \(X\) and \(C_r(X)^+\) be its group completion. The Lawson homology groups of \(X\) are defined as \(L_rH_{2r+i}(X, {\mathbb Z}):=\pi_i(C_r(X)^+)\).NEWLINENEWLINEIn connection with results of \textit{E. M. Friedlander} [Ann. Sci. ENS 28, No. 3, 317--343 (1995; Zbl 0854.14006)], the author proves the surprising fact that Grothendieck's standard conjectures over fields of characteristic zero are equivalent to the surjectivity of the natural map from Lawson to singular homology NEWLINE\[NEWLINEL_rH_a(X) \to H_a(X, {\mathbb Z})NEWLINE\]NEWLINE for all smooth projective complex varieties \(X\) and all \(a \geq \mathrm{dim}\,X+r\). In fact, Suslin conjectures that this map is even an isomorphism.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].