2 Proof of Theorem 1.1

We follow the classical arguments of Gyoja [Reference Kashiwara18], Kashiwara [Reference Malgrange23], Malgrange [Reference Gyoja16], and Sabbah [Reference Mebkhout and Sabbah25]. For the theory of D-modules, we refer to [Reference Dimca12], [Reference Hotta, Takeuchi and Tanisaki17], [Reference Kashiwara20], [Reference Kashiwara and Schapira21], and [Reference Mebkhout and Sabbah25] and use freely the notions and the terminologies in them. Let $\mathbb {C} [s,t]$ be the $\mathbb {C}$ -algebra generated by the two elements $s,t$ satisfying the relation $ts=(s+1)t$ , that is, $[t,s]=t$ . Similarly, we define $\mathbb {C} [s, t^{\pm }]$ , $\mathcal {D}_X [s,t]$ and $\mathcal {D}_X [s, t^{\pm }]$ . Then there exists a natural isomorphism

(2.1) $$ \begin{align} \mathbb{C} [s, t^{\pm}] \overset{\sim}{\longrightarrow} \mathbb{C} [t, \partial_t] \left[ \frac{1}{t} \right] \qquad (s \longmapsto - \partial_t t) \end{align} $$

of $\mathbb {C}$ -algebras (see [Reference Gyoja16]) and the one

(2.2) $$ \begin{align} \mathcal{D}_X [s, t^{\pm}] \overset{\sim}{\longrightarrow} ( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t]) \left[ \frac{1}{t} \right] \end{align} $$

of $\mathcal {D}_X$ -algebras associated with it. In the product space $X \times \mathbb {C}$ , we define a hypersurface $Z \subset X \times \mathbb {C}$ by

(2.3) $$ \begin{align} Z= \{ (x,t) \in X \times \mathbb{C} \ | \ tG(x)-F(x)=0 \}. \end{align} $$

Note that Z is the closure of the graph of the meromorphic function $f= \frac {F}{G}: X \setminus G^{-1}(0) \longrightarrow \mathbb {C}$ in $X \times \mathbb {C}$ . Let

(2.4) $$ \begin{align} \mathcal{H}^1_{[Z]}( \mathcal{O}_{X \times \mathbb{C}}) \simeq \frac{ \mathcal{O}_{X \times \mathbb{C}} [ \frac{1}{tG-F} ]}{ \mathcal{O}_{X \times \mathbb{C}} } \end{align} $$

be the first local cohomology sheaf of $\mathcal {O}_{X \times \mathbb {C}}$ along $Z \subset X \times \mathbb {C}$ and define a regular holonomic $\mathcal {D}_{X \times \mathbb {C}}$ -module $\mathcal {M}$ by

(2.5) $$ \begin{align} \mathcal{M} : = \{ \mathcal{H}^1_{[Z]}( \mathcal{O}_{X \times \mathbb{C}}) \} \left[ \frac{1}{G} \right] \simeq \frac{\mathcal{O}_{X \times \mathbb{C}} [ \frac{1}{(tG-F)G} ] }{\mathcal{O}_{X \times \mathbb{C}} [ \frac{1}{G} ]}, \end{align} $$

which is endowed with the canonical section

(2.6) $$ \begin{align} \delta (t-f(x)):= \left[ \frac{1}{t-f(x)} \right]= \left[ \frac{G(x)}{tG(x)-F(x)} \right] \in \mathcal{M}. \end{align} $$

Unlike the classical case where $f$ is holomorphic, this section does not necessarily generate $\mathcal {M}$ over $\mathcal {D}_{X \times \mathbb {C}}$ (see Lemma 3.2). Nevertheless, as in [Reference Malgrange23] and [Reference Gyoja16], for any nonnegative integer $m \geq 0$ , there exists an isomorphism

(2.7) $$ \begin{align} \mathcal{D}_X [s, t^{\pm}] \left( \frac{1}{G^m} f^{s} \right) \overset{\sim}{\longrightarrow} ( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t]) \left[ \frac{1}{t} \right] \left( \frac{1}{G^m} \delta (t-f(x)) \right) \end{align} $$

( $\frac {1}{G^m} f^{s} \longmapsto \frac {1}{G^m} \delta (t-f(x))$ ) on a neighborhood of $F^{-1}(0) \subset X$ which is linear over $\mathcal {D}_X [s, t^{\pm }] \simeq ( \mathcal {D}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t, \partial _t]) [ \frac {1}{t}]$ . Since there is no nonzero section of $\mathcal {M}$ supported in $G^{-1}(0) \times \mathbb {C} \subset X \times \mathbb {C}$ by Hilbert’s nullstellensatz, to show (2.7), it suffices to compare the annihilators of the generators of its both sides on $X \setminus G^{-1}(0)$ . Here, the right-hand side of (2.7) is understood to be a subsheaf of $( \mathcal {M} |_{ \{ t=0 \} } ) [ \frac {1}{t}]$ and the multiplication by t on it corresponds to the action $s \longmapsto s+1$ on the left-hand side (see, e.g., [Reference Gyoja16] for details). Restricting the isomorphism (2.7) to a subsheaf, we obtain an isomorphism

(2.8) $$ \begin{align} \mathcal{D}_X [s] \left( \frac{1}{G^m} f^{s} \right) \overset{\sim}{\longrightarrow} \mathcal{D}_X [- \partial_t t] \left( \frac{1}{G^m} \delta (t-f(x)) \right). \end{align} $$

Now, let us consider the V-filtration $\{ V_j( \mathcal {D}_{X \times \mathbb {C}} ) \}_{j \in \mathbb {Z}}$ of $\mathcal {D}_{X \times \mathbb {C}}$ along the hypersurface $\{ t=0 \} = X \times \{ 0 \} \subset X \times \mathbb {C}$ . Similarly, we define a filtration $\{ V_j( \mathcal {D}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t, \partial _t] ) \}_{j \in \mathbb {Z}}$ of $\mathcal {D}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t, \partial _t] \subset \mathcal {D}_{X \times \mathbb {C}} |_{ \{ t=0 \} }$ . Denote the section

(2.9) $$ \begin{align} \frac{1}{G^m} \delta (t-f(x)) \in \mathcal{M} |_{ \{ t=0 \} } \end{align} $$

of $\mathcal {M} |_{ \{ t=0 \} }$ simply by $\sigma _m$ . Then, by $t \cdot \delta (t-f)=f \cdot \delta (t-f)$ , we obtain isomorphisms

(2.10) $$ \begin{align} V_0( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \sigma_m & \simeq \sum_{k=0}^{+ \infty} \mathcal{D}_X [s] \left( \frac{1}{G^m} f^{s+k} \right), \end{align} $$

(2.11) $$ \begin{align} V_{-1}( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \sigma_m & \simeq \sum_{k=1}^{+ \infty} \mathcal{D}_X [s] \left( \frac{1}{G^m} f^{s+k} \right). \end{align} $$

This implies that the $V_0( \mathcal {D}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t, \partial _t] )$ -module

(2.12) $$ \begin{align} \mathcal{K} := \frac{V_{0}( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \sigma_m }{V_{-1}( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \sigma_m } \end{align} $$

is isomorphic to

(2.13) $$ \begin{align} \frac{ \sum_{k=0}^{+ \infty} \mathcal{D}_X [s] ( \frac{1}{G^m} f^{s+k} ) }{ \sum_{k=1}^{+ \infty} \mathcal{D}_X [s] ( \frac{1}{G^m} f^{s+k} ) }. \end{align} $$

Here, we used the identification

(2.14) $$ \begin{align} V_{0}( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \simeq \mathcal{D}_X [s,t] \end{align} $$

given by $- \partial _t t \longmapsto s$ . Moreover, by Lemma 2.1, there also exists an isomorphism

(2.15) $$ \begin{align} ( \mathcal{O}_{X \times \mathbb{C}} |_{ \{ t=0 \} } ) \otimes_{\mathcal{O}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t]} \mathcal{K} \simeq \mathcal{K}^{\infty} := \frac{ V_0( \mathcal{D}_{X \times \mathbb{C}} )\sigma_m }{V_{-1}( \mathcal{D}_{X \times \mathbb{C}} )\sigma_m}. \end{align} $$

By the classical result on the specializability of $\mathcal {M}$ along $\{ t=0 \}$ , there exists a nonzero polynomial $b(s) \in \mathbb {C} [s]$ such that

(2.16) $$ \begin{align} b( - \partial_t t ) \sigma_m \in V_{-1}( \mathcal{D}_{X \times \mathbb{C}} ) \sigma_m. \end{align} $$

This condition is equivalent to the one that the image

(2.17) $$ \begin{align} \mathcal{G} := \mathrm{Im} [ b( - \partial_t t ) : \mathcal{K}^{\infty} \longrightarrow \mathcal{K}^{\infty} ] \end{align} $$

is zero. Note that the sheaf homomorphism

(2.18) $$ \begin{align} b( - \partial_t t ) : \mathcal{K} \longrightarrow \mathcal{K} \end{align} $$

is $\mathcal {O}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t]$ -linear and the above one in (2.17) is obtained by applying the tensor product $( \mathcal {O}_{X \times \mathbb {C}} |_{ \{ t=0 \} } ) \otimes _{\mathcal {O}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t]} ( \cdot )$ to it. Since $\mathcal {O}_{X \times \mathbb {C}} |_{ \{ t=0 \} } $ is flat over $\mathcal {O}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t]$ , we thus obtain an isomorphism

(2.19) $$ \begin{align} \mathcal{G} \simeq ( \mathcal{O}_{X \times \mathbb{C}} |_{ \{ t=0 \} } ) \otimes_{\mathcal{O}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t]} \mathrm{Im} [b( - \partial_t t ) : \mathcal{K} \longrightarrow \mathcal{K} ]. \end{align} $$

By $\mathcal {G} \simeq 0$ and the faithfully flatness of $\mathcal {O}_{X \times \mathbb {C}} |_{ \{ t=0 \} } $ over $\mathcal {O}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t]$ , we obtain also

(2.20) $$ \begin{align} \mathrm{Im} [b( - \partial_t t ) : \mathcal{K} \longrightarrow \mathcal{K} ] \simeq 0. \end{align} $$

It follows from the previous description (2.13) of $\mathcal {K}$ that we have the desired condition

(2.21) $$ \begin{align} b(s) \left( \frac{1}{G^m} f^{s} \right) \in \sum_{k=1}^{+ \infty} \mathcal{D}_X [s] \left( \frac{1}{G^m} f^{s+k} \right). \end{align} $$

This completes the proof. $\Box $

Proof. By our construction of the regular holonomic $\mathcal {D}_{X \times \mathbb {C}}$ -module $\mathcal {M}$ in the proof of Theorem 1.1, there exists a natural morphism

(2.22) $$ \begin{align} \Phi : \mathcal{M}_0 := \frac{( \mathcal{O}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t] ) [ \frac{1}{(tG-F)G} ] }{( \mathcal{O}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t] ) [ \frac{1}{G} ]} \longrightarrow \mathcal{M} |_{ \{ t=0 \} } \end{align} $$

of $\mathcal {O}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t]$ -modules. Since $F, G \in \mathcal {O}_X$ are coprime each other, the same is true also for $tG-F, G \in \mathcal {O}_{X \times \mathbb {C}}$ , and hence the morphism $\Phi $ is injective. Therefore, for $j=0,-1$ , the $V_{0}( \mathcal {D}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t, \partial _t] )$ -module $V_{j}( \mathcal {D}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t, \partial _t] ) \sigma _m \subset \mathcal {M} |_{ \{ t=0 \} }$ is isomorphic to the image of the morphism

(2.23) $$ \begin{align} V_{j}( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \longrightarrow \mathcal{M}_0 \qquad (P \longmapsto P \sigma_m ). \end{align} $$

By the isomorphisms

(2.24) $$ \begin{align} ( \mathcal{O}_{X \times \mathbb{C}} |_{ \{ t=0 \} } ) \otimes_{\mathcal{O}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t]} V_{j}( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \simeq V_j( \mathcal{D}_{X \times \mathbb{C}} ) |_{ \{ t=0 \} } \qquad (j=0,-1), \end{align} $$

(2.25) $$ \begin{align} ( \mathcal{O}_{X \times \mathbb{C}} |_{ \{ t=0 \} } ) \otimes_{\mathcal{O}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t]} \mathcal{M}_0 \simeq \mathcal{M} |_{ \{ t=0 \} } \end{align} $$

and the flatness of $\mathcal {O}_{X \times \mathbb {C}} |_{ \{ t=0 \} } $ over $\mathcal {O}_X \otimes _{\mathbb {C}_X} \mathbb {C}_X[t]$ , we obtain isomorphisms

$$ \begin{align*} V_j( \mathcal{D}_{X \times \mathbb{C}} ) \sigma_m = & \mathrm{Im}[ V_j( \mathcal{D}_{X \times \mathbb{C}} )|_{ \{ t=0 \} } \longrightarrow \mathcal{M} |_{ \{ t=0 \} } ] \\ \simeq & ( \mathcal{O}_{X \times \mathbb{C}} |_{ \{ t=0 \} } ) \otimes_{\mathcal{O}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t]} \mathrm{Im}[ V_j( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \longrightarrow \mathcal{M}_0 ] \\ \simeq & ( \mathcal{O}_{X \times \mathbb{C}} |_{ \{ t=0 \} } ) \otimes_{\mathcal{O}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t]} V_j( \mathcal{D}_X \otimes_{\mathbb{C}_X} \mathbb{C}_X[t, \partial_t] ) \sigma_m \qquad (j=0,-1). \end{align*} $$

Then the assertion immediately follows.

3 Proof of Theorem 1.4

First of all, we shall recall the classical theory of Kashiwara–Malgrange filtrations. For more precise explanations on them, we refer to [Reference Kashiwara19] and [Reference Mebkhout and Sabbah25]. We assume first that $\mathcal {M}$ is a general regular holonomic $\mathcal {D}_{X \times \mathbb {C}}$ -module on the product of a complex manifold X and $\mathbb {C}_t$ . Set $\theta =t \partial _t \in \mathcal {D}_{X \times \mathbb {C}}$ and for a section $\sigma \in \mathcal {M}$ of $\mathcal {M}$ denote by $p_{\sigma }(s) \in \mathbb {C} [s]$ the minimal polynomial such that

(3.1) $$ \begin{align} p_{\sigma}( \theta ) \sigma \in V_{-1}( \mathcal{D}_{X \times \mathbb{C}} ) \sigma. \end{align} $$

Furthermore, we set

(3.2) $$ \begin{align} \mathrm{ord}_{ \{ t=0 \} } ( \sigma ):= p_{\sigma}^{-1}( 0 ) \subset \mathbb{C}. \end{align} $$

On the set $\mathbb {C}$ of complex numbers, let us consider the lexicographic order $\geq $ defined by

(3.3) $$ \begin{align} z \geq w \ \Longleftrightarrow \ \mathrm{Re} z> \mathrm{Re} w \quad \text{or} \quad \mathrm{Re} z = \mathrm{Re} w, \ \mathrm{Im} z \geq \mathrm{Im} w. \end{align} $$

Then, for $\alpha \in \mathbb {C}$ , we define a $V_0( \mathcal {D}_{X \times \mathbb {C}} )$ -submodule $V_{\alpha }( \mathcal {M} )$ of $\mathcal {M}$ by

(3.4) $$ \begin{align} V_{\alpha}( \mathcal{M} )= \{ \sigma \in \mathcal{M} \ | \ \mathrm{ord}_{ \{ t=0 \} } ( \sigma ) \geq - \alpha -1 \}. \end{align} $$

We can easily see that there exists a finite subset $A \subset \{ z \in \mathbb {C} \ | \ -1 \leq z <0 \}$ such that, for any section $\sigma \in \mathcal {M}$ of $\mathcal {M}$ , we have

(3.5) $$ \begin{align} \mathrm{ord}_{ \{ t=0 \} } ( \sigma ) \subset A+ \mathbb{Z}. \end{align} $$

Moreover, for each element $\alpha \in A$ of such $A$ , the filtration $\{ V_{\alpha +j}( \mathcal {M} ) \}_{j \in \mathbb {Z}}$ of $\mathcal {M}$ is a good V-filtration. For $\alpha \in A+ \mathbb {Z}$ , we set

(3.6) $$ \begin{align} V_{< \alpha}( \mathcal{M} )= \bigcup_{\beta < \alpha} V_{\beta}( \mathcal{M} ) = \{ \sigma \in \mathcal{M} \ | \ \mathrm{ord}_{ \{ t=0 \} } ( \sigma )>- \alpha -1 \} \end{align} $$

and

(3.7) $$ \begin{align} \mathrm{gr}^V_{\alpha}( \mathcal{M} )= V_{\alpha}( \mathcal{M} )/ V_{< \alpha}( \mathcal{M} ). \end{align} $$

Then $\mathrm {gr}^V_{\alpha }( \mathcal {M} )$ is a regular holonomic $\mathcal {D}_X$ -module and we can easily show that there exists $N \gg 0$ such that

(3.8) $$ \begin{align} ( \theta + \alpha +1)^N \mathrm{gr}^V_{\alpha}( \mathcal{M} ) =0. \end{align} $$

The following lemma is well known to the specialists.

Now, we return to the situation in the proof of Theorem 1.1. Namely, for the meromorphic function $f= \frac {F}{G}$ , we have

(3.12) $$ \begin{align} \mathcal{M} \simeq \frac{\mathcal{O}_{X \times \mathbb{C}} [ \frac{1}{(tG-F)G} ] }{\mathcal{O}_{X \times \mathbb{C}} [ \frac{1}{G} ]} \end{align} $$

and

(3.13) $$ \begin{align} \sigma_m = \frac{1}{G^m} \delta (t-f(x)) = \left[ \frac{G}{(tG-F)G^m} \right] \in \mathcal{M} \qquad (m \geq 0). \end{align} $$

Then we have the following result, whose proof is inspired from Sabbah’s exposition [Reference Sabbah35].

Proof. Set $g:=(tG(x)-F(x)) \cdot G(x) \in \mathcal {O}_{X \times \mathbb {C}}$ , and let $b_g(s) \in \mathbb {C} [s]$ be its Bernstein–Sato polynomial. Then by [Reference Saito36, Th. 0.4], for any root $\alpha \in \mathbb {Q}$ of $b_g(s)$ , we have

(3.14) $$ \begin{align} - \mathrm{dim} (X \times \mathbb{C})= - \mathrm{dim} X -1 < \alpha <0. \end{align} $$

Moreover, for any $k \geq 1$ , there exists $P_k(s) \in \mathcal {D}_{X \times \mathbb {C}} [s]$ such that

(3.15) $$ \begin{align} b_g(s-k) \cdots \cdots b_g(s-2) b_g(s-1) g^{s-k} = P_k(s) g^s. \end{align} $$

Set $n:= \mathrm {dim} X$ . Then, by substituting $s$ in the above formula by $-n$ , we see that for any $k \geq 1$ the meromorphic function $g^{-n-k}$ is a nonzero constant multiple of $P_k(-n) g^{-n}$ . This implies that

(3.16) $$ \begin{align} \mathcal{M} \simeq \frac{\mathcal{O}_{X \times \mathbb{C}} [ \frac{1}{(tG-F)G} ] }{\mathcal{O}_{X \times \mathbb{C}} [ \frac{1}{G} ]} \end{align} $$

is generated by its section

(3.17) $$ \begin{align} \left[ \frac{1}{g^n} \right] = \left[ \frac{1}{(tG-F)^nG^n} \right] \in \mathcal{M} \end{align} $$

over $\mathcal {D}_{X \times \mathbb {C}}$ . On the other hand, the section $\partial _t^{n-1} \sigma _m \in \mathcal {M}$ of $\mathcal {M}$ is a nonzero constant multiple of

(3.18) $$ \begin{align} \left[ \frac{1}{(tG-F)^{n}G^{m-n}} \right] \in \mathcal{M}. \end{align} $$

Therefore, if $m \geq 2n= 2 \mathrm {dim} X$ , it generates $\mathcal {M}$ over $\mathcal {D}_{X \times \mathbb {C}}$ .

Now, let us prove Theorem 1.4. By the proof of Theorem 1.1 and the correspondence $s \longleftrightarrow - \partial _t t = - \theta -1$ , the Bernstein–Sato polynomial $b_{f,m}^{\mathrm {mero}}(s)$ of $f$ coincides with $p_{\sigma _m}(-s-1)$ . This, in particular, implies that we have

(3.19) $$ \begin{align} (b_{f,m}^{\mathrm{mero}})^{-1}(0)= \{ - \lambda -1 \ | \ \lambda \in \mathrm{ord}_{ \{ t=0 \} } ( \sigma_m ) \}. \end{align} $$

Note also that for the $\mathcal {D}_X$ -module $\mathcal {M}$ in the proof of Theorem 1.1, we have an isomorphism

(3.20) $$ \begin{align} \mathrm{DR}_{X \times \mathbb{C}} ( \mathcal{M} ) \simeq R \Gamma_{(X \setminus G^{-1}(0)) \times \mathbb{C}} ( \mathbb{C}_Z ) [n] \end{align} $$

and the nearby cycle sheaf $\psi _t(\mathrm {DR}_{X \times \mathbb {C}} ( \mathcal {M} ) )$ coincides with the meromorphic nearby cycle $\psi _f^{\mathrm {mero}} ( \mathbb {C}_X )$ introduced in [Reference Nguyen and Takeuchi29] up to some shift. Assume first that $m \geq 2 \mathrm {dim} X$ . Then, by Lemma 3.2, the section $\sigma _m \in \mathcal {M}$ generates $\mathcal {M}$ over $\mathcal {D}_{X \times \mathbb {C}}$ and the second assertion of Theorem 1.4 follows from Lemma 3.1, Kashiwara’s isomorphism

(3.21) $$ \begin{align} \bigoplus_{-1 \leq \alpha <0} \mathrm{DR}_X ( \mathrm{gr}^V_{\alpha}( \mathcal{M} ) ) \simeq \psi_t( \mathrm{DR}_{X \times \mathbb{C}} ( \mathcal{M} )) \simeq \bigoplus_{-1 \leq \alpha <0} \psi_{t, \exp (2 \pi i \alpha )} ( \mathrm{DR}_{X \times \mathbb{C}} ( \mathcal{M} )) \end{align} $$

and [Reference Nguyen and Takeuchi29, Lem. 2.1 (iii)]. If we do not have the condition $m \geq 2 \mathrm {dim} X$ , by considering the $\mathcal {D}_{X \times \mathbb {C}}$ -submodule $\mathcal {D}_{X \times \mathbb {C}} \sigma _m \subset \mathcal {M}$ instead of $\mathcal {M}$ itself, we obtain the first assertion of Theorem 1.4. This completes the proof. $\Box $

By the proofs of Theorems 1.1 and 1.4, we obtain the following weak relation among the roots of the b-functions $b_{f,m}^{\mathrm {mero}}(s)$ for various $m \geq 0$ .

Lemma 3.3. Let $m, m^{\prime } \geq 0$ be two nonnegative integers such that $m \geq m^{\prime }$ . Then, for some $l \gg 0$ , we have an inclusion

(3.22) $$ \begin{align} (b_{f,m^{\prime}}^{\mathrm{mero}})^{-1}(0) \subset \bigcup_{i=0}^l \left\{ (b_{f,m}^{\mathrm{mero}})^{-1}(0)-i \right\}. \end{align} $$

Proof. By the proofs of Theorems 1.1 and 1.4, we have $b_{f,m}^{\mathrm {mero}}(s)=p_{\sigma _m}(-s-1)$ and $b_{f,m^{\prime }}^{\mathrm {mero}}(s)= p_{\sigma _{m^{\prime }}}(-s-1)$ . Moreover, by our assumption $m \geq m^{\prime }$ , we have

(3.23) $$ \begin{align} \sigma_{m^{\prime}} = G^{m-m^{\prime}} \cdot \sigma_m \in \mathcal{O}_X \sigma_m \subset V_0( \mathcal{D}_{X \times \mathbb{C}} ) \sigma_m. \end{align} $$

Set $\mathcal {N} := \mathcal {D}_{X \times \mathbb {C}} \sigma _m$ and $\mathcal {N}^{\prime } := \mathcal {D}_{X \times \mathbb {C}} \sigma _{m^{\prime }}$ . Then the V-filtration $\{ V_j( \mathcal {D}_{X \times \mathbb {C}} ) \sigma _m \}_{j \in \mathbb {Z}}$ (resp. $\{ V_j( \mathcal {D}_{X \times \mathbb {C}} ) \sigma _{m^{\prime }} \}_{j \in \mathbb {Z}}$ ) of $\mathcal {N}$ (resp. $\mathcal {N}^{\prime }$ ) is good. By Artin-Rees’s lemma, the V-filtration $\{ U_j( \mathcal {N}^{\prime } ) \}_{j \in \mathbb {Z}}$ of $\mathcal {N}^{\prime }$ defined by

(3.24) $$ \begin{align} U_j( \mathcal{N}^{\prime} ):= \mathcal{N}^{\prime} \cap \left( V_j( \mathcal{D}_{X \times \mathbb{C}} ) \sigma_m \right) \qquad (j \in \mathbb{Z} ) \end{align} $$

is also good and satisfies the condition $\sigma _{m^{\prime }} \in U_0( \mathcal {N}^{\prime } )$ . Then there exists $l \gg 0$ such that

(3.25) $$ \begin{align} U_{-l-1}( \mathcal{N}^{\prime} ) \subset V_{-1}( \mathcal{D}_{X \times \mathbb{C}} ) \sigma_{m^{\prime}} \subset \mathcal{N}^{\prime}. \end{align} $$

This implies that we have

(3.26) $$ \begin{align} p_{\sigma_m}( \theta -l) \cdots p_{\sigma_m}( \theta -1) p_{\sigma_m}( \theta ) \sigma_{m^{\prime}} \in V_{-1}( \mathcal{D}_{X \times \mathbb{C}} ) \sigma_{m^{\prime}}. \end{align} $$

Then the assertion immediately follows.

4 Upper and lower bounds for the roots of b-functions

Recall that in [Reference Kashiwara18], Kashiwara proved that if $f$ is holomorphic, the roots of the Bernstein–Sato polynomial $b_f(s)$ are negative rational numbers. In this section, we prove an analogous result for the meromorphic function $f= \frac {F}{G}$ . We can easily prove that the roots of our b-function $b_{f,m}^{\mathrm {mero}}(s)$ are rational numbers, but their negativity does not follow from our proof. For this reason, here we only give an upper bound

(4.1) $$ \begin{align} (b_{f,m}^{\mathrm{mero}})^{-1}(0) \subset B_{f,m}^{\pi} \subset \mathbb{Q} \qquad (m \geq 0) \end{align} $$

for the set $(b_{f,m}^{\mathrm {mero}})^{-1}(0)$ in terms of resolutions of singularities $\pi $ of $D \subset X$ . The precise statement is as follows. Let $\pi : Y \longrightarrow X$ be a resolution of singularities of the divisor $D= F^{-1}(0) \cup G^{-1}(0) \subset X$ , which means that $\pi : Y \longrightarrow X$ is a proper morphism of n-dimensional complex manifolds such that $\pi ^{-1}(D) \subset Y$ is normal crossing and $\pi |_{Y \setminus \pi ^{-1}(D)}: Y \setminus \pi ^{-1}(D) \longrightarrow X \setminus D$ is an isomorphism. Then we define a meromorphic function g on Y by

(4.2) $$ \begin{align} g:=f \circ \pi = \frac{F \circ \pi}{G \circ \pi}. \end{align} $$

From now on, we fix a nonnegative integer $m \geq 0$ and consider the (local) Bernstein–Sato polynomials of $g$ of order $m$ . At each point $q \in \pi ^{-1}(D)$ of the normal crossing divisor $\pi ^{-1}(D)$ , there exists a local coordinate system $y=(y_1,y_2, \ldots , y_n)$ such that $q=(0,0, \ldots , 0)$ and

(4.3) $$ \begin{align} (F \circ \pi )(y) = \prod_{i=1}^n y_i^{a_i} \quad (a_i \geq 0), \qquad (G \circ \pi )(y) = \prod_{i=1}^n y_i^{b_i} \quad (b_i \geq 0). \end{align} $$

Then we have

(4.4) $$ \begin{align} \left( \frac{1}{G^m} f^{s} \right) (y) = \prod_{i=1}^n y_i^{(a_i-b_i)s-m b_i}. \end{align} $$

It follows that the set $K_q \subset \mathbb {Q}$ of the roots of the (local) Bernstein–Sato polynomial of $g$ at $q$ is explicitly given by

(4.5) $$ \begin{align} K_q= \bigcup_{i: \ a_i> b_i} \left\{ \frac{m b_i}{a_i-b_i} - \frac{k}{a_i-b_i} \ | \ 1 \leq k \leq a_i-b_i \right\} \subset \mathbb{Q} \end{align} $$

(see, e.g., [Reference Kashiwara20, Lem. 6.10]). It is clear that this set $K_q$ does not depend on the choice of the local coordinates. For the point $x_0 \in D$ , its inverse image $\pi ^{-1}(x_0) \subset \pi ^{-1}(D)$ being compact, we obtain a finite subset

(4.6) $$ \begin{align} K:= \bigcup_{q \in \pi^{-1}(x_0)} K_q \subset \mathbb{Q}. \end{align} $$

Theorem 4.1. For any $m \geq 0$ , the roots of the (local) Bernstein–Sato polynomial $b_{f,m}^{\mathrm {mero}}(s)$ of $f$ at $x_0 \in D$ are contained in the set

(4.7) $$ \begin{align} B_{f,m}^{\pi} := \bigcup_{l=0,1,2, \ldots} \left( K-l \right) = \{ r-l \ | \ r \in K, \ l=0,1,2, \ldots \} \subset \mathbb{Q}. \end{align} $$

In particular, for $m=0$ , the roots of $b_{f,0}^{\mathrm {mero}}(s)$ are negative rational numbers.

Proof. Our proof is similar to the one in [Reference Kashiwara18] for the case where $f$ is holomorphic. But we also need some new ideas to treat the meromorphic case. Recall that for the section $\sigma := \sigma _m \in \mathcal {M}$ (see (2.5)) of the regular holonomic $\mathcal {D}_{X \times \mathbb {C}}$ -module $\mathcal {M}$ , we denote by $p_{\sigma }(s) \in \mathbb {C} [s]$ , the minimal polynomial $p(s) \not = 0$ such that

(4.8) $$ \begin{align} p( \theta ) \sigma \in V_{-1}( \mathcal{D}_{X \times \mathbb{C}} ) \sigma \end{align} $$

and we have $b_{f,m}^{\mathrm {mero}}(s) = p_{\sigma }(-s-1)$ . Let $i: Y \longrightarrow Y \times X$ ( $y \longmapsto (y, \pi (y))$ ) be the graph embedding by $\pi $ and $p: Y \times X \rightarrow X$ ( $(y, x) \longmapsto x$ ), the projection such that $\pi = p \circ i$ . We set also

$$ \begin{align*} \tilde{i} :=i \times \mathrm{id}_{\mathbb{C}} : Y \times \mathbb{C} \longrightarrow (Y \times X) \times \mathbb{C}, \\ \tilde{p} :=p \times \mathrm{id}_{\mathbb{C}} : (Y \times X) \times \mathbb{C} \longrightarrow X \times \mathbb{C,} \end{align*} $$

so that we have $\tilde {\pi } := \pi \times \mathrm {id}_{\mathbb {C}} = \tilde {p} \circ \tilde {i}$ . As in the case of the meromorphic function $f= \frac {F}{G}$ , we define a regular holonomic $\mathcal {D}_{Y \times \mathbb {C}}$ -module $\mathcal {N}$ associated with $g= \frac {F \circ \pi }{G \circ \pi }$ and its section

(4.9) $$ \begin{align} \tau := \frac{1}{\{ (G \circ \pi )(y) \}^m} \delta (t-g(y)) \in \mathcal{N}. \end{align} $$

Then the roots of its minimal polynomial $p_{\tau }(s) \in \mathbb {C} [s]$ such that

(4.10) $$ \begin{align} p_{\tau}( \theta ) \tau \in V_{-1}( \mathcal{D}_{Y \times \mathbb{C}} ) \tau \end{align} $$

is contained in the set $\{ -r-1 \ | \ r \in K \} \subset \mathbb {Q}$ . Since $\tilde {\pi } : Y \times \mathbb {C} \longrightarrow X \times \mathbb {C}$ is an isomorphism over $(Y \setminus \pi ^{-1}(D)) \times \mathbb {C} \simeq (X \setminus D) \times \mathbb {C}$ , the section $\tau \in \mathcal {N}$ is naturally identified with $\sigma \in \mathcal {M}$ there. Let

(4.11) $$ \begin{align} \mathbf{D} \tilde{i}_* \mathcal{N} \simeq H^0 \mathbf{D} \tilde{i}_* \mathcal{N} = \tilde{i}_* \left( \mathcal{D}_{(Y \times X) \times \mathbb{C} \leftarrow Y \times \mathbb{C}} \otimes_{\mathcal{D}_{Y \times \mathbb{C}}} \mathcal{N} \right) \end{align} $$

be the direct image of $\mathcal {N}$ by $\tilde {i}$ and

(4.12) $$ \begin{align} \widetilde{\tau} := 1_{(Y \times X) \times \mathbb{C} \leftarrow Y \times \mathbb{C}} \otimes \tau \in \mathbf{D} \tilde{i}_* \mathcal{N}, \end{align} $$

its section defined by $\tau \in \mathcal {N}$ . Then it is easy to see that the minimal polynomial $p_{\widetilde {\tau }}(s) \in \mathbb {C} [s]$ such that

(4.13) $$ \begin{align} p_{\widetilde{\tau}}( \theta ) \widetilde{\tau} \in V_{-1}( \mathcal{D}_{(Y \times X) \times \mathbb{C}} ) \widetilde{\tau} \end{align} $$

is equal to $p_{\tau }(s)$ . Let us consider the $\mathcal {D}_{Y \times \mathbb {C}}$ -submodule $\mathcal {N}_0:= \mathcal {D}_{Y \times \mathbb {C}} \tau \subset \mathcal {N}$ of $\mathcal {N}$ generated by $\tau \in \mathcal {N}$ . Then we have $\widetilde {\tau } \in \mathbf {D} \tilde {i}_* \mathcal {N}_0 \subset \mathbf {D} \tilde {i}_* \mathcal {N}$ and $\mathcal {D}_{(Y \times X) \times \mathbb {C}} \widetilde {\tau } = \mathbf {D} \tilde {i}_* \mathcal {N}_0$ . Hence, we can define a good V-filtration $\{ U_j( \mathbf {D} \tilde {i}_* \mathcal {N}_0 ) \}_{j \in \mathbb {Z}}$ of $\mathbf {D} \tilde {i}_* \mathcal {N}_0$ by

(4.14) $$ \begin{align} U_j( \mathbf{D} \tilde{i}_* \mathcal{N}_0 ):= V_j( \mathcal{D}_{(Y \times X) \times \mathbb{C}} ) \widetilde{\tau} \qquad (j \in \mathbb{Z} ). \end{align} $$

Then it is easy to see that, for any $j \in \mathbb {Z}$ , we have

(4.15) $$ \begin{align} p_{\tau}( \theta +j) U_j( \mathbf{D} \tilde{i}_* \mathcal{N}_0 ) \subset U_{j-1}( \mathbf{D} \tilde{i}_* \mathcal{N}_0 ). \end{align} $$

Let us consider the relationship between $p_{\widetilde {\tau }}(s) = p_{\tau }(s)$ and $p_{\sigma }(s)= b_{f,m}^{\mathrm {mero}}(-s-1)$ . For this purpose, let

(4.16) $$ \begin{align} \mathcal{M}^{\prime} := H^0 \mathbf{D} \widetilde{\pi}_* \mathcal{N}_0 \simeq H^0 \mathbf{D} \tilde{p}_* ( \mathbf{D} \tilde{i}_* \mathcal{N}_0 ) \end{align} $$

be the zeroth direct image of $\mathbf {D} \tilde {i}_* \mathcal {N}_0$ by $\tilde {p}$ and as in Gyoja [Reference Gyoja16, §4.2] define its section $\sigma ^{\prime } \in \mathcal {M}^{\prime }$ to be the image of a section $1_{X \leftarrow Y} \otimes \tau \in \tilde {p}_* [ \Omega ^n_{Y \times X/X} \otimes _{\mathcal {O}_{Y \times X}} \mathbf {D} \tilde {i}_* \mathcal {N}_0 ]$ by the morphism

$$ \begin{align*} \tilde{p}_* \left[ \Omega^n_{Y \times X/X} \otimes_{\mathcal{O}_{Y \times X}} \mathbf{D} \tilde{i}_* \mathcal{N}_0 \right] \longrightarrow H^0 \mathbf{D} \tilde{p}_* ( \mathbf{D} \tilde{i}_* \mathcal{N}_0 ) = \mathcal{M}^{\prime}. \end{align*} $$

For the construction of $1_{X \leftarrow Y} \otimes \tau $ , see [Reference Gyoja16, §4.2] for details. Note that on the open subset $(X \setminus D) \times \mathbb {C}$ of $X \times \mathbb {C}$ , we have $\mathcal {M}^{\prime } = \mathcal {D}_{X \times \mathbb {C}} \sigma = \mathcal {M}$ and $\sigma ^{\prime }$ coincides with $\sigma $ . For $j \in \mathbb {Z}$ , we denote by $U_j( \mathcal {M}^{\prime } ) \subset \mathcal {M}^{\prime }=H^0 \mathbf {D} \tilde {p}_* ( \mathbf {D} \tilde {i}_* \mathcal {N}_0 )$ the image of the natural morphism

(4.17) $$ \begin{align} H^0 R \tilde{p}_* \left\{ \mathrm{DR}_{Y \times X/X} \left( U_j( \mathbf{D} \tilde{i}_* \mathcal{N}_0 ) \right) \right\} \longrightarrow \mathcal{M}^{\prime}. \end{align} $$

Then, by the proof of [Reference Mebkhout and Sabbah25, Th. 4.8.1(1)], $\{ U_j( \mathcal {M}^{\prime } ) \}_{j \in \mathbb {Z}}$ is a good V-filtration of $\mathcal {M}^{\prime }$ ; and for any $j \in \mathbb {Z}$ , we have

(4.18) $$ \begin{align} p_{\tau}( \theta +j) U_j( \mathcal{M}^{\prime} ) \subset U_{j-1}( \mathcal{M}^{\prime} ). \end{align} $$

Moreover, by our construction, the section $\sigma ^{\prime } \in \mathcal {M}^{\prime }$ is contained in $U_0( \mathcal {M}^{\prime } )$ . Let

(4.19) $$ \begin{align} \mathcal{M}^{\prime \prime} := \mathcal{D}_{X \times \mathbb{C}} \sigma^{\prime} \subset \mathcal{M}^{\prime} \end{align} $$

be the $\mathcal {D}_{X \times \mathbb {C}}$ -submodule of $\mathcal {M}^{\prime }$ generated by $\sigma ^{\prime }$ . Then, by Artin–Rees’s lemma, the V-filtration $\{ U_j( \mathcal {M}^{\prime \prime } ) \}_{j \in \mathbb {Z}}$ of $\mathcal {M}^{\prime \prime }$ defined by

(4.20) $$ \begin{align} U_j( \mathcal{M}^{\prime \prime} ) := \mathcal{M}^{\prime \prime} \cap U_j( \mathcal{M}^{\prime} ) \qquad (j \in \mathbb{Z} ) \end{align} $$

is also good, and hence there exists $l \gg 0$ such that

(4.21) $$ \begin{align} U_{-l}( \mathcal{M}^{\prime \prime} ) = \mathcal{M}^{\prime \prime} \cap U_{-l}( \mathcal{M}^{\prime} ) \subset V_{-1} ( \mathcal{D}_{X \times \mathbb{C}} ) \sigma^{\prime}. \end{align} $$

Combining these results together, we get

(4.22) $$ \begin{align} p_{\tau}( \theta -(l-1)) \cdots p_{\tau}( \theta -1) p_{\tau}( \theta ) \sigma^{\prime} \in \mathcal{M}^{\prime \prime} \cap U_{-l}( \mathcal{M}^{\prime} ) \subset V_{-1} ( \mathcal{D}_{X \times \mathbb{C}} ) \sigma^{\prime}. \end{align} $$

This implies that the minimal polynomial $p_{\sigma ^{\prime }} (s) \in \mathbb {C} [s]$ for the section $\sigma ^{\prime } \in \mathcal {M}^{\prime }$ divides the product

(4.23) $$ \begin{align} p_{\tau}( s -(l-1)) \cdots p_{\tau}( s -1) p_{\tau}( s ) \in \mathbb{C} [s]. \end{align} $$

Now, according to Kashiwara [Reference Kashiwara20], there exists an adjunction morphism

(4.24) $$ \begin{align} \mathbf{D} \widetilde{\pi}_* ( \mathbf{D} \widetilde{\pi}^* \mathcal{M} ) \longrightarrow \mathcal{M} \end{align} $$

of $\mathcal {D}_{X \times \mathbb {C}}$ -modules. Since $\mathbf {D} \widetilde {\pi }^* \mathcal {M}$ is isomorphic to $\mathcal {N}$ (use, e.g., the Riemann–Hilbert correspondence) and $\mathcal {N}_0 \subset \mathcal {N}$ , we obtain a morphism

(4.25) $$ \begin{align} \Psi : \mathcal{M}^{\prime} = H^0 \mathbf{D} \widetilde{\pi}_* \mathcal{N}_0 \longrightarrow \mathcal{M} \end{align} $$

of $\mathcal {D}_{X \times \mathbb {C}}$ -modules. Then the section $\Psi ( \sigma ^{\prime } ) \in \mathcal {M}$ of $\mathcal {M}$ coincides with $\sigma \in \mathcal {M}$ on the open subset $(X \setminus D) \times \mathbb {C} \subset X \times \mathbb {C}$ . Moreover, by the isomorphism $\mathcal {M} \simeq \mathcal {D}_X [ \partial _t ]$ on the open subset $(X \setminus G^{-1}(0)) \times \mathbb {C} \subset X \times \mathbb {C}$ , this coincidence can be extended to $(X \setminus G^{-1}(0)) \times \mathbb {C}$ . Here, we used the classical theorem on the unique continuation of holomorphic functions. Since we have $\mathcal {M} \simeq \mathcal {M} [ \frac {1}{G} ]$ , by Hilbert’s nullstellensatz, we get $\Psi ( \sigma ^{\prime } ) = \sigma $ on the whole $X \times \mathbb {C}$ . This implies that the minimal polynomial $p_{\sigma } (s) = b_{f,m}^{\mathrm {mero}}(-s-1)$ divides the one $p_{\sigma ^{\prime }} (s)$ . Now the assertion is clear. This completes the proof.

We have seen that the roots of our b-functions $b_{f,m}^{\mathrm {mero}}(s)$ are rational numbers. Let $\rho : \mathbb {Q} \longrightarrow \mathbb {Q} / \mathbb {Z}$ be the quotient map. Then Lemma 3.3 means that the subset $A_m:= \rho \ \{ (b_{f,m}^{\mathrm {mero}})^{-1}(0) \} \subset \mathbb {Q} / \mathbb {Z}$ increases with respect to $m \geq 0$ . By Theorem 1.4, this sequence is stationary for $m \geq 2 \mathrm {dim} X$ .

Next, we shall give a lower bound for the subsets $(b_{f,m}^{\mathrm {mero}})^{-1}(0) \subset \mathbb {Q}$ . In the proof of Theorem 1.1, we have seen that the minimal polynomial of $s$ acting on the $\mathcal {D}_X$ -module

(4.26) $$ \begin{align} \mathcal{K} \simeq \frac{ \sum_{k=0}^{+ \infty} \mathcal{D}_X [s] ( \frac{1}{G^m} f^{s+k} ) }{ \sum_{k=1}^{+ \infty} \mathcal{D}_X [s] ( \frac{1}{G^m} f^{s+k} ) } \end{align} $$

is equal to our b-function $b_{f,m}^{\mathrm {mero}}(s)$ . Localizing it along the hypersurface $G^{-1}(0) \subset X$ , we obtain a new $\mathcal {D}_X$ -module

(4.27) $$ \begin{align} \mathcal{K} \left[ \frac{1}{G} \right] \simeq \frac{ \{ \mathcal{D}_X [s]( \frac{1}{G^m} f^{s}) \} [ \frac{1}{G} ] }{ \{ \mathcal{D}_X [s]( \frac{1}{G^m} f^{s+1}) \} [ \frac{1}{G} ]} \end{align} $$

on which $s$ still acts. Obviously, we have $b_{f,m}^{\mathrm {mero}}(s)=0$ on $\mathcal {K} [ \frac {1}{G} ]$ . By this observation, we obtain the following result. We denote the localized ring $\mathcal {D}_X [ \frac {1}{G} ]$ simply by $\widetilde {\mathcal {D}}_X$ .

Theorem 4.2. Let $m \geq 0$ be a nonnegative integer. Then there exists a nonzero polynomial $b(s) \in \mathbb {C} [s]$ satisfying the equation

(4.28) $$ \begin{align} b(s) \left( \frac{1}{G^m} f^{s} \right) = \tilde{P} (s) \left( \frac{1}{G^m} f^{s+1} \right) \end{align} $$

for some $\tilde {P} (s) \in \widetilde {\mathcal {D}}_X [s]$ .

Since $\tilde {P} (s) \in \widetilde {\mathcal {D}}_X [s]$ in the equation (4.28) can be rewritten as

(4.29) $$ \begin{align} \tilde{P} (s)= \frac{1}{G^m} \circ \tilde{Q} (s) \circ G^m \qquad ( \tilde{Q} (s) \in \widetilde{\mathcal{D}}_X [s]), \end{align} $$

in fact, the condition on $b(s)$ in Theorem 4.2 is equivalent to the existence of some $\tilde {Q} (s) \in \widetilde {\mathcal {D}}_X [s]$ satisfying the simpler equation

(4.30) $$ \begin{align} b(s) f^{s} = \tilde{Q} (s) f^{s+1} \end{align} $$

independent of $m \geq 0$ . This shows that we have

(4.31) $$ \begin{align} \tilde{b}_{f,0}^{\mathrm{mero}}(s) = \tilde{b}_{f,1}^{\mathrm{mero}}(s) = \tilde{b}_{f,2}^{\mathrm{mero}}(s) = \cdots \cdots \cdots. \end{align} $$

Therefore, we denote $\tilde {b}_{f,m}^{\mathrm {mero}}(s)$ simply by $\tilde {b}_{f}^{\mathrm {mero}}(s)$ . Then, by our construction, for any $m \geq 0$ , our b-function $b_{f,m}^{\mathrm {mero}}(s)$ is divided by the reduced one $\tilde {b}_{f}^{\mathrm {mero}}(s)$ . We thus obtain a lower bound

(4.32) $$ \begin{align} ( \tilde{b}_{f}^{\mathrm{mero}})^{-1}(0) \subset (b_{f,m}^{\mathrm{mero}})^{-1}(0) \subset \mathbb{Q} \end{align} $$

for the subset $(b_{f,m}^{\mathrm {mero}})^{-1}(0) \subset \mathbb {Q}$ . Several authors studied (global) b-functions on algebraic varieties. In particular, the result in [Reference Mebkhout and La Narváez-Macarro24] ensures the existence of b-functions on smooth affine varieties (see [Reference Àlvarez Montaner, Jeffries and Núñez-Betancourt2] for a review on this subject). Since for algebraic $X$ and $f= \frac {F}{G}$ , the variety $X \setminus G^{-1}(0)$ is affine, our Theorem 4.2 could be considered as an analytic counterpart of their result in a very special case.

From now on, we consider the special case where the meromorphic function $f= \frac {F}{G}$ is quasi-homogeneous. More precisely, for a local coordinate system $x=(x_1,x_2, \ldots , x_n)$ of X such that $x_0= \{ x=0 \}$ , we assume that there exist a vector field $v= \sum _{i=1}^n w_i x_i \partial _{x_i} \in \mathcal {D}_X$ ( $w=(w_1,w_2, \ldots , w_n) \in \mathbb {Z}_{\geq 0}^n \setminus \{ 0 \}$ is a weight vector) and $d_1,d_2 \in \mathbb {Z}_{>0}$ with $d:=d_1-d_2 \not = 0$ such that

(4.33) $$ \begin{align} vF=d_1 \cdot F, \quad vG=d_2 \cdot G, \quad vf=d \cdot f \not= 0. \end{align} $$

Let us calculate $\tilde {b}_{f}^{\mathrm {mero}}(s)$ of such f following the arguments in [Reference Kashiwara20, §6.4]. First, by the condition $vf=d \cdot f$ ( $d \not = 0$ ), we have isomorphisms

(4.34) $$ \begin{align} \widetilde{\mathcal{D}}_X [s] f^{s} \simeq \widetilde{\mathcal{D}}_X f^{s}, \quad \mathcal{K} \left[ \frac{1}{G} \right] \simeq \frac{ \widetilde{\mathcal{D}}_X f^{s} }{ \widetilde{\mathcal{D}}_X f^{s+1} }, \end{align} $$

and for our reduced b-function $\tilde {b}_{f}^{\mathrm {mero}}(s)$ , there exists $\tilde {P} \in \widetilde {\mathcal {D}}_X$ such that

(4.35) $$ \begin{align} \tilde{b}_{f}^{\mathrm{mero}}(s) f^s= \tilde{P} f^{s+1}. \end{align} $$

In this situation, by the proof of [Reference Kashiwara20, Lem. 6.6], we see that $\mathcal {K} [ \frac {1}{G} ]$ is a holonomic $\mathcal {D}_X$ -module and Theorem 4.2 can be proved also by using the trick in the proof of [Reference Kashiwara20, Th. 6.7]. If we set $s=-1$ in (4.35), we obtain

(4.36) $$ \begin{align} \tilde{b}_{f}^{\mathrm{mero}}(-1)= f \tilde{P}(1). \end{align} $$

Restricting this equality to the subset $F^{-1}(0) \setminus G^{-1}(0) \subset X \setminus G^{-1}(0)$ , we see that $\tilde {b}_{f}^{\mathrm {mero}}(-1)=0$ . Namely, for a nonzero polynomial $\tilde {\beta }_{f}^{\mathrm {mero}}(s) \in \mathbb {C} [s]$ , we have

(4.37) $$ \begin{align} \tilde{b}_{f}^{\mathrm{mero}}(s)= (s+1) \cdot \tilde{\beta}_{f}^{\mathrm{mero}}(s). \end{align} $$

On the other hand, by (4.36), we have $\tilde {P}(1)=0$ , and hence $\tilde {P} \in \sum _{i=1}^n \widetilde {\mathcal {D}}_X \partial _{x_i}$ . Namely, there exist $\tilde {Q}_i \in \widetilde {\mathcal {D}}_X$ ( $1 \leq i \leq n$ ) such that $\tilde {P} = \sum _{i=1}^n \tilde {Q}_i \partial _{x_i}$ . Moreover, if we set

(4.38) $$ \begin{align} f_i:=f_{x_i}= \frac{\partial f}{\partial x_i} = \frac{F_{x_i}G-FG_{x_i}}{G^2} \quad (1 \leq i \leq n), \end{align} $$

then we have

(4.39) $$ \begin{align} \partial_{x_i}f^{s+1}=(s+1)f_if^s \quad (1 \leq i \leq n). \end{align} $$

Therefore, we obtain

(4.40) $$ \begin{align} \tilde{\beta}_{f}^{\mathrm{mero}}(s) f^s = \sum_{i=1}^n \tilde{Q}_i f_if^s. \end{align} $$

Conversely, for $\tilde {Q}_i \in \widetilde {\mathcal {D}}_X$ ( $1 \leq i \leq n$ ) satisfying this equality, the differential operator $\tilde {P} = \sum _{i=1}^n \tilde {Q}_i \partial _{x_i} \in \widetilde {\mathcal {D}}_X$ satisfies the one (4.35). Consequently, our $\tilde {\beta }_{f}^{\mathrm {mero}}(s) \not = 0$ is the minimal polynomial $b(s) \in \mathbb {C} [s]$ satisfying the condition $b(s)f^s \in \sum _{i=1}^n \widetilde {\mathcal {D}}_X f_if^s$ . Since we have

(4.41) $$ \begin{align} v(f_if^s)=(d-w_i+d \cdot s)(f_if^s) \quad (1 \leq i \leq n), \end{align} $$

$\sum _{i=1}^n \widetilde {\mathcal {D}}_X f_if^s$ is a $\widetilde {\mathcal {D}}_X [s]$ -submodule of $\widetilde {\mathcal {D}}_X f^s \simeq \widetilde {\mathcal {D}}_X [s] f^s$ . Set $h_i:= F_{x_i}G-FG_{x_i}=G^2f_i\in \mathcal {O}_X$ ( $1 \leq i \leq n$ ). Then the $\widetilde {\mathcal {D}}_X$ -module

(4.42) $$ \begin{align} \widetilde{\mathcal{R}}:= \frac{ \widetilde{\mathcal{D}}_X f^{s} }{\sum_{i=1}^n \widetilde{\mathcal{D}}_X f_if^{s} } \simeq \frac{ \widetilde{\mathcal{D}}_X f^{s} }{\sum_{i=1}^n \widetilde{\mathcal{D}}_X h_if^{s} } \end{align} $$

has an action of $s$ and the minimal polynomial of $s$ on it is equal to $\tilde {\beta }_{f}^{\mathrm {mero}}(s)$ .

Proof. Let us consider the coherent $\mathcal {D}_X$ -module

(4.43) $$ \begin{align} \mathcal{S} := \frac{ \mathcal{D}_X }{\sum_{i=1}^n \mathcal{D}_X h_i} \end{align} $$

and its localization

(4.44) $$ \begin{align} \widetilde{\mathcal{S}}:= \mathcal{S} \left[ \frac{1}{G} \right] \simeq \frac{ \widetilde{\mathcal{D}}_X }{\sum_{i=1}^n \widetilde{\mathcal{D}}_X h_i}. \end{align} $$

Note that $\widetilde {\mathcal {R}}$ is a quotient of $\widetilde {\mathcal {S}}$ . Since $f= \frac {F}{G}$ is quasi-homogeneous of degree $d=d_1-d_2 \not = 0$ , there is no singular point of f in $X \setminus (F^{-1}(0) \cup G^{-1}(0))$ . Then by the smoothness of $f^{-1}(0)=F^{-1}(0) \setminus G^{-1}(0) \subset X \setminus G^{-1}(0)$ , we have

(4.45) $$ \begin{align} \mathrm{Sing} \ f = \{ x \in X \setminus G^{-1}(0) \ | \ h_1(x)=h_2(x)= \cdots =h_n(x)=0 \} = \emptyset. \end{align} $$

This implies that the support of the coherent $\mathcal {D}_X$ -module $\mathcal {S}$ is contained in $G^{-1}(0) \subset X$ . Then, by Hilbert’s nullstellensatz, we get $\widetilde {\mathcal {S}}=0$ ; and hence, $\widetilde {\mathcal {R}}=0$ .

By using [Reference Nguyen and Takeuchi29, Th. 3.3 and Cor. 3.4], we can construct many examples of $f= \frac {F}{G}$ satisfying the conditions in Proposition 4.5 and having a monodromy eigenvalue $\not = 1$ at the point $x_0 \in X$ . By Theorem 1.4, for such $f$ , we thus obtain

(4.46) $$ \begin{align} b_{f,m}^{\mathrm{mero}}(s) \not= \tilde{b}_{f}^{\mathrm{mero}}(s)=s+1 \quad (m \geq 2 \mathrm{dim} X). \end{align} $$

Namely, in the situation of Proposition 4.5 the reduced b-function $\tilde {b}_{f}^{\mathrm {mero}}(s)$ captures only the tiny (trivial) part $s+1$ of $b_{f,m}^{\mathrm {mero}}(s)$ for $m \geq 2 \mathrm {dim} X$ .

5 Multiplier ideals for meromorphic functions

In this section, we define multiplier ideal sheaves for the meromorphic function $f= \frac {F}{G}$ and study their basic properties. Recall that multiplier ideals for holomorphic functions were introduced by Nadel [Reference Nadel28]. For their precise properties, we refer to the excellent book [Reference Lazarsfeld22] by Lazarsfeld. For the meromorphic function $f= \frac {F}{G}$ , we define them as follows. Denote by $\mathrm {L}^1_{\mathrm {loc}}$ , the set of locally integrable functions on X.

Definition 5.1. For a positive real number $\alpha>0$ , we define an ideal $\mathcal {J} (X, f)_{\alpha } \subset \mathcal {O}_X$ of $\mathcal {O}_X$ by

(5.1) $$ \begin{align} \mathcal{J} (X, f)_{\alpha}:= \left\{ h \in \mathcal{O}_X \ | \ \frac{|h|^{2}}{|f|^{2 \alpha}} = \frac{|h|^{2} \cdot |G|^{2 \alpha}}{|F|^{2 \alpha}} \in \mathrm{L}^1_{\mathrm{loc}} \right\} \end{align} $$

and call it the multiplier ideal of $f$ of order $\alpha>0$ .

Let $\pi : Y \longrightarrow X$ be a resolution of singularities of the divisor $D= F^{-1}(0) \cup G^{-1}(0) \subset X$ as in §4. Here, we assume, moreover, that the meromorphic function $g= \frac {F \circ \pi }{G \circ \pi }$ has no point of indeterminacy on the whole $Y$ . Such a resolution $\pi : Y \longrightarrow X$ always exists. Let $\mathrm {div} \ g$ be the divisor on $Y$ defined by $g$ . Then there exist two effective divisors $(\mathrm {div} \ g)_+$ and $(\mathrm {div} \ g)_-$ such that

(5.2) $$ \begin{align} \mathrm{div} \ g = (\mathrm{div} \ g)_+ - (\mathrm{div} \ g)_-. \end{align} $$

By our assumption, their supports, which we denote by $g^{-1}(0)$ and $g^{-1}( \infty )$ , respectively, are disjoint from each other. By using such a resolution $\pi : Y \longrightarrow X$ , we can easily see that for $\alpha ^{\prime }> \alpha >0$ , we have $\mathcal {J} (X, f)_{\alpha ^{\prime }} \subset \mathcal {J} (X, f)_{\alpha }$ . Then, as in the case where f is holomorphic, we can define the jumping numbers of the multiplier ideals $\{ \mathcal {J} (X, f)_{\alpha } \}_{\alpha> 0}$ . In the situation as above, we have $g^{-1}( \infty ) \subset (G \circ \pi )^{-1}(0)$ but $g: Y \setminus (G \circ \pi )^{-1}(0) \longrightarrow \mathbb {C}$ can be extended to a holomorphic function $\tilde {g} : Y \setminus g^{-1}( \infty ) \longrightarrow \mathbb {C}$ . Let

(5.3) $$ \begin{align} \iota_{\tilde{g}} : Y \setminus g^{-1}( \infty ) \longrightarrow Y \times \mathbb{C} \qquad (y \longmapsto (y, \tilde{g} (y))) \end{align} $$

be the graph embedding defined by $\tilde {g}$ . From now, we shall use the terminologies of mixed Hodge modules. For example, regarding the holonomic $\mathcal {D}_{X \times \mathbb {C}}$ -module $\mathcal {M}$ as a mixed Hodge module on $X \times \mathbb {C}$ , for $\alpha \in \mathbb {Q}$ and $p \in \mathbb {Z}$ , we set

(5.4) $$ \begin{align} F_pV_{\alpha} \mathcal{M} = F_p \mathcal{M} \cap V_{\alpha} \mathcal{M}. \end{align} $$

We denote the normal crossing divisor $(G \circ \pi )^{-1}(0)$ in $Y$ by $E$ and consider the regular holonomic $\mathcal {D}_Y$ -module $\mathcal {O}_Y(*E)$ as a mixed Hodge module. Then its Hodge filtration $\{ F_p \mathcal {O}_Y(*E) \}_{p \in \mathbb {Z}}$ satisfies the condition

(5.5) $$ \begin{align} F_p \mathcal{O}_Y(*E) \simeq 0 \quad (p<0), \qquad F_0 \mathcal{O}_Y(*E) \not=0. \end{align} $$

Moreover, $F_0 \mathcal {O}_Y(*E) \subset \mathcal {O}_Y(*E)$ is the subsheaf of $\mathcal {O}_Y(*E)$ consisting of meromorphic functions on $Y$ having poles of order $\leq 1$ only along $E \subset Y$ . See Mustata–Popa [Reference Mustata and Popa27, Chap. D] for the details about the Hodge filtration of $\mathcal {O}_Y(*E)$ . We denote the restriction of $F_0 \mathcal {O}_Y(*E) \simeq \mathcal {O}_Y(E)$ to $Y \setminus g^{-1}( \infty )$ simply by $\mathcal {E}$ . Then the following proposition can be proved just by following the arguments in Budur–Saito [Reference Budur and Saito9] (see also [Reference Budur7, §§3 and 4] for more precise explanations). We set $Y^{\circ }:= Y \setminus g^{-1}( \infty )$ and

(5.6) $$ \begin{align} \varpi :=( \pi \times \mathrm{id}_{\mathbb{C}} ) \circ \iota_{\tilde{g}} : Y^{\circ} \longrightarrow X \times \mathbb{C} \qquad (y \longmapsto ( \pi (y) , \tilde{g} (y))). \end{align} $$

Let $K_{Y/X}$ be the relative canonical divisor of $\pi : Y \longrightarrow X$ .

Proposition 5.2. Let $\alpha>0$ be a positive real number. Then, for $0< \varepsilon \ll 1$ , there exists an isomorphism

(5.7) $$ \begin{align} F_1V_{- \alpha} \mathcal{M} \simeq \varpi_* \left\{ \mathcal{O}_{Y^{\circ}}(K_{Y/X}) \otimes \mathcal{E} \cap \mathcal{O}_{Y^{\circ}} \left(- \lfloor ( \alpha - \varepsilon ) (\mathrm{div} \ g)_+ \rfloor \right) \right\}. \end{align} $$

Now, let $\mathrm {pr}_X: X \times \mathbb {C} \longrightarrow X$ be the projection. Then there exists an injective homomorphism of sheaves

(5.8) $$ \begin{align} \gamma : \mathcal{O}_X \longrightarrow (\mathrm{pr}_X)_* \mathcal{M} \qquad (h \longmapsto h \cdot \sigma_0), \end{align} $$

by which we regard $\mathcal {O}_X$ as a subsheaf of $(\mathrm {pr}_X)_* \mathcal {M}$ . By Proposition 5.2 and the local integrability condition in Definition 5.1, we obtain the following analog for meromorphic functions of Budur–Saito [Reference Budur and Saito9, Th. 0.1] (see also [Reference Budur7] for details). Note that by Proposition 5.2, for any $\alpha>0$ , we have

(5.9) $$ \begin{align} \mathcal{O}_X \cap (\mathrm{pr}_X)_* F_1V_{- \alpha} \mathcal{M} = \mathcal{O}_X \cap (\mathrm{pr}_X)_* V_{- \alpha} \mathcal{M}. \end{align} $$

Theorem 5.3. Let $\alpha>0$ be a positive real number. Then we have

(5.10) $$ \begin{align} \mathcal{J} (X, f)_{\alpha} = \mathcal{O}_X \cap (\mathrm{pr}_X)_* V_{< - \alpha} \mathcal{M} = \{ h \in \mathcal{O}_X \ | \ \mathrm{ord}_{\{ t=0 \}}(h \cdot \sigma_0)> \alpha -1 \}. \end{align} $$

By this theorem and

(5.11) $$ \begin{align} \mathrm{ord}_{\{ t=0 \}}(h \cdot \sigma_0) \subset \bigcup_{i=0,1,2, \ldots} \left\{ \mathrm{ord}_{\{ t=0 \}}( \sigma_0) +i \right\} \qquad (h \in \mathcal{O}_X) \end{align} $$

(see the proof of Lemma 3.3), we immediately obtain the following generalization of the celebrated theorem of Ein–Lazarsfeld–Smith–Varolin [Reference Ein, Lazarsfeld, Smith and Varolin13] to meromorphic functions.

Corollary 5.4. The jumping numbers of the multiplier ideals $\{ \mathcal {J} (X, f)_{\alpha } \}_{\alpha>0}$ are contained in the set

(5.12) $$ \begin{align} \bigcup_{i=0,1,2, \ldots} \left\{ -(b_{f,0}^{\mathrm{mero}})^{-1}(0)+i \right\} \subset \mathbb{Q}_{>0}. \end{align} $$

Moreover, the minimal jumping number $\alpha>0$ is equal to the negative of the largest root of $b_{f,0}^{\mathrm {mero}}(s)$ .