Hodge filtration and Hodge ideals for \(\mathbb{Q}\)-divisors with weighted homogeneous isolated singularities or convenient non-degenerate singularities (Q2152627)

From the Introduction: Let \(D\) be an integral and reduced effective divisor on a smooth complex variety \(X\). Let \(\mathcal{O}_X(*D)\) be the sheaf of rational functions with poles along \(D\). This is also a left \(\mathscr{D}_X\)-module underlying the mixed Hodge module \(j_*\mathbb{Q}_U^H[n]\), where \(U=X\backslash D\) and \(j:U\hookrightarrow X\) is the inclusion map. Any \(\mathscr{D}_X\)-module associated to a mixed Hodge module has a good filtration \(F_\bullet\), the Hodge filtration of the mixed Hodge module [\textit{M. Saito}, Publ. Res. Inst. Math. Sci. 26, No. 2, 221--333 (1990; Zbl 0727.14004)]. To study the Hodge filtration of \(\mathcal{O}_X(*D)\), it seems more convenient to consider a series of ideal sheaves, defined by \textit{M. Mustaţă} and \textit{M. Popa} [Hodge ideals. Providence, RI: American Mathematical Society (AMS) (2019; Zbl 1442.14004)], which can be considered to be a generalization of multiplier ideals of divisors. The Hodge ideals \(\{I_k(D)\}_{k\in\mathbb{N}}\) of the divisor \(D\) are defined by: \[ F_k\mathcal{O}_X(*D)=I_k(D)\otimes_{\mathcal{O}_X}\mathcal{O}_X\big{(}(k+1)D\big{)},~\forall~k\in\mathbb{N}. \] It turns out that \(I_0(D)=\mathscr{J}\big{(}(1-\epsilon)D\big{)}\), the multiplier ideal of the divisor \((1-\epsilon)D\), \(0<\epsilon\ll1\). The Hodge filtration \(F_\bullet\) is usually hard to describe. However, it does have an explicit formula in the case when \(D\) is defined by a reduced weighted homogeneous polynomial \(f\) that has an isolated singularity at the origin, which is proved by \textit{M. Saito} [Mosc. Math. J. 9, No. 1, 151--181 (2009; Zbl 1196.14015)]. To state Saito's result and to clarify the notations for this paper, we denote: \begin{itemize} \item \(\mathcal{O}=\mathbb{C}\{x_1,\ldots, x_n\}\) the ring of germs of holomorphic function for local coordinates \(x_1,\ldots, x_n\). \item \(f:(\mathbb{C}^n,0)\to (\mathbb{C},0)\) a germ of holomorphic function that is quasihomogeneous, i.e., \(f\in \mathcal{J}(f)=(\frac{\partial f}{\partial x_1},\ldots, \frac{\partial f}{\partial x_n})\), and with isolated singularity at the origin. \textit{K. Saito} [Invent. Math. 14, 123--142 (1971; Zbl 0224.32011)] showed that after a biholomorphic coordinate change, we can assume \(f\) is a weighted homogeneous polynomial with an isolated singularity at the origin. We will keep this assumption for \(f\) unless otherwise stated. \item \(w=w(f)=(\#1_1,\ldots,\#1_{\#2} wn)\) the weights of the weighted homogeneous polynomial \(f\). \item \(g:(\mathbb{C}^n,0)\to(\mathbb{C},0)\) a germ of a holomorphic function, and we write \[g=\sum_{A\in\mathbb{N}^n}g_Ax^A,\] where \(A=(a_1,\ldots, a_n)\), \(g_A\in\mathbb{C}\) and \(x^A=x_1^{a_1}\cdots x_n^{a_n}\). \item \(\rho(g)\) the weight of an element \(g\in\mathcal{O}\) defined by \[ \rho(g)=\left(\sum_{i=1}^mw_i\right)+\inf\{\langle w, A\rangle: g_A\neq 0\}.\tag{1} \] The weight function \(\rho\) defines a filtration on \(\mathcal{O}\) as \[ \mathcal{O}^{>k}=\{u\in\mathcal{O}: \rho(u)>k\}; \] \[ \mathcal{O}^{\ge k}=\{u\in\mathcal{O}:\rho(u)\ge k\}. \] \end{itemize} Now we can state the formula proved by \textit{M. Saito} [Mosc. Math. J. 9, No. 1, 151--181 (2009; Zbl 1196.14015), Theorem 0.7], namely: \[ F_k\mathcal{O}_X(*D)=\sum_{i=0}^kF_{k-i}\mathscr{D}_X\left(\frac{\mathcal{O}^{\ge i+1}}{f^{i+1}}\right),~\forall~ k\in\mathbb{N}.\tag{2} \] Since we can now construct a Hodge filtration on analogous \(\mathscr{D}_X\)-modules associated to any effective \(\mathbb{Q}\)-divisor \(D\), it is natural to ask if it satisfies a similar formula in the case when \(D\) is supported on a hypersurface defined by such a polynomial \(f\). From now on, we set the divisor to be \(D=\alpha Z\), where \(0<\alpha\le 1\) and \(Z=(f=0)\) is an integral and reduced effective divisor defined by \(f\), a weighted homogeneous polynomial with an isolated singularity at the origin. In this case, the associated \(\mathscr{D}_X\)-module is the well-known twisted localization \(\mathscr{D}_X\)-module \(\mathcal{M}(f^{1-\alpha}):=\mathcal{O}_X(*Z)f^{1-\alpha}\) \big{(}see more details in [\textit{M. Mustaţǎ} and \textit{M. Popa}, J. Éc. Polytech., Math. 6, 283--328 (2019; Zbl 1427.14045)] about how to construct the Hodge filtration \(F_\bullet\mathcal{M}(f^{1-\alpha})\)\big{)}. With new ingredients from [\textit{M. Mustaţă} and \textit{M. Popa}, Forum Math. Sigma 8, Paper No. e19, 41 p. (2020; Zbl 1451.14055)], where this Hodge filtration is compared to the \(V\)-filtration on \(\mathcal{M}(f^{1-\alpha})\), we can generalize Saito's formula and prove the following theorem: Theorem 1. If \(D=\alpha Z\), where \(0<\alpha\le 1\) and \(Z=(f=0)\) is an integral and reduced effective divisor defined by \(f\), a weighted homogeneous polynomial with an isolated singularity at the origin, then we have \[ F_k\mathcal{M}(f^{1-\alpha})=\sum_{i=0}^kF_{k-i}\mathscr{D}_X\cdot\left(\frac{\mathcal{O}^{\ge \alpha+i}}{f^{i+1}}f^{1-\alpha}\right), \] where the action \(\cdot\) of \(\mathscr{D}_X\) on the right hand side is the action on the left \(\mathscr{D}_X\)-module \(\mathcal{M}(f^{1-\alpha})\) defined by \[D\cdot(wf^{1-\alpha}):=\left(D(w)+w\frac{(1-\alpha)D(f)}{f}\right)f^{1-\alpha}, ~\text{ for any }~D\in \textrm{Der}_{\mathbb{C}}\mathcal{O}_X. \] Notice that if we set \(\alpha=1\), Theorem 1 recovers Saito's formula (2) mentioned above. For any polynomial \(f\) with an isolated singularity at the origin, it is well known that the Jacobian algebra \[ \mathcal{A}_f:=\mathbb{C}\{\#1_1,\ldots,\#1_{\#2} xn\}/(\partial_1f,\ldots, \partial_nf) \] is a finite dimensional \(\mathbb{C}\)-vector space. Fix a monomial basis \(\{\#1_1,\ldots,\#1_{\#2} v\mu\}\) for this vector space, where \(\mu\) is the dimension of \(\mathcal{A}_f\) (usually called the Milnor number of \(f\)). With this notation, applying Theorem 1, we can give a formula for the Hodge filtration that is easier to use in practice: Corollary 1. If \(D=\alpha Z\), where \(0<\alpha\le 1\) and \(Z=(f=0)\) is an integral and reduced effective divisor defined by \(f\), a weighted homogeneous polynomial with an isolated singularity at the origin, then we have \[ F_0\mathcal{M}(f^{1-\alpha})=f^{-1}\cdot \mathcal{O}^{\ge \alpha}f^{1-\alpha} \] and \[ F_k\mathcal{M}(f^{1-\alpha})=(f^{-1}\cdot \sum_{v_j\in \mathcal{O}^{\ge k+1+\alpha}}\mathbb{C} v_j) f^{1-\alpha}+F_1\mathscr{D}_X\cdot F_{k-1}\mathcal{M}(f^{1-\alpha}). \] Alternatively, in terms of Hodge ideals these formulas say that \[ I_0(D)=\mathcal{O}^{\ge\alpha} \] and \[ I_{k+1}(D)=\sum_{v_j\in \mathcal{O}^{\ge k+1+\alpha}}\mathbb{C}v_j+\sum_{1\le i\le n,a\in I_{k}(D)}\mathcal{O}_X\big{(}f\partial_ia-(\alpha+k)a\partial_if\big{)}. \] In particular, the Hodge filtration is fully computable since it is good and hence can be determined by finitely many terms. In order to do this effectively, \textit{M. Saito} [Mosc. Math. J. 9, No. 1, 151--181 (2009; Zbl 1196.14015)] introduced the following measure of the complexity of the Hodge filtration: The \textit{generating level} of any \(\mathscr{D}_X\)-module \((\mathcal{M}, F_{\bullet})\) with a good filtration is the smallest integer \(k\) such that \[ F_l\mathscr{D}_X\cdot F_k\mathcal{M}=F_{k+l}\mathcal{M}\text{ for all }l\ge0. \] Note that such a \(k\) always exist by definition. In the case \(D\) is integral and reduced, defined by a weighted homogeneous polynomial with an isolated singularity at the origin, Saito proves that the generating level of \(\mathcal{O}_X(*D)\) is \([n-\tilde{\alpha}_f-1]\), where \(\tilde{\alpha}_f\) is the minimal exponent (see [\textit{M. Mustaţă} and \textit{M. Popa}, Forum Math. Sigma 8, Paper No. e19, 41 p. (2020; Zbl 1451.14055), Section 6]) which is also called the microlocal log canonical threshold (see [\textit{M. Saito}, Mosc. Math. J. 9, No. 1, 151--181 (2009; Zbl 1196.14015), Theorem 0.7]). Popa conjectured in [\textit{M. Popa}, in: Proceedings of the international congress of mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1--9, 2018. Volume II. Invited lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 781--806 (2018; Zbl 1441.14066), Question 5.10] that a similar result should hold in the \(\mathbb{Q}\)-divisor setup we considered here. Using Theorem 1, we prove this conjecture is true: Corollary 2. If \(D=\alpha Z\), where \(0<\alpha\le 1\) and \(Z=(f=0)\) is an integral and reduced effective divisor defined by \(f\), a weighted homogeneous polynomial with an isolated singularity at the origin, then the generating level of \(\mathcal{M}(f^{1-\alpha})\) is \([n-\tilde{\alpha}_f-\alpha]\). One perspective for studying Hodge ideals, introduced by Saito, is to consider the induced microlocal \(V\)-filtration on \(\mathcal{O}_X\) associated to \(f\). When \(D\) is a reduced and integral divisor, Saito showed in [\textit{M. Saito}, ``Hodge ideals and microlocal \(V\)-filtration'', Preprint, \url{arXiv:1612.08667}] that \[I_k(D)=\tilde{V}^{k+\alpha}\mathcal{O}_X\mod (f).\] When we consider an effective \(\mathbb{Q}\)-divisor \(D=\alpha H\), in [\textit{M. Mustaţă} and \textit{M. Popa}, Forum Math. Sigma 8, Paper No. e19, 41 p. (2020; Zbl 1451.14055), Definition 3.1] the authors define another series of ideal sheaves \(\{\tilde{I}_k(D)\}_{k\in\mathbb{N}}\). In this setting, when \(0<\alpha\le 1\), it is easy to see that \(\tilde{I}_k(D)=\tilde{V}^{k+\alpha}\mathcal{O}_X\). In the case when \(D=\alpha\cdot Z\), where \(0<\alpha\le 1\), \(Z=\operatorname{div}(f)\) reduced and \(f\in\mathcal{O}_X(X)\), the authors [\textit{M. Mustaţă} and \textit{M. Popa}, Forum Math. Sigma 8, Paper No. e19, 41 p. (2020; Zbl 1451.14055), Theorem A\('\)] obtained a formula for \(I_k(D)\) in terms of the \(V\)-filtration on \(\mathcal{M}(f^{1-\alpha})\), from which in particular it follows that \(I_k(D)=\tilde{I}_k(D)\mod (f)\). It is natural to wonder to what extent equality holds without modding out by \((f)\). Although \(I_0(D)=\tilde{I}_0(D)\) is always true, \(I_k(D)\) and \(\tilde{I}_k(D)\) are usually not the same for \(k\ge 1\) (see, e.g. [\textit{M. Saito}, ``Hodge ideals and microlocal \(V\)-filtration'', Preprint, \url{arXiv:1612.08667}, Remark (ii) in \S 2.4] for \(\alpha=1\); see also [\textit{M. Popa}, in: Proceedings of the international congress of mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1--9, 2018. Volume II. Invited lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 781--806 (2018; Zbl 1441.14066), Remark 9.8] for examples of \(\mathbb{Q}\)-divisors). However, if \(D=\alpha Z\) with \(0<\alpha\le1\) and \(Z\) is defined by a weighted homogeneous polynomial with an isolated singularity at the origin, we can give a criterion for equality between \(I_k(D)\) and \(\tilde{I}_k(D)\): Proposition 1. If \(D=\alpha Z\), where \(0<\alpha\le 1\) and \(Z=(f=0)\) is an integral and reduced effective divisor defined by \(f\), a weighted homogeneous polynomial with an isolated singularity at the origin, then for any \(k\in\mathbb{N}\), if \(I_k(D)=\tilde{I}_k(D)=(\#1_1,\ldots,\#1_{\#2} xn)^m\) for some \(m\in\mathbb{N}\), then \(I_{k+1}(D)= \tilde{I}_{k+1}(D)\). When \(k=0\), we prove that the converse statement is true, i.e., if \(I_0(D)=\tilde{I}_0(D)\neq (\#1_1,\ldots,\#1_{\#2} xn)^m \) for any \(m\in\mathbb{N}\), then \(I_1(D)\neq \tilde{I}_1(D)\). So it seems plausible to make the following: Conjecture 1. If \(D=\alpha Z\), where \(0<\alpha\le 1\) and \(Z=(f=0)\) is an integral and reduced effective divisor defined by \(f\), a weighted homogeneous polynomial with an isolated singularity at the origin, then for any \(k\in\mathbb{N}\), \(I_{k+1}(D)= \tilde{I}_{k+1}(D)\) if and only if \(I_k(D)=\tilde{I}_k(D)=(\#1_1,\ldots,\#1_{\#2} xn)^m\) for some \(m\in \mathbb{N}\).