Recent zbMATH articles in MSC 32S40https://zbmath.org/atom/cc/32S402021-06-15T18:09:00+00:00WerkzeugBernstein-Sato varieties and annihilation of powers.https://zbmath.org/1460.140482021-06-15T18:09:00+00:00"Bath, Daniel"https://zbmath.org/authors/?q=ai:bath.danielLet \(X\) be a smooth scheme over \(\mathbb{C}\) or analytic variety of dimension \(n\). Take \(r\) different global functions \(f_1,\ldots,f_r\in\mathcal{O}_X\) and let \(f=f_1\cdots f_r\). Associated with that family of functions, we can consider the \(\mathcal{O}_X[S]\)-module \(\mathcal{O}_X[f^{-1},S]\cdot F^S\), where \(F\) and \(S\) are multi-index notation for \((f_1,\ldots,f_r)\) and \((s_1,\ldots,s_r)\) and \(F^S\) is the symbol \(\prod_if_i^{s_i}\). Such an object can be endowed with a natural structure of \(\mathcal{D}_X[S]\)-module, extending the one over \(\mathcal{O}_X[S]\).
Analogously as in the case \(r=1\), we can consider the Bernstein-Sato ideal \(B_F=\mathbb{C}[S]\cap\langle\mathcal{D}_X[S]F+\operatorname{ann}_{\mathcal{D}_X[S]} F^S\rangle\). The paper under review is a generalization to the multivariable case of some of the results of [\textit{L. Narváez Macarro}, Adv. Math. 281, 1242--1273 (2015; Zbl 1327.14090)] and [\textit{U. Walther}, Invent. Math. 207, No. 3, 1239--1287 (2017; Zbl 1370.14022)] under certain conditions on \(f\).
We say that \(f\) is strongly Euler-homogeneous at some point \(x\in X\) if there is a logarithmic vector field \(E_x\in\operatorname{Der}_{X,x}(-\log f)\) such that \(E_x\bullet f=f\) and \(E_x\) vanishes at \(x\). We also say that \(f\) is tame if the projective dimension of the logarithmic \(k\)-forms \(\Omega_X^k(\log f)\) is at most \(k\) at each \(x\in X\). Last, the author defines \(f\) to be Saito-holonomic if certain stratification that we can consider on \(X\), arising from logarithmic derivations, is locally finite.
Under those three conditions, the author proves the first main result of the paper: the annihilator \(\operatorname{ann}_{\mathcal{D}_X[S]} F^S\) is generated by derivations, namely by the elements \(\delta-\sum_is_i(\delta\bullet f_i)/f_i\), where \(\delta\in\operatorname{Der}_{X}(-\log f)\). Using such result, the author goes on proving the generalized \(-\frac{n}{d}\) conjecture as stated by \textit{Budur} [Ann. Inst. Fourier 65, No. 2, 549--603 (2015; Zbl 1332.32038)] for central, essential, indecomposable, tame hyperplane arrangements in \(\mathbb{C}^n\).
Let now \(A=(a_1,\ldots,a_r)\in\mathbb{C}^r\) be an \(r\)-uple and let \(x\in X\) be a point and consider the \(\mathcal{D}_{X,x}\)-module map \[\nabla_A:\frac{\mathcal{D}_{X,x}[S]F^S}{(S-A)\mathcal{D}_{X,x}[S]F^S}\rightarrow \frac{\mathcal{D}_{X,x}[S]F^S}{(S-(A-1))\mathcal{D}_{X,x}[S]F^S},\] induced by sending \(F^S\) to \(F^{S+1}\). The other main result of the paper is that if \(\nabla_A\) is injective, then it is surjective too, whenever \(f\) is strongly Euler-homogeneous, Saito-holonomic and tame. Moreover, if \(f\) is also reduced, and free instead of just tame, the converse statement holds. This is used to study the dual of the \(\mathcal{D}_{X,x}[S]\)-modules from above and the cohomology support loci of \(f\) (defined in the text).
Reviewer: Alberto Castaño Domínguez (Sevilla)