\(V\)-filtrations and minimal exponents for local complete intersections (Q6547190)

Let \(X\) be a smooth and irreducible complex algebraic variety. For a hypersurface \(Z\) inside \(X\), its \textit{minimal exponent} \(\tilde{\alpha}(Z)\) is an important singularity invariant introduced by \textit{M. Saito} [Bull. Soc. Math. Fr. 122, No. 2, 163--184 (1994; Zbl 0810.32004)]. By the results of Kollár and Lichtin, it is related to log canonical threshold as \(\textrm{lct}(X,Z)=\min\left\{\tilde{\alpha}(Z),1\right\}\). Let us also recall its connection with the singularity level. For arbitrary closed subscheme \(Z\) of \(X\), the local cohomology sheaves \(\mathcal{H}_Z^q(\mathcal{O}_X)\) underlie mixed Hodge modules in Saito's theory, in particular they are endowed with the Hodge filtration \(F_{\bullet}\). They also carry an Order filtration \(O_{\bullet}\). When \(Z\) is a reduced hypersurface, the only non-vanishing local cohomoloy is \(\mathcal{H}_Z^1(\mathcal{O}_X)\), in this case \(F_k\mathcal{H}_Z^1(\mathcal{O}_X)\subseteq O_k\mathcal{H}_Z^1(\mathcal{O}_X)\) for \(k\geq 0\), and Saito showed that \[F_k\mathcal{H}_Z^1(\mathcal{O}_X)=O_k\mathcal{H}_Z^1(\mathcal{O}_X) \textrm{ for all }k\leq p\iff\tilde{\alpha}(Z)\geq p+1.\]\N\NIn this article, the authors introduce and study a generalization of the minimal exponent when \(Z\) is a local complete intersection in \(X\). Given a local complete intersection \(Z\) defined by \(f_1,\dots,f_r\in \mathcal{O}_X(X)\), the authors consider the graph embedding \(i:X\to X\times\mathbf{A}^r\) associated to \(\mathbf{f}=(f_1,\dots,f_r)\). The definition of \(\tilde{\alpha}(Z)\) is through the \(V\)-filtration \((V^{\gamma}B_{\mathbf{f}})_{\gamma\in\mathbf{Q}}\) on \(B_{\mathbf{f}}:=\bigoplus_{\beta\in\mathbf{Z}_{\geq 0}^r}\mathcal{O}_X\partial_t^{\beta}\delta_{\mathbf{f}}\). \N\NIt follows that, as in the hypersurface case, \(\textrm{lct}(X,Z)=\min\left\{\tilde{\alpha}(Z),r\right\}\). Moreover, if we write \(Y=X\times \mathbf{A}^r\) with coordinates \(y_1,\dots,y_r\) on \(\mathbf{A}^r\), then Theorem 1.1 of the article shows that \(\tilde{\alpha}(Z)=\tilde{\alpha}(g|_U)\) where \(g=\sum f_iy_i\in\mathcal{O}_Y(Y)\) and \(U=X\times \left(\mathbf{A}^r\backslash\left\{0\right\}\right)\). Theorem 1.2 proves various properties of a local version of this newly defined minimal exponent. Further, if we define the singularity level as \[p(Z)=\textrm{sup}\left\{k\geq 0\mid F_k\mathcal{H}_Z^r(\mathcal{O}_X)=O_k\mathcal{H}_Z^r(\mathcal{O}_X)\right\}\] with the convention that this is \(-1\) if the set is empty, then Theorem 1.3 shows that for a local complete intersection closed subscheme \(Z\) of pure codimension \(r\), one has \(p(Z)=\max\left\{\lfloor \tilde{\alpha}(Z)\rfloor-r,-1\right\}\). Theorem 1.4 gives a description of the Hodge filtration on \(\mathcal{H}_Z^r(\mathcal{O}_X)\) using the \(V\)-filtration on \(B_{\mathbf{f}}\). The article also proves a conjecture of \textit{M. Mustaţă} and \textit{M. Popa} [Forum Math. Pi 10, Paper No. e22, 58 p. (2022; Zbl 1511.14010)] to show that for \(Z\) as above, it has rational singularities if and only if \(\tilde{\alpha}(Z)>r\), in particular \(F_1\mathcal{H}_Z^r(\mathcal{O}_X)=O_1\mathcal{H}_Z^r(\mathcal{O}_X)\) implies \(Z\) has rational singularities.\N\NFinally, the authors ask if for a local complete intersection \(Z\), we have \(\tilde{\alpha}(Z)=\tilde{\gamma}(Z)\) where \(\tilde{\gamma}\) is the negative of the largest root of the reduced Bernstein--Sato polynomial of \(Z\) (an affirmative answer to which would prove another conjecture of Mustaţă-Popa). This question has been subsequently answered in the affirmative by the second named author [\textit{B. Dirks}, ``Some applications of microlocalization for local complete intersection subvarieties'', Preprint, \url{arXiv:2310.15277}].