May 25, 2026
In this article, we prove the strong monodromy conjecture for complex hyperplane arrangements by proving a conjecture of Budur, Mustaţă and Teitler that \(-n/d\) is a root of the \(b\)-function of an irreducible essential and central hyperplane arrangement \(f\) of degree \(d\) on \(\mathbb{C}^n\).
We first recall the monodromy conjecture and the special case of hyperplane arrangements. Let \(f\in \mathbb{C}[z_1,\dots,z_n]\) be a polynomial such that \(f(0)=0\). Let \(X = \mathbb{A}^n_\mathbb{C}\), and denote by \(D=(f=0)\) the divisor of \(f\), counting multiplicities. Take a log resolution \(\mu\colon Y\to X\) of \(D\). Define integers \(a_i\) and \(k_i\) by \[\mu^* D = \sum_{i\in S} a_i E_i, \quad K_{Y/X} = \sum_{i\in S} k_i E_i,\] where the \(E_i\) are irreducible divisors, and \(K_{Y/X}\) is the relative canonical divisor of \(\mu\). The (local) topological zeta function of \(f\) is \[Z_f(s)\mathrel{\vcenter{:}}= \sum_{I\subseteq S} \chi(E^o_I\cap \mu^{-1}(0))\prod_{i\in I}\dfrac{1}{a_is+k_i+1},\] where \(E^o_I=\bigcap_{i\in I}E_i\setminus \bigcup_{j\in S \setminus I}E_j\) and \(\chi\) is the topological Euler characteristic. Denef and Loeser [1] proved that \(Z_f(s)\) is independent of the choice of log resolution.
The poles of \(Z_f(s)\) are conjectured to have geometric and algebraic meaning.
Conjecture 1 ([1], Topological Monodromy Conjecture). If \(c\) is a pole of the topological zeta function \(Z_f(s)\), then:
\((\)Weak version\():\) \(\exp ( 2\pi i c)\) is an eigenvalue of the monodromy action on the cohomology of the Milnor fiber.
\((\)Strong version\():\) \(c\) is a root of the Bernstein–Sato polynomial of \(f\).
It is well known that, by a classical result of Kashiwara and Malgrange, the strong version implies the weak version. The above conjecture is one of the most important conjectures in singularity theory. It remains wide open in general.
A testing ground for Conjecture 1 is the case of hyperplane arrangements, i.e., \(f = \prod_{i} f_i\), where each \((f_i=0)\) defines a hyperplane in \(\mathbb{C}^n\). Budur, Mustaţă, and Teitler [2] studied Conjecture 1 in this case. They proved the weak version for hyperplane arrangements and also reduced the strong version to the following conjecture. We use \(b_f(s)\) to denote the Bernstein–Sato polynomial (or \(b\)-function) of \(f\). A hyperplane arrangement is central if all hyperplanes pass through the origin \(0\), and it is essential if \(\{0\}\) is a stratum of the intersection lattice. A central hyperplane arrangement is irreducible if there is no linear change of coordinates on \(\mathbb{C}^n\) such that \(f\) can be written as a product of two non-constant polynomials in disjoint sets of variables.
Conjecture 2 (Budur, Mustaţă, and Teitler). If \(f\in \mathbb{C}[z_1,\dots,z_n]\) is a polynomial of degree \(d\) such that \((f=0)\) defines an irreducible, essential, central hyperplane arrangement, then \(b_f(-\frac{n}{d})=0\).
Theorem 3 (Budur, Mustaţă, and Teitler). Conjecture 2 implies the strong version of Conjecture 1 for hyperplane arrangements.
Conjecture 2 is known for reduced \(f\) with \(n\le 3\) [3], for Weyl hyperplane arrangements [4], for tame hyperplane arrangements [5] (see also [6]), and for \(f\) with generic multiplicities [7]. It was claimed by Davis and Yang [8] that they proved Conjecture 2 by using the theory of Hodge modules.
In this paper, we prove Conjecture 2 in general, and hence Conjecture 1 for hyperplane arrangements, without using Hodge theory. More precisely, we prove:
Theorem 4. Let \(f\) be a polynomial of degree \(d\) (not necessarily reduced) such that \((f=0)\) defines an irreducible, essential, central hyperplane arrangement. Then \(-k/d\) is a root of \(b_f(s)\) for \(k=n,n+1,\dots,d\). In particular, Conjecture 2 holds.
The proof relies on the theory of algebraic relative (holonomic) \(\mathscr{D}\)-modules. Specifically, we prove the commutativity of duality functors with proper direct images for algebraic relative \(\mathscr{D}\)-modules (see Proposition 5), which serves as an algebraic analogue of a fundamental result by Schapira and Schneiders [9] for analytic relative \(\mathscr{D}\)-modules. Another key ingredient is the Beilinson–Bernstein construction of \(\mathscr{D}\)-module nearby cycles (see §3). By combining this commutation property, the Beilinson–Bernstein nearby cycles, and resolution of singularities, we first provide an alternative proof of Kashiwara’s rationality theorem for \(b\)-functions [10], as well as Lichtin’s refinement (Theorem 9). The strategy underlying our proof of Theorem 9 establishes the framework for proving Theorem 4. Furthermore, to better control the section “\(f^s\)” before and after passing to a resolution, we must also consider the special direct images of \(\mathscr{D}\)-modules (cf.[11]). We employ the wonderful model of De Concini and Procesi as the canonical log resolution of the hyperplane arrangement. Exploiting the wonderful model, we prove that the standard direct image and the special direct image of the \(!\)-extension of Beilinson-Bernstein on the resolution coincide globally (see Lemma 4). This crucial identification allows us to trace the section “\(f^s\)” back to the resolution, thereby completing the proof of Theorem 4. In the proof, we also utilize the propagation property of cohomology jumping loci for rank one local systems on the hyperplane arrangement complement [12] to establish the non-vanishing result in Lemma 3.
A broader version of Conjecture 2, concerning the zeroes of Bernstein-Sato ideals associated with hyperplane arrangements, is given in [13]. One can check that the proof of Theorem 4 readily extends to this general case.
AI statements 1. This work was partially assisted by artificial intelligence, primarily through the web-based version of Gemini 3.1 Pro Deep Think. Specifically, the AI model proposed the proof strategy for Lemma 4.5, including the application of the wonderful model of hyperplane arrangements. The author subsequently refined and finalized the proof presented in this paper. Transcripts of the AI interactions are available upon request.
Acknowledgment 1. The author thanks Nero Budur for sharing his conjecture with him about five years ago. He also thanks Peng Zhou for helpful discussions. This work was supported by a start-up grant from Zhejiang University.
We first recall the theory of algebraic relative \(\mathscr{D}\)-modules. Let \(X\) be a smooth algebraic variety over \(\mathbb{C}\) and let \(R\) be an integral commutative \(\mathbb{C}\)-algebra such that \(R_q\) is a regular local ring for every prime ideal \(q\in \mathrm{Spec }R\). Set \[\mathscr{A}^R=\mathscr{A}^R_X\mathrel{\vcenter{:}}= \mathscr{D}_X\otimes_\mathbb{C}R,\] where \(\mathscr{D}_X\) is the sheaf of rings of algebraic differential operators. Similarly to \(\mathscr{D}_X\), \(\mathscr{A}^R\) is a coherent and Noetherian sheaf of rings.
If \(\mathcal{N}\) is a (left or right) \(\mathscr{A}^R\)-module and \(q\in \mathrm{Spec }R\), then since \(R\) is central in \(\mathscr{A}^R\), \[\mathcal{N}_q\mathrel{\vcenter{:}}=\mathcal{N}\otimes_R R_q\] becomes an \(\mathscr{A}^R_q \simeq \mathscr{A}^{R_q}\)-module, where \(R_q\) is the localization of \(R\) at \(q\). We then define \[\mathrm{supp}(\mathcal{N})=\mathrm{supp}^R(\mathcal{N})\mathrel{\vcenter{:}}=\{q\in \mathrm{Spec }R\mid \mathcal{N}_q\neq 0\}\subseteq \mathrm{Spec }R\] to be the support of \(\mathcal{N}\) as an \(R\)-module.
Since \(R\) is a \(\mathbb{C}\)-algebra, we have a natural inclusion \(\mathscr{D}_X\hookrightarrow \mathscr{A}^R\), and thus \(\mathscr{A}^R\)-modules are, in particular, \(\mathscr{D}_X\)-modules. For a left \(\mathscr{A}^R\)-module \(\mathcal{M}\), \[\mathcal{M}^r\mathrel{\vcenter{:}}= \mathcal{M}\otimes_{\mathscr{O}_X}\omega_X\] is the corresponding right \(\mathscr{D}_X\)-module of \(\mathcal{M}\) under the side-change operation, where \(\omega_X\) is the canonical line bundle. Since \(R\) is central, \(\mathcal{M}^r\) is naturally a right \(\mathscr{A}^R\)-module. Therefore, the side-change operation for \(\mathscr{D}\)-modules extends naturally to \(\mathscr{A}^R\)-modules.
For a right \(\mathscr{A}^R\)-module \(\mathcal{N}\) (or, more generally, a bounded complex of right \(\mathscr{A}^R\)-modules), we define its dual by \[\mathbb{D}(\mathcal{N})\mathrel{\vcenter{:}}= \mathcal{R}\mathcal{H}om_{\mathscr{A}^R}(\mathcal{N},\omega_X\otimes_{\mathscr{O}_X}\mathscr{A}^R);\] similarly, for a left \(\mathscr{A}^R\)-module \(\mathcal{M}\), we define \[\mathbb{D}(\mathcal{M})\mathrel{\vcenter{:}}= \mathcal{R}\mathcal{H}om_{\mathscr{A}^R}(\mathcal{M},\mathscr{A}^R\otimes_{\mathscr{O}_X}\omega_X^{-1}).\] It is clear that duality and the side-change operation commute, i.e., \[\mathbb{D}(\mathcal{M})\otimes_{\mathscr{O}_X}\omega_X\simeq \mathbb{D}(\mathcal{M}^r).\]
Let \(g\colon Y\to X\) be a morphism between smooth algebraic varieties, and let \(\mathcal{N}^\bullet\) be a bounded complex of right \(\mathscr{A}^R_Y\)-modules. Then the relative \(\mathscr{A}^R\)-module direct image is defined as \[g_+(\mathcal{N}^\bullet)\mathrel{\vcenter{:}}= Rg_*(\mathcal{N}^\bullet\otimes^L_{\mathscr{A}_Y^R}g^*\mathscr{A}^R_X)\simeq Rg_*(\mathcal{N}^\bullet\otimes^L_{\mathscr{D}_Y}g^*\mathscr{D}_X),\] where the isomorphism follows because \(\mathscr{A}_Y^R=\mathscr{D}_Y\otimes_\mathbb{C}R\) and \(g^*\mathscr{A}^R_X\simeq g^*\mathscr{D}_X\otimes_\mathbb{C}R\). This means that the relative direct image of \(\mathcal{N}^\bullet\) is precisely its direct image as a complex of \(\mathscr{D}\)-modules.
Proposition 5. Let \(g\colon Y\to X\) be a proper morphism between smooth algebraic varieties. Then we have a natural isomorphism of functors \[g_+\mathbb{D}\xrightarrow{\simeq} \mathbb{D}g_+\colon D^b_c(\mathscr{A}^R_Y)^{\mathrm{op}} \longrightarrow D^b_c(\mathscr{A}^R_X)^{\mathrm{op}},\] where \(D^b_c(\mathscr{A}^R_\bullet)^{\mathrm{op}}\) is the derived category of bounded complexes of right \(\mathscr{A}^R_\bullet\)-modules with coherent cohomology sheaves.
Proof. First, similar to the classical case (i.e., the case when \(R=\mathbb{C}\)), both \(g_+\) and \(\mathbb{D}\) preserve coherence (cf.[14]).
By definition, the trace map in [14] provides the right-module version of the trace map: \[\mathrm{Tr}_g^{\mathrm{op}}\colon g_+(\omega_Y)[\dim Y]\longrightarrow \omega_X[\dim X].\] Similarly to the proof of [14], the trace map \(\mathrm{Tr}_g^{\mathrm{op}}\) then induces a canonical isomorphism \[g_+\mathbb{D}\xrightarrow{\simeq} \mathbb{D}g_+.\] ◻
Remark 6. The above proposition can be seen as a simplified algebraic analogue of the fundamental result [9] for analytic relative \(\mathscr{D}\)-modules.
In this section, we recall the Beilinson–Bernstein construction of \(\mathscr{D}\)-module nearby cycles for regular functions on smooth algebraic varieties [15]. See also [16] and [17], [18]. We present the algebraic right-module version of the construction here for the application of this paper.
Let \(X\) be a smooth algebraic variety over \(\mathbb{C}\), and let \(f\) be a non-invertible regular function on \(X\). We consider the left \(\mathscr{A}^{\mathbb{C}[s]}\)-module \[\mathcal{M}\mathrel{\vcenter{:}}= j_*(\mathscr{O}_U[s]f^s)=\mathscr{O}_X[s,1/f]f^s\] and \(\mathcal{M}^r=\mathcal{M}\otimes_{\mathscr{O}_X}\omega_X\), which is the corresponding right \(\mathscr{A}^{\mathbb{C}[s]}\)-module under the side-change operation, where \(j\colon U\mathrel{\vcenter{:}}= X\setminus D\hookrightarrow X\) is the open embedding and \(D\) is the effective divisor defined by \((f=0)\). If \(v\) is a global section of \[\omega_X(*D)\mathrel{\vcenter{:}}=\varinjlim_{k\to \infty} \omega_X(kD)\] such that \(v\cdot\mathscr{D}_X|_U=\omega_U\), then we consider the coherent \(\mathscr{A}^{\mathbb{C}[s]}\)-submodule \[\mathcal{N}\mathrel{\vcenter{:}}= f^sv\cdot\mathscr{D}_X[s]\subseteq \mathcal{M}^r.\] More generally, if \(\mathcal{E}\subseteq \omega_X(*D)\) is a coherent \(\mathscr{O}_X\)-submodule such that \(\mathcal{E}\cdot \mathscr{D}_X|_U=\omega_U\), then we similarly obtain a coherent \(\mathscr{A}^{\mathbb{C}[s]}\)-module \[\mathcal{N}(\mathcal{E})\mathrel{\vcenter{:}}= f^s\mathcal{E}\cdot \mathscr{D}_X[s]\subseteq\mathcal{M}^r.\] Then, for \(l\in \mathbb{Z}\), we write \[\mathcal{N}(lD)\mathrel{\vcenter{:}}=\mathcal{N}(\mathscr{O}_X(lD)\cdot v)=f^{s-l}v\cdot\mathscr{D}_X[s]\subseteq \mathcal{M}^r.\] It is clear that \[\mathcal{N}(lD)\hookrightarrow \mathcal{N}((l+k)D)\] for \(k\in \mathbb{Z}_{\ge0}\).
The \(b\)-function of \(f\) and \(v\), denoted by \(b_{f,v}\), is the monic generator of the annihilator ideal in \(\mathbb{C}[s]\): \[{\mathrm{Ann}}_{\mathbb{C}[s]}\left(\frac{\mathcal{N}}{\mathcal{N}(-D)}\right)\subseteq \mathbb{C}[s].\] By substitution, we immediately see that for every \(k\in \mathbb{Z}\), \[\label{eq:subk} b_{f,v}(s+k)=b_{f,f^kv}(s).\tag{1}\] In the case when \(X=\mathbb{A}^n_\mathbb{C}=\mathrm{Spec }\mathbb{C}[z_1,\dots,z_n]\) and \(v=dz_1\wedge dz_2\wedge \dots \wedge dz_n\), since the action of \(\mathbb{C}[s]\) and the side-change operation commute, \(b_{f,v}\) is exactly the Bernstein–Sato polynomial \(b_f\) of \(f\in \mathbb{C}[z_1,\dots,z_n]\).
Lemma 1. With the notation as above, we have \[Z(b_{f,v})=\mathrm{supp}\left(\frac{\mathcal{N}}{\mathcal{N}(-D)}\right),\] where \(Z(b_{f,v})\subseteq \mathrm{Spec }\mathbb{C}[s]\) is the algebraic set defined by \(b_{f,v}\), i.e., the set of all roots of \(b_{f,v}\).
Proof. Take a finite affine open covering \(X=\bigcup_i V_i\) such that \(\omega_{V_i}\simeq \mathscr{O}_{V_i}\cdot e_i\) and \(v|_{V_i}=h_i\cdot e_i\) for some \(h_i\in \mathscr{O}_X(V_i)\). By [19], \[\frac{\mathscr{D}_{V_i}[s]h_i\cdot f^s}{\mathscr{D}_{V_i}[s]h_i\cdot f^{s+1}}\] is algebraic relative holonomic (cf.[20]). By [20], we know \[Z(b_{f,h_i})=\mathrm{supp}\left(\frac{\mathscr{D}_{V_i}[s]h_i\cdot f^s}{\mathscr{D}_{V_i}[s]h_i\cdot f^{s+1}}\right),\] where \(b_{f,h_i}\) is the monic generator of \({\mathrm{Ann}}_{\mathbb{C}[s]}\left(\frac{\mathscr{D}_{V_i}[s]h_i\cdot f^s}{\mathscr{D}_{V_i}[s]h_i\cdot f^{s+1}}\right)\). Applying the side-change operation, we then obtain \[Z(b_{f|_{V_i},v|_{V_i}})=Z\left({\mathrm{Ann}}_{\mathbb{C}[s]}\left(\left.\frac{\mathcal{N}}{\mathcal{N}(-D)}\right|_{V_i}\right)\right)=\mathrm{supp}\left(\left.\frac{\mathcal{N}}{\mathcal{N}(-D)}\right|_{V_i}\right).\] By definition, we know \[{\mathrm{Ann}}_{\mathbb{C}[s]}\left(\frac{\mathcal{N}}{\mathcal{N}(-D)}\right)=\bigcap_i {\mathrm{Ann}}_{\mathbb{C}[s]}\left(\left.\frac{\mathcal{N}}{\mathcal{N}(-D)}\right|_{V_i}\right)\] and \[\mathrm{supp}\left(\frac{\mathcal{N}}{\mathcal{N}(-D)}\right)=\bigcup_i \mathrm{supp}\left(\left.\frac{\mathcal{N}}{\mathcal{N}(-D)}\right|_{V_i}\right).\] Since \(\mathbb{C}[s]\) is a PID, we thus obtain \[Z(b_{f,v})=\mathrm{supp}\left(\frac{\mathcal{N}}{\mathcal{N}(-D)}\right).\] ◻
It is well known that the local \(b\)-function \(b_{f|_{V_i},v|_{V_i}}\) is always nonzero and non-constant if \(f|_{V_i}\) is not invertible. The above proof also shows that \(b_{f,v}\) is the least common multiple of all the \(b_{f|_{V_i},v|_{V_i}}\); in particular, the global \(b\)-function \(b_{f,v}\) is non-constant.
Fixing an arbitrary \(\alpha\in \mathbb{C}\simeq \mathrm{Spec }\mathbb{C}[s]\), we use \(\mathfrak{m}=\mathfrak{m}_\alpha\subseteq \mathbb{C}[s]\) to denote the maximal ideal corresponding to \(\alpha\). We define \[j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\mathrel{\vcenter{:}}= \mathbb{D}j_*\big(\mathbb{D}(\mathcal{M}^r_{\mathfrak{m}}|_U)\big)\] and \[j_!(\mathcal{M}^r_\eta |_U)\mathrel{\vcenter{:}}= \mathbb{D}j_*\big(\mathbb{D}(\mathcal{M}^r_\eta|_U)\big),\] where \(\eta\in \mathrm{Spec }\mathbb{C}[s]\) is the generic point.
Since taking duals and the side-change operation commute, by [18], we have the following right-module version of a fundamental result of Beilinson–Bernstein; see also [15], [16], and [17].
Theorem 7 (Beilinson–Bernstein). With the notation and assumptions as above, we have:
\(j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\stackrel{\mathrm{q.i.}}{\simeq}\mathcal{H}^0\big(j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\big)\), i.e., \(j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\) is a sheaf;
the natural morphism \(j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\hookrightarrow j_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\) is injective;
\(j_*(\mathcal{M}^r_{\mathfrak{m}}|_U)=\mathcal{M}^r_{\mathfrak{m}}=\mathcal{N}(lD)_{\mathfrak{m}}\) for \(l\gg 0\);
\(j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)=\mathcal{N}(-lD)_{\mathfrak{m}}\) for \(l\gg 0\);
forgetting the \(\mathbb{C}[s]\)-module structure, \(\Psi_{f}^\alpha\mathrel{\vcenter{:}}=\frac{j_*(\mathcal{M}^r_{\mathfrak{m}}|_U)}{j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)}\) is a regular holonomic \(\mathscr{D}_X\)-module;
\(j_!(\mathcal{M}^r_\eta|_U)= j_*(\mathcal{M}^r_\eta|_U)\).
The sheaf \(\Psi_{f}^\alpha\) is the right-module version of the \(\alpha\)-nearby cycles of \(f\), which corresponds to the topological nearby cycles of the constant sheaf \(\mathbb{C}_X\) along \(f\) under the Riemann–Hilbert correspondence (see [15] and [18]).
By Theorem 7, \(\frac{\mathcal{N}(kD)_{\mathfrak{m}}}{\mathcal{N}((k-1)D)_{\mathfrak{m}}}\) is always a subquotient of \(\Psi^\alpha_f\) for every \(k\in \mathbb{Z}\). Applying Eq. 1 and Lemma 1 together gives:
Corollary 8. For \(\alpha\in\mathbb{C}\), the condition that \(b_{f,v}(\alpha+k)=0\) for some \(k\in \mathbb{Z}\) is equivalent to \(\Psi^\alpha_f\neq 0\).
We fix a non-invertible regular function \(f\) on \(X=\mathbb{A}^n_\mathbb{C}\) and a log resolution \(\mu\colon Y\to X\), along with the notation from §1.1. Kashiwara [10] proved that the roots of \(b_f\) are negative rational numbers. Lichtin refined Kashiwara’s result into the following form:
Theorem 9 ([21], Thm. 5). All the roots of \(b_f\) are of the form \[-\frac{k_i+l+1}{a_i}\] for \(i\in S\) and \(l\in \mathbb{Z}_{\ge0}\).
As a warm-up, we first give an alternative proof of the above theorem using the nearby cycles constructed in §3.
Proof. We set \(\widetilde{f}=\mu^*f\), \(v=dz_1\wedge dz_2\wedge \dots \wedge dz_n\), and \(\widetilde{v}=\mu^*v\in \mu^*\omega_X\hookrightarrow\omega_Y\). We consider \[\widetilde{\mathcal{N}}\mathrel{\vcenter{:}}= \widetilde{f}^s\widetilde{v}\cdot \mathscr{D}_Y[s]\subseteq \widetilde{j}_*(\mathcal{M}^r|_U),\] where \(\widetilde{j}\colon U\hookrightarrow Y\) is the open embedding. Then we have a commutative diagram \[\begin{tikzcd} U\arrow[r,"\widetilde{j}"]\arrow[d,"\mathrm{id}"]& Y \arrow[d,"\mu"]\\ U\arrow[r,"j"]& X \end{tikzcd}\] Since both \(j\) and \(\widetilde{j}\) are affine open embeddings, we have \(j_+=j_*\) and \(\widetilde{j}_+=\widetilde{j}_*\). By functoriality, the above diagram yields that for every \(\alpha\in \mathbb{C}\), \[\mu_+(\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U))\simeq j_*(\mathcal{M}^r_{\mathfrak{m}}|_U),\] where \(\mathfrak{m}=\mathfrak{m}_\alpha\) is the maximal ideal corresponding to \(\alpha\). Similarly, by Proposition 5, we also know \[\label{eq:birapushforminex} \mu_+(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U))\simeq j_!(\mathcal{M}^r_{\mathfrak{m}}|_U).\tag{2}\] By Theorem 7(2), we have a short exact sequence \[0\to \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\longrightarrow\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\longrightarrow \Psi_{\widetilde{f}}^\alpha\to 0.\] Applying \(\mu_+\) yields \[\label{eq:pushfnearby} \mu_+(\Psi_{\widetilde{f}}^\alpha)\simeq \Psi_{f}^\alpha.\tag{3}\]
We take a coordinate system \((y_1,\dots,y_n)\) on an open neighborhood \(V\subseteq Y\) locally around some \(p\in E\) such that \[\widetilde{f}|_V=u\cdot\prod_{i\in S_p\subseteq S}y_i^{a_i}\quad \mathrm{and} \quad \widetilde{v}|_V=u'\prod_{i\in S_p}y_i^{k_i}dy_1\wedge dy_2\wedge \dots \wedge dy_n\] with \(u,u'\) being invertible regular functions on \(V\). Then a local calculation shows that the roots of \(b_{\widetilde{f},\widetilde{v}}\) are of the form \[-\frac{k_i+l+1}{a_i}\] for \(i\in S\) and \(l\in \mathbb{Z}_{\ge0}\). Corollary 8 and Equation 3 together imply that the roots of \(b_f=b_{f,v}\) are of the form \[-\frac{k_i+l+1}{a_i}\] for \(i\in S\) and \(l\in \mathbb{Z}\).
Now, we need to exclude all \(l < 0\). By Theorem 7(4), we have \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)=\widetilde{\mathcal{N}}(-k\mu^*D)_{\mathfrak{m}}\] for \(k\gg 0\). Suppose \(\alpha= -\frac{k_i+l+1}{a_i}\) for some \(l<0\). Then \(\alpha\) is not a root of \(b_{\widetilde{f},\widetilde{v}}(s+k)\) for any \(k\ge 0\). By Equation 1 and Lemma 1, we know \[\label{eq:spj33} \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)=\widetilde{\mathcal{N}}_{\mathfrak{m}}.\tag{4}\] Now, we consider the global section \(w=\widetilde{f}^s\widetilde{v}\otimes 1\) of the sheaf \(\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y} \mu^*\mathscr{D}_X\) (and the sheaf \(\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{O}_Y} \mu^*\mathscr{D}_X\)), which induces morphisms in the derived category of \(\mu^{-1}\mathscr{O}_X\)-modules \[\mathscr{O}_Y\to\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{O}_Y}^L\mu^*\mathscr{D}_X\to\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}^L\mu^*\mathscr{D}_X\to \widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes^L_{\mathscr{D}_Y} \mu^*\mathscr{D}_X,\] since \(\mathscr{D}_Y\) is flat over \(\mathscr{O}_Y\). Applying \(R\mu_*\), we further have the induced morphisms \[\mathscr{O}_X\to R\mu_*\mathscr{O}_Y\to R\mu_*\big(\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes^L_{\mathscr{D}_Y} \mu^*\mathscr{D}_X\big)\simeq j_*(\mathcal{M}^r_{\mathfrak{m}}|_U).\] Then \(1\in \mathscr{O}_X(X)\) yields a global section \(w\) of \(j_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\). Clearly, \(w|_U=f^sv|_U\), and thus \(w-f^sv\) is a section of \(j_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\) supported on \(D\). Since \(j_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\) has no nonzero sections supported on \(D\) by construction, we know \(w=f^sv\). If such an \(\alpha\) is a root of \(b_{f,v}\), then we have \[\frac{\mathcal{N}_{\mathfrak{m}}}{\mathcal{N}(-D)_{\mathfrak{m}}}\neq 0\] by Lemma 1, and thus by Theorem 7(4), \[\frac{\mathcal{N}_{\mathfrak{m}}}{j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)}\neq 0.\] Since \(\mathcal{N}_{\mathfrak{m}}\) is generated by \(f^sv\), we know in particular that \(f^sv\notin j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\). However, by Equation 4 , we have a factorization \[\begin{tikzcd} \mathscr{O}_X\arrow[rr]\arrow[rd]&& j_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\\ & j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\arrow[ru, hook].& \end{tikzcd}\] Then \(w=f^sv\) is also a section of \(j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\), which is a contradiction. Consequently, \(\alpha= -\frac{k_i+l+1}{a_i}\) is not a root of \(b_f=b_{f,v}\) for all \(l<0\). ◻
Corollary 10. For every \(\alpha\in \mathbb{Q}\cap[-1,0)\), we have \[j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)=\mathcal{N}(-D)_{\mathfrak{m}}=f^{s+1}v\cdot\mathscr{D}_X[s]_{\mathfrak{m}},\] where \(\mathfrak{m}\subseteq \mathbb{C}[s]\) is the maximal ideal corresponding to \(\alpha\).
Proof. By Theorem 7(3), it suffices to prove \[\frac{\mathcal{N}(-kD)_{\mathfrak{m}}}{\mathcal{N}(-(k+1)D)_{\mathfrak{m}}}=0\] for all \(k\in \mathbb{Z}_{>0}\). If \(\frac{\mathcal{N}(-kD)_{\mathfrak{m}}}{\mathcal{N}(-(k+1)D)_{\mathfrak{m}}}\neq 0\) for some \(k>0\), then by Lemma 1, Equation 1 , and substitution, we conclude \[b_f(\alpha+k)=0.\] Since \(\alpha\in \mathbb{Q}\cap[-1,0)\) and \(k\ge 1\), \(\alpha+k>0\) is a positive root of \(b_f(s)\), which contradicts Theorem 9. ◻
Now, we prove Theorem 4. Assume \(f\in \mathbb{C}[z_1,\dots,z_n]\) is a polynomial of degree \(d\) such that \((f=0)\) defines an irreducible, essential, central hyperplane arrangement \(A\). Let \(L(A)\) be the intersection lattice of \(A\), and let \(\widetilde{L}(A)\) denote the set of all dense edges in \(L(A)\) (see, for instance, [2] for definitions). It is clear that \(d>n\). Take \(\alpha=-k/d\) for some \(k\in \{n,n+1,\dots,d-1\}\). We may further assume \[f=\prod_{i}f_i^{d_i}\] such that each \(f_i\) is a linear polynomial. For each \(W\in \widetilde{L}(A)\), we define \[d_W=\sum_{f_i|_W\equiv 0}d_i \quad \mathrm{and} \quad f_W=\prod_{f_i|_W\equiv 0}f_i^{d_i}.\] We also set \(r_W\) to be the codimension of \(W\in L(A)\). For example, \(d_{\{0\}}=d\) and \(r_{\{0\}}=n\). We define \[\Lambda_\alpha=\left\{W\in \widetilde{L} (A)\mathrel{\Big|} \alpha=-k/d_W \mathrm{ for some } k\in \{r_W,r_W+1,\dots,d_W-1\}\right\}.\] Since \(\{0\}\in \Lambda_\alpha\), \(\Lambda_\alpha\) is nonempty. We then take a \(W_0\in \Lambda_{\alpha}\) such that \[r_{W_0}=\min\{r_W\mid W\in \Lambda_\alpha\}.\] Since the global \(b\)-function is always a multiple of the local one, by discarding all hyperplanes \((f_i=0)\) that do not contain \(W_0\), we can replace \(f\) by \(f_{W_0}\) and \(X\) by \(X/W_0\), keeping the same notation. Therefore, we know that if \(W\) is a dense edge and \(W\neq \{0\}\), then \(\alpha\neq -k/d_W\) for any \(k\in\{r_W,r_W+1,\dots,d_W-1\}\); i.e., \(\Lambda_\alpha=\{\{0\}\}\).
Let \(\mu\colon Y\to X\) be the log resolution obtained by successively blowing up the (strict transforms of the) unions of the dense edges of dimension \(m\) for \(m=0,1,\dots,n-2\). By [22], \(\mu\) is indeed a log resolution of \(A\). Then \[K_{Y/X}=\sum_{W\in \widetilde{L}(A)}(r_W-1)E_W \quad \mathrm{and} \quad \mu^*D=\sum_{W\in \widetilde{L}(A)}d_WE_W.\] Set \(\widetilde{f}=\mu^*(f)\), \(v=dz_1\wedge\dots\wedge dz_n\), and \(\widetilde{v}=\mu^*v\). As in the proof of Theorem 9, we have \[\widetilde{\mathcal{N}}=\widetilde{f}^s\widetilde{v}\cdot \mathscr{D}_Y[s]\subseteq \widetilde{j}_*(\mathcal{M}^r|_U).\] Then, in a small open neighborhood \(V=V_p\subseteq Y\) of a point \(p\in \bigcup_{W\in \widetilde{L}(A)}E_W\), we have \[\label{eq:localwtfhyp} \widetilde{\mathcal{N}}|_V=u^su'\prod_{W\in \widetilde{L}_p(A)} y_{W}^{d_Ws+r_W-1}dy\cdot \mathscr{D}_V[s],\tag{5}\] where \(u,u'\) are invertible regular functions on \(V\), \(y_W\) is the local coordinate defining \(E_W\), \(dy\) is the local volume form, and \(\widetilde{L}_p(A)\mathrel{\vcenter{:}}=\{W\in \widetilde{L}(A)\mid E_W\cap V\neq \emptyset\}\). Since \(u\) and \(u'\) are invertible, we have an isomorphism of sheaves of rings \[\mathscr{D}_V[s]\simeq u^{-s}(u')^{-1}\cdot \mathscr{D}_V[s]\cdot u'u^{s},\] which induces a \(\mathscr{D}_V[s]\)-bimodule isomorphism \(\mathscr{D}_V[s]\simeq u^{-s}(u')^{-1}\cdot \mathscr{D}_V[s]\). Using this isomorphism, we may safely assume \(u=u'=1\) for simplicity.
For an arbitrary rational number \(\beta\in \mathbb{Q}\), we use \(\mathfrak{n}\subseteq \mathbb{C}[s]\) to denote the maximal ideal corresponding to \(\beta\). Then, by Theorem 7, we have \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)=\widetilde{f}^{s+k}\widetilde{v}\cdot \mathscr{D}_Y[s]_{\mathfrak{n}}\subseteq \widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\] for \(k\gg 0\). Since \(\alpha\neq -k/d_W\) for any \(k\in\{r_W,r_W+1,\dots,d_W-1\}\) and for any \(W\in \widetilde{L}(A)\), the local expression 5 and Theorem 7(4) yield \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\simeq \widetilde{f}^s\mathscr{O}_Y(-E_{\{0\}})\widetilde{v}\cdot\mathscr{D}_Y[s]_{\mathfrak{m}},\] where \(\mathfrak{m}\) is the maximal ideal corresponding to \(\alpha=-k/d\). Therefore, we have a short exact sequence \[\label{eq:sesupstairnearby} 0\to \widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\mathrel{\vcenter{:}}=\frac{\widetilde{\mathcal{N}}_{\mathfrak{m}}}{\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)}\longrightarrow \Psi_{\widetilde{f}}^\alpha\longrightarrow \frac{\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)}{\widetilde{\mathcal{N}}_{\mathfrak{m}}}\to 0.\tag{6}\]
By Theorem 7(3) and 5 , locally around \(p\in V\), the quotient \(\frac{\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)}{\widetilde{\mathcal{N}}_{\mathfrak{m}}}\) has a finite filtration such that each successive quotient is of the form \[\frac{y_{W'}^{-1}\prod_{W\in \widetilde{L}_p(A)} y_{W}^{d_Ws+r_W-1-c_W}dy\cdot\mathscr{D}_V[s]_{\mathfrak{m}}}{\prod_{W\in \widetilde{L}_p(A)} y_{W}^{d_Ws+r_W-1-c_W}dy\cdot\mathscr{D}_V[s]_{\mathfrak{m}}}\] with \(0\le c_W< r_{W}-1\) for every \(W\in \widetilde{L}_p(A)\), where \(W'\) runs over the set \(\widetilde{L}(A)\). We only require \(c_W< r_{W}-1\) because \(\alpha<0\). By construction, we know that a successive quotient with respect to \(W'\) is nonzero if and only if \(-\frac{r_{W'}-1-c_{W'}}{d_{W'}}=\alpha\). In particular, all the successive quotients with respect to the edge \(\{0\}\) are zero. Discarding all zero quotients, we isolate a nonzero successive quotient \(\mathcal{Q}_{W'}\) (depending only on \(W'\)) such that \[\label{eq:localquotient} \mathcal{Q}_{W'}\simeq\frac{y_{W'}^{-1}\prod_{W\in \widetilde{L}_p(A)} y_{W}^{d_Ws+r_W-1-c_W}dy\cdot\mathscr{D}_V[s]}{\prod_{W\in \widetilde{L}_p(A)} y_{W}^{d_Ws+r_W-1-c_W}dy\cdot\mathscr{D}_V[s]}\tag{7}\] for \(W'\in \Gamma_\alpha\mathrel{\vcenter{:}}=\left\{W\in \widetilde{L}(A)\mathrel{\big|} -\frac{r_{W}-1-c_{W}}{d_{W}}=\alpha \mathrm{ for some integer } 0 \le c_W<r_W-1\right\}\); necessarily we have \(c_{W'} = -d_{W'}\alpha - r_{W'} + 1\), and for the other \(c_W\) one can make suitable choices. Similarly, for \(p\in E_{\{0\}}\) we have \[\label{eq:localisoQ} \widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}|_V\simeq \frac{y^{k-n}_{\{0\}}\prod_{\substack{W'\in \widetilde{L}(A) \\ W'\neq \{0\}}}y_{W'}^{a_{W'}}\prod_{W\in \widetilde{L}_p(A)} y_{W}^{d_Ws+r_W-1}dy\cdot\mathscr{D}_V[s]}{y^{k-n+1}_{\{0\}}\prod_{\substack{W'\in \widetilde{L}(A) \\ W'\neq \{0\}}}y_{W'}^{a_{W'}}\prod_{W\in \widetilde{L}_p(A)} y_{W}^{d_Ws+r_W-1}dy\cdot\mathscr{D}_V[s]}\tag{8}\] if \(a_{W'}> 0\) for every \(W'\in \widetilde{L}(A)\setminus\{\{0\}\}\).
Lemma 2. With the notation and assumptions as above, we have \[\mathcal{H}^i\mu_+\left(\frac{\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)}{\widetilde{\mathcal{N}}_{\mathfrak{m}}}\right)=0\] for all \(i<0\).
Proof. Locally around \(p\), we consider \(\mathcal{Q}_{W'}\) for every \(W'\in \Gamma_\alpha\). By Kashiwara’s equivalence and Equation 7 , \(\mathcal{Q}_{W'}\) corresponds to a (left) regular holonomic \(\mathscr{D}_{E_{W'}}\)-module generated by \(\bar e\) subject to the relations \[(y_W\partial_{y_W})\cdot \bar e=d_W\cdot \alpha+r_W-1-c_W\] for \(W\neq W'\). In particular, if \(\{0\}\in \widetilde{L}_p(A)\), then \[(y_{\{0\}}\partial_{y_{\{0\}}})\cdot \bar e=-1-c_{\{0\}}<0.\] Therefore, \(\mathcal{Q}_{W'}\) is an \(\mathscr{O}_Y(*E_{\{0\}})=\varinjlim \mathscr{O}_Y(kE_{\{0\}})\)-module. By construction, we then know \(\frac{\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)}{\widetilde{\mathcal{N}}_{\mathfrak{m}}}\) is also an \(\mathscr{O}_Y(*E_{\{0\}})\)-module. Since \(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\) is supported on \(E_{\{0\}}\), using 6 we conclude \[\Psi_{\widetilde{f}}^\alpha\otimes_{\mathscr{O}_Y} \mathscr{O}_Y(*E_{\{0\}})\simeq \frac{\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)}{\widetilde{\mathcal{N}}_{\mathfrak{m}}}\] and \[\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\simeq R\Gamma_{[E_{\{0\}}]}(\Psi_{\widetilde{f}}^\alpha).\] By [11] and 3 , we have a distinguished triangle \[R\Gamma_{[\{0\}]}(\Psi_f^\alpha)\longrightarrow \Psi_f^\alpha\longrightarrow \mu_+\left(\frac{\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)}{\widetilde{\mathcal{N}}_{\mathfrak{m}}}\right)\xrightarrow{+1}.\] Taking the associated long exact sequence, the required statement thus follows. ◻
Lemma 3. With the notation and assumptions as in Lemma 2, we have \[\mathcal{H}^i\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}})=0\] for all \(i<0\) and \(\mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}})\neq 0\).
Proof. Lemma 2, the short exact sequence 6 , and 3 together give us \[\mathcal{H}^i\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}})=0 \quad \mathrm{for} \quad i<0.\]
Now we prove non-vanishing. By Kashiwara’s equivalence and Equation 8 , locally around \(p\in E_{\{0\}}\), \(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\) corresponds to a (left) regular holonomic \(\mathscr{D}_{E_{\{0\}}}\)-module \(\mathcal{E}\) generated by \(e\) subject to the relations \[(y_W\partial_{y_W})\cdot e=d_W\cdot \alpha+r_W-1+a_W\] for \(W\neq \{0\}\). We can choose \(a_W\) such that \(d_W\cdot \alpha+r_W-1+a_W>1\) for every \(W\neq \{0\}\). Write \(E^o_{\{0\}}=E_{\{0\}}\setminus \bigcup_{W\neq \{0\}} E_W\) and \(\widetilde{j}'\colon E^o_{\{0\}}\hookrightarrow E_{\{0\}}\). The flat sections of \(\mathcal{E}|_{E^o_{\{0\}}}\) give a rank-one local system \(\mathcal{L}\). The residue conditions \(d_W\cdot \alpha+r_W-1+a_W>1\) along \(E_{\{0\}}\cap E_{W}\) then imply \[\mathrm{DR}(\mathcal{E})\simeq \widetilde{j}'_!\mathcal{L}[n-1];\] taking the dual as in [14] yields this isomorphism. By [14], \(i_*R\widetilde{\mu}_*(\widetilde{j}'_!\mathcal{L}[n-1])\) is the complex corresponding to \(\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}})\) under the Riemann–Hilbert correspondence (using the de Rham functor), where \(\widetilde{\mu}\colon E_{\{0\}}\to \{0\}\) is the morphism induced by \(\mu\) and \(i\colon \{0\}\hookrightarrow X\) is the closed embedding. By the Riemann–Hilbert correspondence (see, for instance, [14]), it is then enough to prove that the compactly supported cohomology group satisfies \[H_c^{n-1}(E^o_{\{0\}},\mathcal{L})\neq 0.\] By Poincaré–Verdier duality, it suffices to prove \[H^{n-1}(E^o_{\{0\}},\mathcal{L})\neq 0.\] By construction, we have \(E^o_{\{0\}}\simeq \widetilde{U}\mathrel{\vcenter{:}}=\mathbb{P}(\mathbb{C}^n)\setminus \mathbb{P}(A)\). The non-vanishing is a direct consequence of the propagation property of cohomology jumping loci of rank-one local systems on \(\widetilde{U}\). More precisely, by [12], we know that if \(H^{n-1}(\widetilde{U},\mathcal{L})=0\), then \(H^{p}(\widetilde{U},\mathcal{L})=0\) for every \(p\le n-1\). It is well known that \(\widetilde{U}\) has the homotopy type of a finite CW complex of real dimension \(n-1\). Thus, we have the Euler characteristic \(\chi(\widetilde{U},\mathcal{L})=0\). Since \(A\) is irreducible, the topological Euler characteristic \(\chi(\widetilde{U})\neq 0\) by [22]. However, we also know \(\chi(\widetilde{U},\mathcal{L})=\chi(\widetilde{U})\) (by, for instance, [23]), which is a contradiction. ◻
In §2, we discussed direct images of relative \(\mathscr{D}\)-modules. Now, we require the special direct images (cf.[11]): \[\mu_*(\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X) \quad \mathrm{and} \quad \mu_*(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\] where \(\mu_*\) is the (underived) sheaf-theoretic direct image functor. Since \(\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\) is an \(\mathscr{O}_Y[1/\widetilde{f}]\)-module, by (5.2.3) in [14] we obtain a quasi-isomorphism \[\label{eq:torvanj9542} \widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\stackrel{\mathrm{q.i.}}{\simeq}\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\simeq \widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X.\tag{9}\] Therefore, we have \[\mu_*(\widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\simeq j_*(\mathcal{M}^r_{\mathfrak{n}}|_U).\]
To make the local coordinates of \(Y\) more transparent, we recall the wonderful model of De Concini and Procesi for hyperplane arrangements [24]. We consider the open embedding \[X\setminus A\hookrightarrow X\times \prod_{W\in \widetilde{L}(A),\dim W\le n-2}\mathbb{P}(X/W).\] Then \(Y\) is isomorphic to the closure of the above open embedding by [24] (see also [25]). Let \[\mathcal{S}\subseteq \mathcal{G}\mathrel{\vcenter{:}}=\{W^\perp\subseteq (\mathbb{C}^n)^*\mid W\in \widetilde{L}(A), \dim W\le n-2\}\] be a \(\mathcal{G}\)-nested subset, and let \(b\) be a marked basis of \((\mathbb{C}^n)^*\) adapted to \(\mathcal{S}\) such that \[b\simeq\{x_W\mid W^\perp\in \mathcal{S}\}\cup\{x_1,\dots, x_{n-|\mathcal{S}|}\},\] where \(\{x_W\mid W^\perp\in \mathcal{S}\}\) is the marked part and \(\{x_1,\dots, x_{n-|\mathcal{S}|}\}\) is the unmarked part; see [24] for definitions and constructions. In particular, \(b\) gives a coordinate system \(x=(x_v)_{v\in b}\) on \(X=\mathbb{C}^n\). Then, \(\mathcal{S}\) determines an affine open patch \(\mathcal{U}_\mathcal{S}^b\subseteq Y\) with coordinates \[y=(y_v)_{v\in b}=\big((y_W)_{W^\perp\in\mathcal{S}},(y_1,\dots,y_{n-|\mathcal{S}|})\big)\] such that \(\mu_b\mathrel{\vcenter{:}}=\mu|_{\mathcal{U}_\mathcal{S}^b}\colon \mathcal{U}_\mathcal{S}^b\to X\) is given by \[x_v=x_{W'}=\prod_{W^\perp\in \mathcal{S}, W'\supseteq W}y_W\] if \(v\in b\) is marked by \(W'^\perp\in\mathcal{S}\), and \[x_v=y_v\cdot\prod_{W^\perp\in \mathcal{S}, x_v(W)\equiv 0}y_W\] for \(i=1,2,\dots,n-|\mathcal{S}|\); i.e., if \(v\in b\) is unmarked. Moreover, we have an open embedding \(\mathcal{U}_\mathcal{S}^b\hookrightarrow Y_b\) into the affine space \(Y_b\simeq\mathbb{C}^b\) with coordinates \((y_v)_{v\in b}\), whose complement is a normal crossing divisor. Then we have a map \(\rho\colon Y_b\to X\) such that \(\rho|_{\mathcal{U}_\mathcal{S}^b}=\mu_b\). This map induces an \(n\times n\) matrix \(C=[c_{iv}]=[c_v]_{v\in b}\), where \(c_v\) is the \(v\)-th column, whose entries are either \(0\) or \(1\), such that we can uniformly rewrite \[\label{eq:monomialDCP} \rho^*x_v=\prod_{i\in b}y_i^{c_{iv}}\tag{10}\] for every \(v\in b\). Notice that \(\det C=1\).
Let us list some useful properties of the wonderful model:
The marked coordinate \(y_W\) defines the \(\mu\)-exceptional divisor \(E_W\) for \(W^\perp\in\mathcal{S}\).
The coordinate \(y_v\) defines the strict transform of the divisor defined by \(x_v\) for an unmarked \(v\in b\).
We can always make the coordinates \((x_v)_{v\in b}\) a subset of the union of all the linear factors of \(f\), since \(A\) is irreducible, central, and essential.
All such coordinate patches \(\mathcal{U}_\mathcal{S}^b\) cover \(Y\).
(P1), (P2), and (P3) follow directly by construction and definition. (P4) corresponds to [24].
Lemma 4. With the notation as above, we have a quasi-isomorphism \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\stackrel{\mathrm{q.i.}}{\simeq} \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\] for \(\beta\in \mathbb{Q}\). Moreover, the induced morphism \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\to \widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\] is injective.
Proof. Let \(\mathcal{U}_\mathcal{S}^b\subseteq Y\) be a coordinate patch determined by a nested subset \(\mathcal{S}\) and a marked basis \(b\) adapted to \(\mathcal{S}\). If \(v\in b\) is marked by \(W^\perp\in\mathcal{S}\), then \(y_v=y_W\). If \(v\) is unmarked, then \(y_v=y_W\) for some codimension-one dense edge \(W\in \widetilde{L}(A)\) by (P3). By Theorem 7(4), \(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)|_{\mathcal{U}_\mathcal{S}^b}\) is generated by \[e=\prod_{y_W=y_v \mathrm{ with } v\in b} y_{W}^{d_Ws+r_W-1}dy\] for \(k\gg 0\), subject to the relations \[e\cdot (-\partial_{y_v}y_v)=e\cdot (-\partial_{y_W}y_W)=(d_Ws+k\sum_{i\in b}c_{iv}d_i+r_W-1)e\] if \(v\in b\) (marked or unmarked) satisfies \(y_v=y_W\) for some \(W\in \widetilde{L}(A)\). By the construction of the wonderful model, it is possible that \(y_W=y_v\) is invertible on \(\mathcal{U}_\mathcal{S}^b\) but not invertible on \(Y_b\) for some unmarked \(v\in b\); we keep such \(y_W\) in \(e\) to make computations consistent across all \(\mathcal{U}_\mathcal{S}^b\).
To simplify notation, we define \(\gamma_v\mathrel{\vcenter{:}}= d_Ws+r_W\) if \(y_v=y_W\) for some \(W\in \widetilde{L}(A)\), and write \[\theta^k_v=y_v\partial_{y_v}+\gamma_v+k\sum_{i\in b}c_{iv}d_i\] for \(v\in b\). Then, we have a complex: \[K(\theta_v^k)\colon \mathscr{D}_{\mathcal{U}_\mathcal{S}^b}[s]_{\mathfrak{n}}\xrightarrow{\theta_v^k\cdot}\mathscr{D}_{\mathcal{U}_\mathcal{S}^b}[s]_{\mathfrak{n}}.\] Since the operators \(\theta_v^k\) pairwise commute, we define \(K(\mathscr{D}_{\mathcal{U}_\mathcal{S}^b}[s]_{\mathfrak{n}},\theta_v^k)_{v\in b}\) to be the total complex of the \(\mathscr{D}_{\mathcal{U}_\mathcal{S}^b}[s]_{\mathfrak{n}}\)-tensor product of \(K(\theta_v^k)\) as \(v\) runs over all \(v\in b\) (thus, it is a Koszul complex; see, for instance, [26]). By [27], for all \(k\gg0\), \(K(\mathscr{D}_{\mathcal{U}_\mathcal{S}^b}[s]_{\mathfrak{n}},\theta_v^k)_{v\in b}\) gives a free resolution of \(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)|_{\mathcal{U}_\mathcal{S}^b}\). Then, \(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_Y}\mu^*\mathscr{D}_X|_{\mathcal{U}_\mathcal{S}^b}\) is quasi-isomorphic to \[K(\mu^*_b\mathscr{D}_{X}[s]_{\mathfrak{n}},\theta_v^k)_{v\in b}\mathrel{\vcenter{:}}= K(\mathscr{D}_{\mathcal{U}_\mathcal{S}^b}[s]_{\mathfrak{n}},\theta_v^k)_{v\in b}\otimes_{\mathscr{D}_{\mathcal{U}_\mathcal{S}^b}}\mu^*\mathscr{D}_X\] for all \(k\gg 0\). Since \(\mu_b=\rho|_{\mathcal{U}_\mathcal{S}^b}\), we also use the open embedding \(\widetilde{j}_\mathcal{S}\colon \mathcal{U}_\mathcal{S}^b\hookrightarrow Y_b\). Similarly, \(\widetilde{j}_{\mathcal{S}!}(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_{Y_b}}\rho^*\mathscr{D}_X\) is quasi-isomorphic to \[K(\rho^*\mathscr{D}_{X}[s]_{\mathfrak{n}},\theta_v^k)_{v\in b}\mathrel{\vcenter{:}}= K(\mathscr{D}_{Y_b}[s]_{\mathfrak{n}},\theta_v^k)_{v\in b}\otimes_{\mathscr{D}_{Y_b}}\rho^*\mathscr{D}_X\] for all \(k\gg 0\).
Applying the differential \(d\rho\), we have \[d\rho(y_i\partial_{y_i})=\sum_{v\in b} c_{iv}\rho^*x_v\cdot\rho^*\partial_{x_v}=\sum_{v\in b} c_{iv}\prod_{j\in b}y_j^{c_{jv}}\xi_v\] for every \(i\in b\), where we write \(\xi=(\xi_v)_{v\in b}\mathrel{\vcenter{:}}=(\rho^*\partial_{x_v})_{v\in b}\). Clearly, we have the decomposition \[\rho^*\mathscr{D}_X\simeq \bigoplus_{a,h\in \mathbb{N}^b}\mathbb{C}\cdot y^a\xi^h\] where we abbreviate \(y^a=\prod_{v\in b} y_v^{a_v}\) and \(\xi^h=\prod_{v\in b} \xi_v^{h_v}\). Then, for \(v\in b\), \[\theta^k_v(y^a\xi^h)=\left(a_v+\gamma_v+k\sum_{i\in b}c_{iv}d_i\right)y^a\xi^h+\sum_{i\in b} c_{vi}y^{a+c_i}\xi_i\xi^h.\] For \(w\in \mathbb{Z}^b\simeq\mathbb{Z}^n\), we define \[W_w=\bigoplus_{a,h\in \mathbb{N}^b, \, a-C\cdot h=w}\mathbb{C}\cdot y^a\xi^h.\] Thus, \[\rho^*\mathscr{D}_X[s]_{\mathfrak{n}}\simeq \bigoplus_{w\in \mathbb{Z}^b} W_w[s]_{\mathfrak{n}},\] and \(\theta^k_v\) preserves \(W_w[s]_{\mathfrak{n}}\) for every \(w\in \mathbb{Z}^b\) and \(k\in \mathbb{Z}\). Then, \(\widetilde{j}_{\mathcal{S}!}(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_{Y_b}}\rho^*\mathscr{D}_X\) has the induced decomposition, whose \(w\)-th direct summand is \[K(W_w[s]_{\mathfrak{n}}, \theta^k_v)_{v\in b}\] for all \(k\gg 0\). Since \(a=w+C\cdot h\in \mathbb{N}^b\), \(W_w\) is isomorphic to a monomial ideal \(I_w\subseteq \mathbb{C}[\xi]\) via \(y^{w+C\cdot h}\xi^h\mapsto \xi^h\). By Dickson’s Lemma (see, for instance, [26]), \(I_w\) is finitely generated. Then, the \(\theta_v^k\)-action on \(I_w[s]_{\mathfrak{n}}\) becomes \[\theta^k_v(\xi^h)=\left(w_v+\sum_{i\in b}c_{vi}h_i+\gamma_v+k\sum_{i\in b}c_{iv}d_i\right)\xi^h+\sum_{i\in b} c_{vi}\xi_i\xi^h,\] and thus \[\theta_v^k=(w_v+\gamma_v+k\sum_{i\in b}c_{iv}d_i)+\sum_{i\in b}c_{vi}(\xi_i\partial_{\xi_i}+\xi_i).\] Therefore, we know \[K(W_w[s]_{\mathfrak{n}}, \theta^k_v)_{v\in b}\stackrel{\mathrm{q.i.}}{\simeq}K(I_w[s]_{\mathfrak{n}}, \theta^k_v)_{v\in b}.\] Then, set \[(R^k_i)_{i\in b}=C^{-1}\cdot (\theta_v^k)_{v\in b},\] and thus \[R^k_i=(\widetilde{\gamma}_i+kd_i)+(\xi_i\partial_{\xi_i}+\xi_i),\] with \((\widetilde{\gamma}_i)_{i\in b}=C^{-1}\cdot (\gamma_v+w_v)_{v\in b}\in (\mathbb{Z}[s])^b\) and \(C^{-1}\cdot (\gamma_v)_{v\in b}\in (\mathbb{Z}_{\ge0}[s])^b\). Since the Koszul complex is functorial, \(C^{-1}\) acts on \(K(I_w[s]_{\mathfrak{n}},\theta_v^k)_{v\in b}\). Thus, \[C^{-1}\cdot K(I_w[s]_{\mathfrak{n}},\theta_v^k)_{v\in b}\simeq K(I_w[s]_{\mathfrak{n}},R_i^k)_{i\in b}.\] To prove that \(\widetilde{j}_{\mathcal{S}!}(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_{Y_b}}\rho^*\mathscr{D}_X\) is quasi-isomorphic to its \(0\)-th cohomology, it is enough to prove that for every \(w\in \mathbb{Z}^b\), \(K(I_w[s]_{\mathfrak{n}},R_i^k)_{i\in b}\) is quasi-isomorphic to its \(0\)-th cohomology for \(k\gg 0\).
For a fixed \(w\in \mathbb{Z}^b\), we now prove that \(K(I_w[s]_{\mathfrak{n}},R_i^k)_{i\in b}\) is quasi-isomorphic to its \(0\)-th cohomology for \(k\gg 0\). By construction, we can find \(k\gg 0\) such that \[\widetilde{\gamma}_i+kd_i+l\notin \mathfrak{n}\] for all \(l\in \mathbb{Z}_{\ge 0}\) and for every \(i\in b\). Such a \(k\) depends on \(w\). We first assume that \(I_w\) is principal and generated by \(\xi^h\). Then, the morphism \[I_w[s]_{\mathfrak{n}}\xrightarrow{R_i^k} I_w[s]_{\mathfrak{n}}\] is injective, and its image gives the equivalence relations in its cokernel: \[\xi_i\xi^{h+h'}+(\widetilde{\gamma}_i+kd_i+h_i+h'_i)\xi^{h+h'}\equiv 0\] for \(h'\in \mathbb{Z}_{\ge 0}^b\). Since \(\widetilde{\gamma}_i+kd_i+h_i+h'_i\in \mathbb{C}[s]_{\mathfrak{n}}\) is invertible, the complex \[I_w[s]_{\mathfrak{n}}\xrightarrow{R_i^k} I_w[s]_{\mathfrak{n}}\] is quasi-isomorphic to its cokernel, which is isomorphic to \(\xi^h\cdot\mathbb{C}[s]_{\mathfrak{n}}[\xi_j]_{j\neq i}\). By induction, we conclude that \(K(I_w[s]_{\mathfrak{n}},R_i^k)_{i\in b}\) is quasi-isomorphic to \(\xi^h\cdot\mathbb{C}[s]_{\mathfrak{n}}\simeq \mathbb{C}[s]_{\mathfrak{n}}\). If \(I_{w'}\subseteq I_w\) is another principal monomial ideal, then we similarly obtain an isomorphism \[H^0\big(K(I_{w'}[s]_{\mathfrak{n}},R_i^k)_{i\in b}\big)\simeq H^0\big(K(I_{w}[s]_{\mathfrak{n}},R_i^k)_{i\in b}\big)\] induced by \(I_{w'}\subseteq I_w\). In general, assume \(I_w=J+P\), where \(J\) is a monomial ideal generated by \(l-1\) monomials, and \(P\) is a principal monomial ideal. The intersection \(J \cap P\) is generated by the least common multiples of their generators (see [26]), meaning it has at most \(l-1\) monomial generators. We have a short exact sequence: \[0 \to (J\cap P)[s]_{\mathfrak{n}} \xrightarrow{x \mapsto (x, -x)} J[s]_{\mathfrak{n}} \oplus P[s]_{\mathfrak{n}} \xrightarrow{(u,v) \mapsto u+v} I_w[s]_{\mathfrak{n}} \to 0.\] Taking the Koszul complexes and considering the induced long exact sequence in cohomology, by the induction hypothesis we know \(H^q\big(K(I_{w}[s]_{\mathfrak{n}},R_i^k)_{i\in b}\big)=0\) for \(q\le -2\). For \(q=-1\), we have a morphism \[H^{0}\big(K((J\cap P)[s]_{\mathfrak{n}},R_i^k)_{i\in b}\big) \xrightarrow{x \mapsto (x, -x)}H^{0}\big(K(J[s]_{\mathfrak{n}},R_i^k)_{i\in b}\big)\oplus H^{0}\big(K(P[s]_{\mathfrak{n}},R_i^k)_{i\in b}\big)\] whose kernel is \(H^{-1}\big(K(I_w[s]_{\mathfrak{n}},R_i^k)_{i\in b}\big)\). By the induction hypothesis again, this morphism is isomorphic to the diagonal map \[\mathbb{C}[s]_{\mathfrak{n}}\xrightarrow{x \mapsto (x, -x)}\mathbb{C}[s]_{\mathfrak{n}}\oplus\mathbb{C}[s]_{\mathfrak{n}},\] and thus it is injective. Therefore, \(H^{-1}\big(K(I_w[s]_{\mathfrak{n}},R_i^k)_{i\in b}\big)=0\). We thus conclude \[\widetilde{j}_{\mathcal{S}!}(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_{Y_b}}\rho^*\mathscr{D}_X\stackrel{\mathrm{q.i.}}{\simeq} \widetilde{j}_{\mathcal{S}!}(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_{Y_b}}\rho^*\mathscr{D}_X,\] and after restricting to \(\mathcal{U}_\mathcal{S}^b\), \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_Y}\mu^*\mathscr{D}_X|_{\mathcal{U}_{\mathcal{S}}^b}\stackrel{\mathrm{q.i.}}{\simeq} \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X|_{\mathcal{U}_{\mathcal{S}}^b}.\] Since the patches \(\mathcal{U}_{\mathcal{S}}^b\) cover \(Y\), globally we have \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes^L_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\stackrel{\mathrm{q.i.}}{\simeq} \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X.\]
Since \(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\) is torsion-free and thus flat over \(\mathbb{C}[s]_{\mathfrak{n}}\), \[\frac{\mathbb{C}[s]_{\mathfrak{n}}}{\mathfrak{n}\cdot \mathbb{C}[s]_{\mathfrak{n}}}\otimes_{\mathbb{C}[s]_{\mathfrak{n}}}^L\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\stackrel{\mathrm{q.i.}}{\simeq} \frac{\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)}{\mathfrak{n}\cdot \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)}.\] After evaluating \(s=\beta\), the same argument yields \[\frac{\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)}{\mathfrak{n}\cdot \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)}\otimes^L_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\stackrel{\mathrm{q.i.}}{\simeq} \frac{\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)}{\mathfrak{n}\cdot \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X.\] Therefore, we have \[\frac{\mathbb{C}[s]_{\mathfrak{n}}}{\mathfrak{n}\cdot \mathbb{C}[s]_{\mathfrak{n}}}\otimes_{\mathbb{C}[s]_{\mathfrak{n}}}^L\big(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\big)\stackrel{\mathrm{q.i.}}{\simeq} \frac{\mathbb{C}[s]_{\mathfrak{n}}}{\mathfrak{n}\cdot \mathbb{C}[s]_{\mathfrak{n}}}\otimes_{\mathbb{C}[s]_{\mathfrak{n}}}\big(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\big).\] As a consequence, \(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\) is flat and thus torsion-free over \(\mathbb{C}[s]_{\mathfrak{n}}\). Localizing at the generic point of \(\mathrm{Spec }\mathbb{C}[s]_{\mathfrak{n}}\), by Theorem 7(6), the natural map \[\widetilde{j}_!(\mathcal{M}^r_{\eta}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\longrightarrow \widetilde{j}_*(\mathcal{M}^r_{\eta}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\] is an isomorphism. By torsion-freeness, \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\longrightarrow \widetilde{j}_*(\mathcal{M}^r_{\mathfrak{n}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\] is injective. ◻
Now, we complete the proof of Theorem 4. Returning to \(\alpha=-k/d\), by the definition of the special direct images, we have the following commutative diagram: \[\begin{tikzcd}[column sep=small, font=\scriptsize] 0\arrow[r] & j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\arrow[r]\arrow[d,"\simeq"] & \mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}_{\mathfrak{m}})\arrow[r]\arrow[d] & \mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}) \arrow[r]\arrow[d] & 0 \\ 0\arrow[r] & \mu_*(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\arrow[r] & \mu_*(\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\arrow[r] & \mu_*(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\arrow[r] &0. \end{tikzcd}\] The first vertical morphism is an isomorphism by Lemma 4. The first row is exact by Lemma 3. By Lemma 2, we know \[\mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}_{\mathfrak{m}})\hookrightarrow j_*(\mathcal{M}^r_{\mathfrak{m}}|_U).\] In particular, \(\mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}_{\mathfrak{m}})\) has no torsion submodules supported on \(D\). Since \(\mu\) is an isomorphism away from \(D\), the kernel of the second vertical morphism is supported on \(D\). We thus conclude that the second vertical morphism is injective. By construction, we have the following factorization: \[\begin{tikzcd} \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\arrow[rr]\arrow[rd]& & \widetilde{j}_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X \\ &\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\arrow[ru].& \end{tikzcd}\] The injectivity from Lemma 4 then implies that \[\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\to \widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\] is also injective. We then have the following short exact sequence: \[0\to \widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\longrightarrow\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\longrightarrow \widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\to 0.\] Applying \(\mu_*\), since \(R\mu_*(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\simeq \mu_+(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U))\simeq j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\), the second row in the diagram is also exact. By the snake lemma, the third vertical morphism \(\mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}})\to \mu_*(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\) is injective. By Lemma 3, we know \[\mu_*(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\neq 0.\]
As in the proof of Theorem 9, since \(\mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}_{\mathfrak{m}})\hookrightarrow j_*(\mathcal{M}^r_{\mathfrak{m}}|_U)\), the morphism \[\mathscr{O}_X=\mu_*\mathscr{O}_Y\to \mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}_{\mathfrak{m}}),\quad 1\mapsto w=f^sv\] gives a morphism \[\mathscr{O}_X\to \mu_*(\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X).\] By construction, the morphism \(\mathscr{O}_X\to \mu_*(\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\) is induced by the global section \(\widetilde{f}^s\widetilde{v}\otimes 1\) of \(\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\). Therefore, we have \(f^sv=\mu_*(\widetilde{f}^s\widetilde{v}\otimes 1)\), since the second vertical morphism is injective. If the image of \(f^sv\) in \(\mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}})\) is \(0\), then by the injectivity of the third vertical morphism, the image of \(\mu_*(\widetilde{f}^s\widetilde{v}\otimes 1)\) in \(\mu_*(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X)\) is also \(0\). Therefore, \(\widetilde{f}^s\widetilde{v}\otimes 1\) is a global section of \(\widetilde{j}_!(\mathcal{M}^r_{\mathfrak{m}}|_U)\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\). But, \(\widetilde{f}^s\widetilde{v}\otimes 1\) generates \(\widetilde{\mathcal{N}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X\) as an \((\mathscr{O}_Y[s]_\mathfrak m,\mu^{-1}\mathscr{D}_X)\)-bimodule. We thus conclude \[\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}}\otimes_{\mathscr{D}_Y}\mu^*\mathscr{D}_X=0,\] which is a contradiction. Thus, the image of \(f^sv\) in \(\mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}})\) is not \(0\). By Lemma 2, we know \[\mathcal{H}^0\mu_+(\widetilde{\mathcal{N}}^{E_{\{0\}}}_{\mathfrak{m}})\hookrightarrow \Psi_f^\alpha.\] Thus, the \(\mathscr{D}_X[s]_{\mathfrak{m}}\)-submodule generated by the image of \(f^sv\) satisfies \[\frac{f^sv\cdot\mathscr{D}_X[s]_{\mathfrak{m}}}{j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)}\neq 0.\] By Corollary 10, we obtain \[\frac{f^sv\cdot\mathscr{D}_X[s]_{\mathfrak{m}}}{f^{s+1}v\cdot\mathscr{D}_X[s]_{\mathfrak{m}}}=\frac{f^sv\cdot\mathscr{D}_X[s]_{\mathfrak{m}}}{j_!(\mathcal{M}^r_{\mathfrak{m}}|_U)}\neq 0.\] By Lemma 1, we conclude \(b_f(\alpha)=0\), which completes the proof of Theorem 4.