Motivic weightless complex and the relative Artin motive (Q5965070)

In this paper, the author constructs and studies a motivic sheaf \(\mathrm{EM}_X\), the motivic weightless complex, attached to a variety \(X\) over a field.NEWLINENEWLINELet \(X\) be an algebraic variety over a field \(k\), with either \(k=\mathbb{C}\) or \(k\) finite. In [J. Algebra 424, 147--189 (2015; Zbl 1327.14110)], the author and \textit{A. Nair} introduced the weightless complex \(\mathrm{EC}_X\in D^b(X)\), where \(D^b(X)\) denotes a derived category of mixed sheaves on \(X\) (more precisely, the derived category of mixed Hodge modules if \(k=\mathbb{C}\) and the derived category of mixed \(\mathbb{Q}_{\ell}\)-adic complexes if \(k\) is finite). The complex \(\mathrm{EC}_X\) is a ring object and fits in a composition NEWLINE\[NEWLINE \mathbb{Q}_X \rightarrow \mathrm{EC}_X\rightarrow \mathrm{IC}_X NEWLINE\]NEWLINE where \(\mathrm{IC}_X\) is the intersection complex of \(X\). In fact, \(\mathrm{EC}_X\) is a weight truncation of \(\mathrm{IC}_X\), in a sense that can be made precise using the work of \textit{S. Morel} [J. Am. Math. Soc. 21, No. 1, 23--61 (2008; Zbl 1225.11073)].NEWLINENEWLINEThe main goal of the paper under review is to lift the weightless complex to triangulated categories of motivic sheaves. For a variety \(X\) over a (now arbitrary) field \(k\), we have a triangulated category \(\mathrm{DM}(X)\) of constructible mixed motivic sheaves with rational coefficients over \(X\). We refer to the paper for details, and only note that the notation is slightly non-standard; the more usual one would be \(\mathrm{DM}_c(X)\) or \(\mathrm{DM}_{\mathrm{gm}}(X)\). These categories satisfy a version of the six operations formalism, and are related to derived categories of classical coefficient systems and their six operation formalisms via realisation functors (unconditionally in the Betti and \(\ell\)-adic case, partially constructed in the Hodge case).NEWLINENEWLINEWe can now state some of the main results. there exists a ring object \(\mathrm{EM}_X\in \mathrm{DM}(X)\) with the following properties. {\parindent=0.7cm \begin{itemize}\item[--] If \(X\) is smooth, then NEWLINE\[NEWLINE\mathrm{EM}_X=\mathbb{Q}_X.NEWLINE\]NEWLINE \item[--] For \(k\) finite and \(\ell\) is a prime different from \(\mathrm{char}(k)\), write \(r_{\ell}:\mathrm{DM}(X)\rightarrow D^b_m(X_{\bar{k}},\mathbb{Q}_l)\) for the \(\ell\)-adic realisation functor into the derived category of mixed \(\mathbb{Q}_l\)-adic sheaves, and \(\mathrm{EC}_X\) for the weightless complex of \(X\) in this context. Then NEWLINE\[NEWLINEr_{\ell}(\mathrm{EM}_X)=\mathrm{EC}_X.NEWLINE\]NEWLINE \item[--] For \(k=\mathbb{C}\), a similar result holds for the Hodge realisation (conditional on its existence and compatibility with the six operations). \item[--] If the motivic intersection complex \(\mathrm{IM}_X\) of [\textit{J. Wildeshaus}, Contemp. Math. 571, 255--276 (2012; Zbl 1405.19001)] exists, then \(\mathrm{EM}_X\) is a weight truncation of \(\mathrm{IM}_X\). NEWLINENEWLINE\end{itemize}} The construction of \(\mathrm{EM}_X\) is achieved by adapting one of the weight truncation \(t\)-structures of S. Morel alluded to above to the motivic context. More precisely, on a subcategory \(\mathrm{DM}^{\mathrm{coh}}(X)\) of cohomological motives, the author constructs a \(t\)-structure \(( ^{w}\mathrm{DM}^{\leq \mathrm{Id}}, ^w \mathrm{DM}^{<\mathrm{Id}})\). This \(t\)-structure, like the ones introduced by S. Morel, is unusual in that both \( ^{w}\mathrm{DM}^{\leq \mathrm{Id}}, ^w \mathrm{DM}^{<\mathrm{Id}}\) are triangulated, so that there is only one truncation functor \(w_{\leq \mathrm{Id}}:\mathrm{DM}^{\mathrm{coh}}(X)\rightarrow ^{w}\mathrm{DM}^{\leq \mathrm{Id}}\). The key requirement is that, for \(X\) regular and \(f:Y\rightarrow X\) proper and smooth (maybe up to a finite universal homeomomorphism in char. \(p\)) with Stein factorisation \(f:Y\rightarrow st(Y)q{\rightarrow}X\), we have NEWLINE\[NEWLINE w_{\leq\mathrm{Id}}f_*\mathbb{Q}_Y\simeq q_* \mathbb{Q}_{st(Y)}. NEWLINE\]NEWLINE The \(t\)-structure is then obtained by a gluing procedure from this basic case. One then defines NEWLINE\[NEWLINE \mathrm{EM}_X:=w_{\leq \mathrm{Id}} j_*\mathbb{Q}_U NEWLINE\]NEWLINE for \(j:V\rightarrow X\) with \(V\) regular dense open in \(X_{\mathrm{red}}\).NEWLINENEWLINEThis paper is closely related to the work of \textit{J. Ayoub} and \textit{S. Zucker} [Invent. Math. 188, No. 2, 277--427 (2012; Zbl 1242.14016)]. In fact, the author shows that his motive \(\mathrm{EM}_X\) is isomorphic to the motive \(\mathbb{E}_X\) of [loc. cit.], which is defined in a rather different way. The other main result of loc. cit. the following: if \(X/\mathbb{C}\) is a connected Shimura variety, \(\bar{X}^{BB}\) is its Baily-Borel compactification and \(\bar{X}^{RBS}\) is its reductive Borel-Serre compactification (with \(\pi:\bar{X}^{RBS}\rightarrow \bar{X}^{BB}\) the canonical morphism), then there is a canonical isomorphism NEWLINE\[NEWLINE r_{\mathrm{Betti}}(\mathbb{E}_{\bar{X}^{BB}})\simeq \pi_* \bar{X}^{RBS}. NEWLINE\]NEWLINE Conditionally on the existence and properties of the Hodge realisation, this last isomorphism can also be obtained by combining the main theorem of the present paper with the computation of the weightless complex \(\mathrm{IC}_{\bar{X}^{BB}}\) in terms of \(\bar{X}^{RBS}\) in [\textit{A. Nair}, ``Mixed structures in Shimura varieties and automorphic forms'', preprint].