2.1 Homogeneity assumptions

We are interested in studying the singular structure of polynomials $f \in R = \mathbb{C}[x_{1}, \dots, x_{n}]$ with strong homogeneity assumptions. First, we have a generalization of homogeneity.

In this case, we say $E = \sum w_{i} x_{i} \partial_{i}$ is the weighted homogeneity of f with respect to the weights $\{w_{i}\}$ ; we also define the weighted degree or $\mathbf{w}$ -degree f to be

\[ \mathrm{deg}_{\mathbf{w}}(f) = \sum_{u_{k} \in \mathbf{u}} w_{k} u_{k},\quad \text{for any $\mathbf{u} \in \mathbb{R}_{\geq 0}^{n}$ in the monomial support of } f. \]

An analytic divisor $D \subseteq X$ is quasi-homogeneous at $\mathfrak{x} \in X$ when we may pick a defining equation f of D at $\mathfrak{x}$ such that f is quasi-homogeneous.

Because we will be working with quasi-homogeneous divisors/polynomials, we will also be working with nonstandard graded polynomial rings R. We briefly touch on the key features of such a grading, following the treatment in [Reference Miller and SturmfelsMS05, 8].

Definition 2.3. Suppose that $f \in R = \mathbb{C}[x_{1}, \dots, x_{n}]$ is quasi-homogeneous with respect to the weights $\mathbf{w} = \{w_{i}\}_{1 \leq i \leq n} \in \mathbb{R}_{>0}^{n}$ . We endow R with a multigrading by defining a monomial $\alpha_{\mathbf{u}} x^{\mathbf{u}} \neq 0$ to have weighted degree or $\mathbf{w}$ -degree

(2.1) \begin{equation} \mathrm{deg}_{\mathbf{w}}(\alpha_{\mathbf{u}} x^{\mathbf{u}}) = \sum_{u_{i} \in \mathbf{u}} w_{i} u_{i} \in \mathbb{R}_{> 0}.\end{equation}

We give elements of $\mathbb{C}$ weighted degree 0. Such a grading of R is a weighted grading or $\mathbf{w}$ -grading of R, with respect to the weights $\mathbf{w} = \{w_{i}\}$ . Whenever working with a quasi-homogeneous polynomial/divisor, the graded structure on R is the one induced by $\mathbf{w}$ ; similarly for any graded R-module M.

For a M a $\mathbf{w}$ -graded R-module, $M_{a} = \{m \in M \mid m \text{ homogeneous, }\mathrm{deg}_{\mathbf{w}}(m) = a \}$ , and M(b) is the $\mathbf{w}$ -graded R-module whose underlying module is M and whose grading is defined by $M(b)_{a} = M_{b + a}$ . Additionally, we define

(2.2) \begin{equation} \mathrm{deg}_{\mathbf{w}}(M) = \{a \in \mathbb{R} \mid M_{a} \setminus 0 \neq \emptyset \}.\end{equation}

By the positivity assumption on the weights, the only elements of R of weighted degree zero are constants. So by [Reference Miller and SturmfelsMS05, Theorem 8.6, Definition 8.7], this weighted grading is a positive multigrading of R. Because graded Nakayama’s lemma still holds in the positive multigraded case [Reference Miller and SturmfelsMS05, p. 155], if M is a finitely generated graded R-module (with respect to our weighted grading) then M has a minimal graded free resolution with respect to this weighted grading [Reference Miller and SturmfelsMS05, Proposition 8.18, 8.3]. Later we will make use of these minimal graded free resolutions (e.g., Proposition 2.16, Definition 2.17) and the consequence that the multigraded Betti numbers are well defined [Reference Miller and SturmfelsMS05, Definition 8.22] (e.g., Proposition 5.6). We also will use graded local duality in this setting (cf. § 2.4).

Finally, we consider an additional homogeneity condition on polynomials stronger than quasi-homogeneity.

Whenever we say $f \in R = \mathbb{C}[x_{1}, \dots, x_{n}]$ is locally quasi-homogeneous we mean f is quasi-homogeneous and f/D are locally quasi-homogeneous, i.e., we are implicitly assuming $f \in R$ is itself a quasi-homogeneous representative of the divisor germ at zero.

Convention 2.6. Whenever we refer to our ‘main hypotheses’ on a polynomial f, we are assuming $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ is reduced and locally quasi-homogeneous. Also, whenever f is honestly homogeneous ( $\mathbf{w} = \mathbf{1}$ ) we substitute $\deg$ for $\mathrm{deg}_{\mathbf{w}}$ .

2.2 Logarithmic derivations

Our analysis of the singularities of polynomials depends on logarithmic data, and transferring between the algebraic and analytic setting. We first introduce logarithmic derivations. These are the syzygies of $\{\partial_{1} \bullet f, \dots, \partial_{n} \bullet f, f \}$ . They induce a logarithmic stratification we need later (cf. Remark 2.10). We end with Proposition 2.11: this describes, under the assumptions used, the highest nonvanishing Ext-modules of the logarithmic derivations in the algebraic category.

Note that locally quasi-homogeneous divisors are defined by the existence of certain ‘nice’ analytic local coordinates locally everywhere. Hence, there is subtlety in translating ‘obvious’ analytic results about locally quasi-homogeneous polynomials into the algebraic category. Proposition 2.11, one of the main goals of this subsection, exemplifies this by establishing an algebraic result that, while immediate in the analytic setup, demands a strategically different proof.

Definition 2.7. Fix $f \in R = \mathbb{C}[x_{1}, \dots, x_{n}]$ . The R-module of (global algebraic) logarithmic derivations along f is

\[ \mathrm{Der}_{R}({-}\!\log f) = \{ \delta \in \mathrm{Der}_{R} \mid \delta \bullet f \in R \cdot (f) \}. \]

We say f is free when $\mathrm{Der}_{R}({-}\!\log f)$ is a free R-module. Also define

\[ \mathrm{Der}_{R}({-}\!\log_{0} f) = \{ \delta \in \mathrm{Der}_{R}({-}\!\log f) \mid \delta \bullet f = 0\}. \]

The $\mathscr{O}_{X}$ -module of (analytic) logarithmic derivations along f is

\[ \mathrm{Der}_{\mathscr{O}_{X}}({-}\!\log f) = \{ \delta \in \mathrm{Der}_{X} \mid \delta \bullet f \in \mathscr{O}_{X} \cdot (f) \}. \]

In the analytic setting we say f is free when $\mathrm{Der}_{\mathscr{O}_{X, \mathfrak{x}}}({-}\!\log f)$ is a free $\mathscr{O}_{X,\mathfrak{x}}$ -module for all $\mathfrak{x} \in X$ . We define the analytic $\mathrm{Der}_{\mathscr{O}_{X,\mathfrak{x}}}(\log_{0} f)$ symmetrically.

Below we introduce the logarithmic stratification of $\mathrm{V}(f)$ . When this stratification is locally finite (we call such divisors Saito holonomic) induction is viable. We use this in Proposition 4.8. This often requires the analytic setting, though note Remark 2.5(c).

We conclude with a simple observation about $\mathrm{Ext}_{R}^{1}(\mathrm{Der}_{R}({-}\!\log f), R)$ provided that $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ is quasi-homogeneous. Note that the R-modules $\mathrm{Ext}_R^{\geq 2}(\mathrm{Der}_R({-}\!\log f), R)$ vanish by Remark 2.9(d).

Proof. Consider the short exact sequence of R-modules

\begin{equation*} 0 \to \mathrm{Der}_R({-}\log f) \to \mathrm{Der}_R \oplus R \to (\partial_1 \bullet f, \partial_2 \bullet f, \partial_3 \bullet f, f) \to 0 \end{equation*}

whose last nontrivial map is given by $(\partial_i \oplus 0) \mapsto \partial_i \bullet f$ and $(0 \oplus 1) \mapsto f$ . Thus,

(2.4) \begin{equation} \mathrm{Ext}_R^1(\mathrm{Der}_R({-}\log f), R) \simeq \mathrm{Ext}_R^3(R / (\partial_1 \bullet f, \partial_2 \bullet f, \partial_3 \bullet f, f), R). \end{equation}

Since $\dim R = 3$ , the rightmost Ext-module of (2.4) is either zero or has zero-dimensional support. By Remark 2.9(d), the vanishing of the modules in (2.4) is equivalent to condition (a).

So if (a) does not hold, $\mathrm{Ext}_R^1(\mathrm{Der}_R({-}\!\log f), R)$ must be nonzero with zero-dimensional support. Since this module is positively graded and the only positively graded maximal ideal is the irrelevant ideal, [Reference Bruns and HerzogBH93, Lemma 1.5.6] implies (b).

2.3 Logarithmic forms and free resolutions

We now turn to logarithmic differential forms, a generalization of logarithmic derivations introduced in [Reference SaitoSai80]. Familiarity with these objects is central to our approach: they are the building blocks of the linchpin complex $U_{f}^{\bullet}$ of Definition 4.1. Note that $U_f^\bullet$ is defined analytically.

We end with a detailed discussion of case where $f \in R =\mathbb{C}[x_{1}, x_{2}, x_{3}]$ is quasi-homogeneous. Here logarithmic forms are naturally $\mathbf{w}$ -graded and we construct explicit free resolutions of logarithmic 1,2-forms in both the algebraic (where these modules are graded) and the analytic setting (Proposition 2.16, Definition 2.17 and Proposition 2.18, Definition 2.19). These are used throughout § 5. In Proposition 2.20, we use the algebraic graded data to describe a finite generating set of $\mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X,0}^{j}(\log f), \mathscr{O}_{X,0})$ , for $j = 1,2$ , as a $\mathbb{C}$ -vector space and attach $\mathbf{w}$ -degree data to each generator. These numerics will also be used in § 5 and are key to our understanding of the module $L_{f}$ introduced in Definition 6.1.

In the previous subsection, we compared analytic facts to algebraic ones. We concluded with Proposition 2.11, establishing a fact in the algebraic category whose analytic variant is ‘obvious.’ Here we reverse course: the main prize is Proposition 2.20 which describes certain analytic $\mathscr{O}_X$ -modules entirely in terms of certain R-modules, the structure of the latter being relatively straightforward.

First, we have the objects at hand.

Definition 2.13. For $f \in R = \mathbb{C}[x_{1}, \dots, x_{n}]$ we define the (global algebraic) logarithmic p-forms along f to be

\[ \Omega_{R}^{p}(\log f) = \biggl\{ \eta \in \frac{1}{f} \Omega_{R}^{p} \mid d(\eta) \in \frac{1}{f} \Omega_{R}^{p+1} \biggr\}. \]

Additionally we define

\[ \Omega_{R}^{p}(\log_{0} f) = \{ \eta \in \Omega_{R}^{p}(\log f) \mid df/f \wedge \eta = 0\}. \]

We have similar constructions in the analytic setting. We define the (analytic) logarithmic p-forms along f to be

\[ \Omega_{X}^{p}(\log f) = \{ \eta \in \Omega_{R}^{p}(f) \mid d(\eta) \in \Omega_{R}^{p+1}(f) \} \]

where $\Omega_{R}^{p}(f) = \Omega_{R}^{p}(D) = \Omega_{R}^{p} \otimes_{\mathscr{O}_{X}} \mathscr{O}_{X}(D)$ in which D is the divisor defined by f. We also define

\[ \Omega_{X,\mathfrak{x}}^{p}(\log_{0} f) = \{\eta \in \Omega_{X,\mathfrak{x}}^{p}(\log f) \mid df/f \wedge \eta = 0\}. \]

Now let us start tracking degree data attached to logarithmic forms of quasi-homogeneous divisors in $R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ . This class of polynomials is our focus for the rest of this subsection and the next. The following is a straightforward construction of graded free resolutions; recall they exist for our $\mathbf{w}$ -grading as discussed in the first subsection.

Proposition 2.16. Let $f \in R \in \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be quasi-homogeneous with E its weighted homogeneity. Let

\[ 0 \to F_{t} \xrightarrow[]{\beta_{t}} F_{t-1} \to \cdots \to F_{1} \xrightarrow[]{\beta_{1}} F_{0} \xrightarrow[]{\beta_{0}} \Omega_{R}^{2}(\log_{0} f) \to 0 \]

be a minimal graded free resolution of $\Omega_{R}^{2}(\log_{0} f)$ . Then $t \leq 1$ . Moreover,

(2.5) \begin{align} 0 \to F_{t} \xrightarrow[]{\beta_{t}} F_{t-1} \to \cdots \to F_{1} &\xrightarrow[]{(\beta_{1}, 0)} F_{0} \oplus R(n-d)\nonumber \\ &\xrightarrow[]{(\beta_{0}, \cdot \iota_{E} dx/f)} \Omega_{R}^{2}(\log_{0} f) \oplus R \cdot \iota_{E} dx/f \to 0 \end{align}

is a graded free resolution of $\Omega_{R}^{2}(\log f)$ , and

(2.6) \begin{equation} 0 \to F_{t} \xrightarrow[]{\beta_{t}} F_{t-1} \to \cdots \to F_{1} \xrightarrow[]{(\beta_{1}, 0)} F_{0} \oplus R \xrightarrow[]{(\alpha_{0}, \cdot df/f)} \iota_{E}\Omega_{R}^{2}(\log_{0} f) \oplus R \cdot df/f \to 0 \end{equation}

is a graded free resolution of $\Omega_{R}^{1}(\log f)$ where $\alpha_{0}(m) = \iota_{E} \beta_{0}(m)$ .

We name the resolutions just constructed.

Definition 2.17. Let $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be quasi-homogeneous with E its weighted homogeneity. We set a minimal graded free resolution of $\Omega_{R}^{2}(\log_{0} f)$ to be

\[ F_{\bullet} = 0 \to F_{1} = \bigoplus_{i} R(b_{i}) \xrightarrow[]{\beta_{1}} F_{0} = \bigoplus_{j} R(a_{j}) \xrightarrow[]{\beta_{0}} \Omega_{R}^{2}(\log_{0} f) \to 0. \]

(When f is free we allow $F_{1} = 0$ , otherwise we do not.) We set the minimal graded free resolutions of Proposition 2.16 to be

(2.7) \begin{equation} P_{\bullet}^{2} = 0 \to P_{1}^{2} = F_{1} \xrightarrow[]{(\beta_{1}, 0)} \to P_{0}^{2} = F_{0} \oplus R\Bigl( - d + \sum_{i} w_{i}\Bigr) \xrightarrow[]{(\beta_{0}, \iota_{E} dx/f)} \Omega_{R}^{2}(\log f) \to 0, \end{equation}

(2.8) \begin{equation} P_{\bullet}^{1} = 0 \to P_{1}^{1} = F_{1} \xrightarrow[]{(\beta_{1}, 0)} P_{0}^{0} = F_{0} \oplus R \xrightarrow[]{(\iota_{E} \beta_{0}, \cdot df/f)} \Omega_{R}^{1}(\log f) \to 0. \end{equation}

We convert Definition 2.17 to the analytic setting.

We name these analytic free resolutions for later use (e.g., Definition 4.11).

Definition 2.19. Let $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be quasi-homogeneous with E its weighted homogeneity. We set the free $\mathscr{O}_{X}$ -module resolutions produced by Proposition 2.18 to be

(2.9) \begin{equation} Q_{\bullet}^{2} = 0 \to Q_{1}^{2} \to Q_{0}^{2} \to \Omega_{X}^{2}(\log f) \to 0, \end{equation}

(2.10) \begin{equation} Q_{\bullet}^{1} = 0 \to Q_{1}^{1} \to Q_{0}^{1} \to \Omega_{X}^{1}(\log f) \to 0. \end{equation}

In particular, $Q_{i}^{j} = (\widetilde{F_{i}^{j}})^{\mathrm{an}}$ and the maps, which are as described in Definition 2.17, are given by matrices with $\mathbf{w}$ -homogeneous polynomial entries. Additionally, $Q_{1}^{2} = Q_{1}^{1}.$

Finally we give our promised analysis of $\mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{X}^{j}(\log f), \mathscr{O}_{X})$ for $j = 1,2$ . The takeaways here are: the module with $j=1$ is naturally identified with the module $j=2$ ; both arise from the algebraic analogs in natural ways; and the stalks of each at zero have naturally induced finite $\mathbb{C}$ -generating sets of ‘homogeneous’ elements, whose degree data is entirely determined by the algebraic analog.

Proposition 2.20. Let $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be quasi-homogeneous. Then $\mathrm{Hom}_{\mathscr{O}_{X}}(Q_{\bullet}^{j}, \mathscr{O}_{X})$ , for $j = 1, 2$ , is obtained by sheafifying (in the algebraic category) and then analytifying the complex $\mathrm{Hom}_{R}(P_{\bullet}^{j}, R)$ . We also have the following isomorphisms.

(2.11)

Moreover, $\mathrm{Ext}_{R}^{1}(\Omega_{R}^{j}(\log f), R)$ is supported at $\mathfrak{m}$ and is a finite $\mathbb{C}$ -vector space with basis a finite set of homogeneous elements $\Delta_{f}$ . Finally, the only potentially nonzero stalk of $\mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{\mathscr{O}_{X}}^{j}(\log f), \mathscr{O}_{X})$ is at the origin, and there it is a finite $\mathbb{C}$ -vector space generated by $\Delta_{f}$ . $\Delta_{f}$ can be chosen the same for each choice of j.

To justify the horizontal isomorphism of (2.11), we observe

\begin{align*} \mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{X}^{j}(\log f), \mathscr{O}_{X}) & \simeq H^{1} \bigl( \mathrm{Hom}_{\mathscr{O}_{X}}(Q_{\bullet}^{j}, \mathscr{O}_{X}) \bigr) \\ & \simeq H^{1} \bigl( \big( \widetilde{\mathrm{Hom}_{R}(P_{\bullet}^{j}, R)} \big)^{\mathrm{an}} \bigr) \\ &\simeq \bigl( \widetilde{ H^{1} \bigl( \mathrm{Hom}_{R}(P_{\bullet}^{j}, R) \bigr) } \bigr)^{\mathrm{an}} \\ &\simeq \bigl( \widetilde{\mathrm{Ext}_{R}^{1}\bigl(\Omega_{R}^{j}(\log f), R\bigr)} \bigr)^{\mathrm{an}}. \end{align*}

Here the first ‘ $\simeq$ ’ is Proposition 2.18; the second ‘ $\simeq$ ’ is what we showed in the first paragraph; the third ‘ $\simeq$ ’ is the fact that (algebraic) sheafification and analytification are exact, and hence commute with taking cohomology; and the fourth ‘ $\simeq$ ’ is Proposition 2.16. As for the vertical isomorphisms of (2.11), it suffices to check the rightmost isomorphism. By the definition of $P_{\bullet}^{j}$ and Proposition 2.16,

(2.12) \begin{align} \mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R) &\simeq \frac{\mathrm{Hom}_{R}(F_{1}, R)}{ \mathrm{im} [\beta_{1}^{T}: \mathrm{Hom}_{R}(F_{0}, R) \to \mathrm{Hom}_{R}(F_{1}, R)]} \nonumber\\ &\simeq \mathrm{Ext}_{R}^{1}(\Omega_{R}^{2}(\log f), R). \end{align}

As for the final statements, (2.12) shows $\mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R)$ and $\mathrm{Ext}_{R}^{1}(\Omega_{R}^{2}(\log f), R)$ are graded isomorphic. Recall that $\Omega_{R}^{2}(\log f) \simeq \mathrm{Der}_{R}(\log f)$ by Remark 2.14(e). Using Proposition 2.11, there is a single finite set of weighted homogeneous elements $\Delta_f$ generating both $\mathrm{Ext}_R^1(\Omega_R^1(\log f), R)$ and $\mathrm{Ext}_R^1(\Omega_R^2(\log f), R)$ as finite-dimensional $\mathbb{C}$ -vector spaces. By Remark 2.9(d) and an analytic version of Remark 2.14(e), the only potentially nonzero stalk of $\mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{\mathscr{O}_{X}}^{j}(\log f), \mathscr{O}_{X})$ is at the origin; by (2.11) this stalk is generated as a finite-dimensional $\mathbb{C}$ -vector space by $\Delta_{f}$ as well.

2.4 Degree data: Milnor algebra and local cohomology

Under the assumption that $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ is quasi-homogeneous, we have emphasized the degree data of $\mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R) \simeq \mathrm{Ext}_{R}^{1}(\Omega_{R}^{2}(\log f), R)$ . Here we translate this degree data into a more familiar form for the logarithmically non-inclined reader. Throughout the subsection we stay in the algebraic setting, though later we can and do use Proposition 2.20 to convert this degree data to the analytic world.

Let $R = \mathbb{C}[x_{1}, \dots, x_{n}]$ with irrelevant ideal $\mathfrak{m}$ and let M be a finite R-module. Define

\[\Gamma_{\mathfrak{m}}^{0} M = \{ m \in M \mid \mathfrak{m}^{\ell} m = 0 \text{ for some } \ell \in \mathbb{Z}_{\geq 0} \}.\]

Then $H_{\mathfrak{m}}^{t}$ is the local cohomology of M with respect to $\mathfrak{m}$ ; it is the tth derived functor of $\Gamma_{\mathfrak{m}}^{0}$ . If R is positively graded and M is also $\mathbf{w}$ -graded, then $H_{\mathfrak{m}}^{t} M$ is also $\mathbf{w}$ -graded thanks to the Cech complex interpretation of local cohomology.

Take $f \in R$ . The Jacobian ideal $(\partial f) \subseteq R$ of f is $(\partial f) = (\partial_{1} \bullet f, \dots, \partial_{n} \bullet f)$ , i.e., the ideal generated by the partial derivatives. When f is quasi-homogeneous with weighted homogeneity $E = \sum w_{i} x_{i} \partial_{i}$ , then $f \in (\partial f)$ since $E \bullet f = \mathrm{deg}_{\mathbf{w}}(f) f$ . We call $R / (\partial f)$ the Milnor algebra of f.

When R is given a positive $\mathbf{w}$ -grading, R is a $^{\star}$ -complete $^{\star}$ -local ring in the sense of Bruns and Herzog [Reference Bruns and HerzogBH93, 1.5, p. 142] and so it admits a version of graded local duality [Reference Bruns and HerzogBH93, Theorem 3.6.19]. In particular, for any finite graded module M over R, we have, for all t, an isomorphism of $\mathbb{C}$ -vector spaces

(2.13) \begin{equation} \Big[\mathrm{Ext}_{R}^{j}\Big(M, R \Big({-}\sum w_{i}\Big)\Big)\Big]_{t} \simeq [\big(H_{\mathfrak{m}}^{n-j} M \big)^{\star}]_{t}\end{equation}

where $w_{i} = \deg(x_{i})$ and $(-)^{\star}$ denotes the graded $\mathbb{C}$ -dual.

There is a dictionary between degree data of Ext-modules of logarithmic forms and local cohomology of the Milnor algebra.

Proposition 2.22. Let $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be quasi-homogeneous with weighted homogeneity $E = \sum w_{i} x_{i} \partial_{i}$ . Then

\begin{align*} \mathrm{deg}_{\mathbf{w}}( \mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R) &= \mathrm{deg}_{\mathbf{w}} ( \mathrm{Ext}_{R}^{1}(\Omega_{R}^{2}(\log f), R) ) \\ &= \mathrm{deg}_{\mathbf{w}} \Big( \mathrm{Ext}_{R}^{1}\Big(\mathrm{Der}_{R}({-}\!\log f) \Big(\mathrm{deg}_{\mathbf{w}}(f) - \sum w_{i}\Big), R\Big) \Big) \\ &= \Big\{-t + 2 \mathrm{deg}_{\mathbf{w}}(f) - 2 \sum w_{i} \mid t \in \mathrm{deg}_{\mathbf{w}} (H_{\mathfrak{m}}^{3} (R/(\partial f)) ) \Big\} \end{align*}

Proof. First note that $\mathrm{deg}_{\mathbf{w}}( \mathrm{Ext}_{R}^{1}(\Omega_{R}^{j}(\log f), R))$ is finite, for either $j = 1, 2$ , by Proposition 2.20. By Proposition 2.16, there is a graded R-isomorphism

(2.14) \begin{equation} \mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R) \simeq \mathrm{Ext}_{R}^{1}(\Omega_{R}^{2}(\log f), R) \end{equation}

which gives the first equality promised.

Recall our grading convention $\mathrm{Der}_{R} = \oplus R \cdot \partial_{i} = \oplus R(w_{i}).$ We have a short exact sequence of graded R-modules

(2.15) \begin{equation} 0 \to \mathrm{Der}_{R}({-}\!\log_{0} f)(- \mathrm{deg}_{\mathbf{w}}(f)) \to \mathrm{Der}_{R}(- \mathrm{deg}_{\mathbf{w}}(f)) \to (\{ \partial_{i} \bullet f \}_{i} ) \to 0. \end{equation}

Here, the last nontrivial map is given by $\partial_{i} \mapsto \partial_{i} \bullet f$ , and the first nontrivial map is induced by the graded inclusion $\mathrm{Der}_{R}({-}\!\log_{0} f) \hookrightarrow \mathrm{Der}_{R}$ . We also have a graded R-isomorphism

(2.16) \begin{align} \Omega_{R}^{2}(\log f)(- \mathrm{deg}_{\mathbf{w}}(f) + \sum w_{i}) \simeq \mathrm{Der}_{R}({-}\!\log f) \end{align}

as described in Remark 2.9(e).

Using all this, we find graded isomorphisms:

(2.17) \begin{align} \mathrm{Ext}_{R}^{1} (\Omega_{R}^{2}(\log f), R) &\simeq \mathrm{Ext}_{R}^{1} \Bigl( \mathrm{Der}_{R}({-}\!\log f)\Bigl(\deg_{w}(f) - \sum w_{i}\Bigr), R \Bigr) \nonumber\\ &\simeq \mathrm{Ext}_{R}^{2} \Bigl( (\partial f)\Bigl(2 \mathrm{deg}_{\mathbf{w}}(f) - \sum w_{i}, R\Bigr) \Bigr) \nonumber \\ &\simeq \mathrm{Ext}_{R}^{2} ((\partial f), R) \Bigl({-} 2 \mathrm{deg}_{\mathbf{w}}(f) + \sum w_{i}\Bigr). \nonumber \\ &\simeq \mathrm{Ext}_{R}^{3}(R / (\partial f), R)\Bigl({-}2 \mathrm{deg}_{\mathbf{w}}(f) + \sum w_{i}\Bigr). \nonumber \\ &\simeq \mathrm{Ext}_{R}^{3}\Bigl(R / (\partial f), R\Bigl(-\sum w_{i}\Bigr)\Bigr)\Bigl({-}2\mathrm{deg}_{\mathbf{w}}(f) + 2 \sum w_{i}\Bigr) \end{align}

Here the first ‘ $\simeq$ ’ is (2.16); the second ‘ $\simeq$ ’ is (2.15) along with the graded isomorphism $\mathrm{Der}_{R}({-}\!\log f) \simeq \mathrm{Der}_{R}({-}\!\log_{0} f) \oplus R \cdot E$ ; and the penultimate ‘ $\simeq$ ’ is the graded short exact sequence $0 \to (\partial f) \to R \to R/(\partial f) \to 0$ . The proposition follows by (2.14) and applying graded local duality (cf. (2.13)) to (2.17).

When f is homogeneous, Proposition 2.22 can be reformulated slightly. Indeed, here R has the standard $\mathbf{1}$ -grading and (by [Reference van Straten and WarmtvSW15, Theorem 4.7], [Reference SernesiSer14, Theorem 3.4]) there is symmetry in the degree data of $H_{\mathfrak{m}}^{0}(R / (\partial f))$ about $(3 \deg(f) - 6)/2$ :

(2.18) \begin{equation} H_{\mathfrak{m}}^{0} (R / (\partial f))_{k} \simeq H_{\mathfrak{m}}^{0} (R / (\partial f))_{3 \deg(f) - 6 - k} \quad (\text{as $\mathbb{C}$-vector spaces}).\end{equation}

Proposition 2.23. Let $f = R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be homogeneous. Then

\begin{align*} \deg( \mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R) &= \deg ( \mathrm{Ext}_{R}^{1}(\Omega_{R}^{2}(\log f), R) \\ &= \deg \big( H_{\mathfrak{m}}^{0} (R / (\partial f)) \big) - \deg(f). \end{align*}

3.1 Bernstein–Sato polynomials

Let us construct the Bernstein–Sato polynomial. For $g \in\mathscr{O}_{X}$ global we can define a left $\mathscr{D}_{X}[s]$ -module structure on $\mathscr{O}_{X}(\star g)[s] g^{s}$ , where $g^{s}$ is a formal symbol. First, extend the $\mathscr{D}_{X}$ -action on $\mathscr{O}_{X}(\star g)$ to a $\mathscr{D}_{X}[s]$ -action on $\mathscr{O}_{X}(\star g)[s]$ by letting $s \in \mathscr{D}_{X}[s]$ act as multiplication and commute with other operators. Then extend these actions to a $\mathscr{D}_{X}[s]$ -module structure on $\mathscr{O}_{X}(\star g)[s] g^{s}$ by the Leibniz rule and letting a derivation act on $g^{s}$ via a naive application of the chain rule. For example, if $h(s) \in \mathscr{O}_{X}[s]$ ,

\[\partial_{i} \bullet h(s)g^{s} = ( (\partial_{i} \bullet h(s) )g^{s}) + ( h(s)s(\partial_{i} \bullet g) g^{s-1}) = \biggl((\partial_{i} \bullet h(s)) + \frac{ s( \partial_{i} \bullet g)}{g}\biggr) g^{s}.\]

For $k \in \mathbb{Z}$ , we identify $g^{s + k} \in \mathscr{O}_{X}(\star f)[s] g^{s}$ with $g^{k} g^{s} \in \mathscr{O}_{X}(\star f)[s] g^{s}$ .

Definition 3.1. With $g \in \mathscr{O}_{X}$ global, define

(3.1) \begin{equation} \mathscr{D}_{X}[s] g^{s+k} = \text{left } \mathscr{D}_{X}[s] \text{ submodule of } \mathscr{O}_{X}(\star g)[s] g^{s} \text{ generated by } g^{s+k}. \end{equation}

Now consider the left $\mathscr{D}_{X}[s]$ -module,

(3.2) \begin{equation} \frac{\mathscr{D}_{X}[s] g^{s}}{\mathscr{D}_{X}[s] g^{s+1}}. \end{equation}

The $\mathbb{C}[s]$ -annihilator of (3.2) is an ideal in $\mathbb{C}[s]$ ; we call its monic generator $b_{g}(s) \in \mathbb{C}[s]$ the Bernstein–Sato polynomial of g. We denote the zeros of the Bernstein–Sato polynomial by $Z(b_{g}(s)) \subseteq \mathbb{C}$ . If we work with $g \in \mathscr{O}_{X,\mathfrak{x}}$ , we can repeat the construction, obtaining $b_{g, \mathfrak{x}}(s)$ , the local Bernstein–Sato polynomial of g at $\mathfrak{x}$ .

More generally, we have the following definition.

Definition 3.3. Let M be a coherent (left or right) $\mathscr{D}_{X}[s]$ -module. We define the $\mathbb{C}[s]$ -ideal

(3.3) \begin{equation} B(M) = \mathrm{ann}_{\mathbb{C}[s]} M \subseteq \mathbb{C}[s]. \end{equation}

If $B(M) \neq 0$ , then its monic generator is the b-function of M. We denote the zeros of B(M) by $Z(B(M)) \subseteq \mathbb{C}$ . We have a similar notion locally at $\mathfrak{x}$ using $M_{\mathfrak{x}}$ .

3.2 Annihilators

Since $\mathscr{D}_{X}[s] g^{s+k}$ is a cyclic module, we have the canonical isomorphism

\[\frac{\mathscr{D}_{X}[s]}{\mathrm{ann}_{\mathscr{D}_{X}[s]}g^{s+k}} \xrightarrow[]{\simeq} \mathscr{D}_{X}[s] g^{s+k}\]

where, of course, $\mathrm{ann}_{\mathscr{D}_{X}[s]} g^{s+k} = \{P(s) \in \mathscr{D}_{X}[s] \mid P(s) \bullet g^{s+k} = 0\}$ .

In certain situations, include those of Convention 2.6, $\mathrm{ann}_{\mathscr{D}_{X}[s]}g^{s+k}$ has as nice as possible a description. To set this up, note that if $\delta \in \mathrm{Der}_{\mathscr{O}_{X}}({-}\!\log g)$ is a logarithmic derivation, then $\delta - (s+k)(\delta \bullet g)/g \in \mathscr{D}_{X}[s]$ and in fact kills $g^{s+k}$ . We make the following definition.

Definition 3.4. We define the left $\mathscr{D}_{X}[s]$ -ideal

(3.4) \begin{equation} \mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} g^{s+k} = \mathscr{D}_{X}[s] \cdot \{ \delta - s\frac{ \delta \bullet g}{g} \mid \delta \in \mathrm{Der}_{\mathscr{O}_{X}}(\log g) \} \subseteq \mathrm{ann}_{\mathscr{D}_{X}[s]} g^{s+k}. \end{equation}

There has been a lot of work studying when the inclusion of (3.4) is actually an equality (cf. [Reference YanoYan78, Reference Calderón-Moreno and Narváez-MacarroCMNM02, Reference Calderón Moreno and Narváez MacarroCMNM09, Reference Narváez MacarroNM15, Reference WaltherWal17]). The most general assumptions are provided in [Reference WaltherWal17, Theorem 3.26]: g is Saito holonomic (Remark 2.10); g is tame (Definition 4.6); g is strongly Euler homogeneous locally everywhere (this is a weaker form of locally quasi-homogenous; cf. [Reference WaltherWal17, Defintion 2.7]). For our purposes, if $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ is locally quasi-homogeneous, then f satisfies these assumptions (cf. Remarks 2.10(d) and 4.7).

3.3 Filtrations and relative holonomicity

There are three filtrations that we sometimes consider: the canonical order filtration $F_{\bullet}^{\mathrm{ord}} \mathscr{D}_{X}$ on $\mathscr{D}_{X}$ , the relative order filtration $F_{\bullet}^{\mathrm{rel}} \mathscr{D}_{X}[s] = F_{\bullet}^{\mathrm{ord}} \mathscr{D}_{X} \otimes_{\mathbb{C}} \mathbb{C}[s]$ on $\mathscr{D}_{X}[s]$ , and the total order filtration $F_{\bullet}^{\sharp} \mathscr{D}_{X}[s] = \oplus_{0 \leq k \leq \bullet} F_{\bullet}^{\mathrm{ord}} \mathscr{D}_{X} \otimes \mathbb{C}[s]_{\bullet - k}$ on $\mathscr{D}_{X}[s]$ . One can define characteristic varieties with respect to all of these filtrations. For example, we denote by $\mathrm{Ch}^{\mathrm{rel}}(-)$ the characteristic variety of the input with respect to the relative order filtration.

Maisonobe introduced [Reference MaisonobeMai23] a useful generalization of holonomic $\mathscr{D}_{X}$ -modules to the case of $\mathscr{D}_{X}[s]$ -modules. While he uses the phrase ‘marjoré par une lagrangienne’ we use the verbiage from the extension of his theory developed in [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a, Reference Budur, van der Veer, Wu and ZhouBvdVWZ21b]. Some, but not all, of the results cited in [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a] can be found in [Reference MaisonobeMai23].

Definition 3.6. We say a coherent (left or right) $\mathscr{D}_{X}[s]$ -module M is relative holonomic at $\mathfrak{x} \in X$ when there exists a small ball $B_{\mathfrak{x}} \subseteq X$ such that the relative characteristic variety $\mathrm{Ch}^{\mathrm{rel}}(M)$ satisfies

\[ \mathrm{Ch}^{\mathrm{rel}}(M) \cap \pi^{-1}(B_{\mathfrak{x}} \times \mathrm{Spec}\ \mathbb{C}[s])= \cup_{\alpha} \Lambda_{\alpha} \times S_{\alpha} \subseteq T^{\star} X \times \mathrm{Spec}\ \mathbb{C}[s] \]

where $\Lambda_{\alpha} \subseteq T^{\star}X$ is a Lagrangian, $S_{\alpha} \subseteq \mathrm{Spec}\ \mathbb{C}[s]$ is an algebraic variety, and $\pi : T^{\star} X \times \mathrm{Spec}\ \mathbb{C}[s] \to X \times \mathrm{Spec}\ \mathbb{C}[s]$ is the projection. M is relative holonomic if it is so at all $\mathfrak{x} \in X$ .

We conclude with an elementary lemma which is contained in both [Reference MaisonobeMai23] and [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a].

3.4 Side-changing operations

As in the case of $\mathscr{D}_{X}$ -modules, there is an equivalence of categories between left $\mathscr{D}_{X}[s]$ -modules and right $\mathscr{D}_{X}[s]$ -modules; this equivalence is given by extending the $\mathscr{D}_{X}$ -module equivalence in the natural way. Given a left $\mathscr{D}_{X}[s]$ -module M, we denote the corresponding right $\mathscr{D}_{X}[s]$ -module by $M^{r}$ ; similarly, given a right $\mathscr{D}_{X}[s]$ -module N, we denote the corresponding left $\mathscr{D}_{X}[s]$ -module by $N^{\ell}$ .

In local coordinates the transformation from left to right $\mathscr{D}_{X}[s]$ -modules is given by the transposition operator

\[\tau \left(\sum x^{I} \partial^{J} s^{a}\right) = (-\partial^{J}) x^{I} s^{a}.\]

Given a set $L \subseteq \mathscr{D}_{X}[s]$ , we denote $\tau(L) = \{\tau(\ell) \mid \ell \in L\}$ . In particular, if $I \subseteq \mathscr{D}_{X}[s]$ is a left $\mathscr{D}_{X}[s]$ -ideal then

\[\left( \frac{\mathscr{D}_{X}[s]}{I} \right)^{r} = \frac{\mathscr{D}_{X}[s]}{\tau(I)}.\]

Of course $\tau$ also gives the transformation from right to left $\mathscr{D}_{X}[s]$ -modules and we have a similar statement relating right cyclic $\mathscr{D}_{X}[s]$ -modules to left ones.

We will use the following easy lemmas about side-changing operators.

Lemma 3.9. In local coordinates $(x, \partial)$ , if $\delta = \sum a_{i} \partial_{i} \in \mathrm{Der}_{\mathscr{O}_{X}}({-}\!\log g) \subseteq \mathscr{D}_{X}$ , then

\[\tau\left(\delta - s \frac{\delta \bullet g}{g}\right) = - \delta -\biggl( \sum_{i} \partial_{i} \bullet a_{i} \biggr) - s \frac{\delta \bullet g}{g}.\]

3.5 Ext-modules and localizations

Let $S \subseteq \mathbb{C}[s]$ be a multiplicatively closed subset; let $T = S^{-1} \mathbb{C}[s]$ . As in the prequel, we may consider left or right modules over $\mathscr{D}_{T} = \mathscr{D}_{X} \otimes_{\mathbb{C}} T$ . Furthermore, we may define the relative order filtration on $\mathscr{D}_{T}$ and the relative characteristic variety in analogous ways. Again we say a coherent (left or right) M $\mathscr{D}_{T}$ -module is relative holonomic when $\mathrm{Ch}^{\mathrm{rel}}(M)$ has a decomposition into products of Lagrangians of $T^{\star}X$ and algebraic varieties in $\mathrm{Spec}\ T$ as in Definition 3.6. Finally, we define the b-function of a left/right $\mathscr{D}_{T}$ -module M as before: it is the T-annihilator of M.

When M is a left (respectively, right) $\mathscr{D}_{T}$ -module, $\mathrm{Ext}_{\mathscr{D}_{T}}^{\bullet}(M, \mathscr{D}_{T})$ has a natural right (respectively, left) $\mathscr{D}_T$ -module structure. We always consider such Ext-modules to have such right/left $\mathscr{D}_{T}$ -module structures. So for us,

\[\{\text{left $\mathscr{D}_{T}$-modules}\} \ni M \mapsto \mathrm{Ext}_{\mathscr{D}_{T}}^{\bullet}(M, \mathscr{D}_{T}) \in \{\text{right $\mathscr{D}_{T}$-modules}\},\]

and symmetrically as well.

We record three more lemmas that will be utilized at the very end of the paper. All but the last are scavenged from [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a]. First we need a consequence of the flatness of localization.

Lemma 3.11. Let M be a relative holonomic left or right $\mathscr{D}_{X}[s]$ -module. Let $S \subseteq \mathbb{C}[s]$ be a multiplicatively closed set and $T = S^{-1} \mathbb{C}[s]$ . Then at any point $\mathfrak{x} \in X$ we have an isomorphism of $\mathscr{D}_{T, \mathfrak{x}} = S^{-1} \mathscr{D}_{X,\mathfrak{x}}$ -modules

\[ S^{-1} \mathrm{Ext}_{\mathscr{D}_{X, \mathfrak{x}}[s]}^{v}(M_{\mathfrak{x}}, \mathscr{D}_{X, \mathfrak{x}}) \simeq \mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{v}(S^{-1} M_{\mathfrak{x}}, \mathscr{D}_{T, \mathfrak{x}}). \]

For the last two lemmas, we introduce some notation. Let M be a left or right $\mathscr{D}_{T}$ -module. We say the grade of M at $\mathfrak{x} \in X$ is v when $\mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{j}( M_{\mathfrak{x}}, \mathscr{D}_{T, \mathfrak{x}}) = 0$ for all $j < v$ and is nonzero when $j = v$ . We say M is v-Cohen–Macaulay at $\mathfrak{x} \in X$ when $\mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{j}( M_{\mathfrak{x}}, \mathscr{D}_{T, \mathfrak{x}}) = 0 \iff j \neq v$ .

Lemma 3.12. Let M be a relative holonomic left or right $\mathscr{D}_{X}[s]$ -module that is $(n+1)$ -Cohen–Macaulay. Let $p(s) \in \mathbb{C}[s]$ and $S = \{1, p(s), p(s)^{2}, \dots \}$ so that $T = S^{-1}\mathbb{C}[s] = \mathbb{C}[s, p(s)^{-1}]$ . Then, for any $\mathfrak{x} \in X$ ,

\[ Z(B(M_{\mathfrak{x}})) \cap \mathrm{Spec}\ \mathbb{C}[s, p(s)^{-1}] = Z(B(S^{-1} M_{\mathfrak{x}})), \]

where we regard $S^{-1} M_{\mathfrak{x}}$ as a $\mathscr{D}_{T, \mathfrak{x}}$ -module.

Proof. Let $\alpha \in \mathrm{Spec}\ \mathbb{C}[s, p(s)^{-1}]$ and $\mathbb{C}_{\alpha}$ the corresponding residue field. By the commutativity of localization we have an isomorphism of $\mathscr{D}_{\mathbb{C}_{\alpha}, \mathfrak{x}}$ -modules

\[ M_{\mathfrak{x}} \otimes_{\mathbb{C}[s]} \mathbb{C}_{\alpha} \simeq S^{-1} M_{\mathfrak{x}} \otimes_{\mathbb{C}[s]} \mathbb{C}_{\alpha}. \]

Now use [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a, Proposition 3.4.3] and the fact that the ( $n+1$ )-Cohen–Macaulay property persists under localizations (Lemma 3.11).

As for $\mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{n+1}(M_{\mathfrak{x}}, \mathscr{D}_{T, \mathfrak{x}})$ , as $\mathscr{D}_{T, \mathfrak{x}}$ is an Auslander ring [Reference BjörkBjö93, IV.Theorem 4.15], by the Auslander condition its grade is at least $n+1$ ; by [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a, Lemma 3.2.4(2), Proposition 3.2.5] it is relative holonomic. Thus it has a b-function by arguing as before.

As for (b), by [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a, Lemma 3.4.1] it suffices to show $\mathrm{Ch}^{\mathrm{rel}} (M_{\mathfrak{x}}) = \mathrm{Ch}^{\mathrm{rel}} (\mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{n+1}(M_{\mathfrak{x}}, \mathscr{D}_{T, \mathfrak{x}})$ ), Using [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a, Lemma 3.2.4(2)] twice, we have

(3.5) \begin{align} \mathrm{Ch}^{\mathrm{rel}} \bigl( \mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{n+1}(\mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{n+1}(M_{\mathfrak{x}}, \mathscr{D}_{T, \mathfrak{x}}), \mathscr{D}_{T, \mathfrak{x}}) \bigr) &\subseteq \mathrm{Ch}^{\mathrm{rel}} \bigl( \mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{n+1}(M_{\mathfrak{x}}, \mathscr{D}_{T, \mathfrak{x}}) \bigr) \nonumber \\ &\subseteq \mathrm{Ch}^{\mathrm{rel}} (M_{\mathfrak{x}}). \end{align}

So if we show that the leftmost and rightmost modules of (3.5) have the same relative characteristic variety, we will have established (b).

Note that $M_{\mathfrak{x}}$ is $(n+1)$ -pure by [Reference Budur, van der Veer, Wu and ZhouBvdVWZ21a, Remark 3.3.2], i.e., every (nonzero) submodule of $M_{\mathfrak{x}}$ has grade $n+1$ . Consider the bidualizing filtration

\[ M_{\mathfrak{x}} = B_{0} \supseteq B_{-1} \supseteq \cdots \]

constructed by Björk in [Reference BjörkBjö93, IV.1.6]. By [Reference BjörkBjö93, IV.Proposition 2.8], this filtration is explicitly given by $M_{\mathfrak{x}} = B_{0} = B_{-1} = \cdots = B_{-(n+1)}$ ; and $0 = B_{-(n+1) - 1} = B_{-(n+1) - 2} = \cdots $ . Then Björk’s short exact sequence [Reference BjörkBjö93, IV.Proposition 2.1] simplifies to

(3.6) \begin{equation} M_{\mathfrak{x}} \simeq \mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{n+1}(\mathrm{Ext}_{\mathscr{D}_{T, \mathfrak{x}}}^{n+1}(M_{\mathfrak{x}}, \mathscr{D}_{T, \mathfrak{x}}), \mathscr{D}_{T, \mathfrak{x}}), \end{equation}

In particular, the leftmost and rightmost relative characteristic varieties of (3.5) agree and (b) is verified.

4.1 The complex

The first step is to construct a new complex of right $\mathscr{D}_{X}[s]$ -modules that is constructed out of logarithmic data of a divisor. This can be thought of as a parametric version of the complex of right $\mathscr{D}_{X}$ -modules $\mathrm{DR}_{X}^{-1} ( M_{X,\log}(L), \nabla^{\alpha})$ considered throughout [Reference Bath and SaitoBS24]. To be slightly more explicit, it is a parametric version of M. Saito’s $\mathrm{DR}^{-1}$ functor [Reference SaitoSai89, Definition 1.3] applied to the twisted logarithmic de Rham complex $(\Omega_{X}(\log f), \nabla^{\lambda})$ . (The differential $\nabla^{\lambda}$ is $d(-) + \wedge \lambda df/f \wedge (-)$ .) By ‘parametric’ we mean by replacing the weight $\lambda$ with a new indeterminate s.

We define the complex for general polynomials f even though we will eventually restrict to the case of $f \in R = \mathbb{C}[x_{1}, x_{2},x_{3}]$ locally quasi-homogeneous.

Definition 4.1. Let $f \in R = \mathbb{C}[x_{1}, \dots, x_{n}]$ and $X = \mathbb{C}^{n}$ . We define the complex $U_{f}^{\bullet}$ of right $\mathscr{D}_{X}[s]$ -modules

(4.2) \begin{align} U_{f}^{\bullet} &= 0 \xrightarrow[]{\nabla_{s}} \Omega_{X}^{0}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{\nabla_{s}} \Omega_{X}^{1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{\nabla_{s}} \nonumber\\ &\quad \cdots \xrightarrow[]{\nabla_{s}} \Omega_{X}^{n}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{\nabla_{s}} 0 \end{align}

where the right $\mathscr{D}_{X}[s]$ -module structure of

(4.3) \begin{equation} U_{f}^{p} = \Omega_{X}^{p}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \end{equation}

is given trivially by multiplication on the right and the differential is locally given by

(4.4) \begin{equation} \nabla_{s}(\eta \otimes P(s)) = \big( d(\eta) \otimes P(s) \big) + \left( \frac{df}{f} \wedge \eta \otimes s P(s) \right) + \biggl( \sum_{1 \leq i \leq n} dx_{i} \wedge \eta \otimes \partial_{i} P(s) \biggr). \end{equation}

The reason why the complex $U_{f}^{\bullet}$ is useful is that under our standard assumptions it resolves its terminal cohomology module. Without any assumptions we can always describe its terminal cohomology module as follows.

Proposition 4.3. We have an isomorphism of right $\mathscr{D}_{X}[s]$ -modules

\[ \frac{U_{f}^{n}}{\mathrm{im} [ \nabla_{s}: U_{f}^{n-1} \to U_{f}^{n}]} \simeq \biggl( \frac{\mathscr{D}_{X}[s]}{\mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1}} \biggr)^{r}. \]

Proof. Let $ \eta = (1/f) \sum_{i} a_{i} \widehat{dx_{i}} \in \Omega_{X}^{n-1}(\log f)$ . Recall (see Remark 2.14(e)) the $\mathscr{O}_{X}$ -isomorphism $\phi: \Omega_{X}^{n-1}(\log f) \to \mathrm{Der}_{X}(- \log f)$ given by

(4.5) \begin{equation} \phi \biggl( \frac{\sum_{i} a_{i} \widehat{dx_{i}}}{f} \biggr) = \sum_{1 \leq i \leq n} (-1)^{i-1} a_{i} \partial_{i}. \end{equation}

We compute $\mathrm{im} [\nabla_{s}: U_{f}^{n-1} \to U_{f}^{n} ]$ , recalling that $U_{f}^{n} = \omega_{X}(D) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s].$ It is enough to consider the effect of $\nabla_{s}$ on $\eta \otimes 1$ . Observe that

(4.6) \begin{align} \nabla_{s}(\eta \otimes 1 ) &= \biggl( \sum_{1 \leq i \leq n} (-1)^{i-1}(\partial_{i} \bullet a_{i}) \frac{dx}{f} \otimes 1 \biggr) + \biggl( \sum_{1 \leq i \leq n} (-1)^{i} a_{i} (\partial_{i} \bullet f) \frac{dx}{f^{2}} \otimes 1 \biggr) \nonumber\\ &\quad + \biggl( \sum_{1 \leq i \leq n} (-1)^{i-1} a_{i} (\partial_{i} \bullet f) \frac{dx}{f^{2}} \otimes s \biggr) + \biggl(\sum_{1 \leq i \leq n} (-1)^{i-1} a_{i} \frac{dx}{f} \otimes \partial_{i} \biggr) \nonumber \\ &= \biggl( \frac{dx}{f} \otimes \biggl( \phi(\eta) + (s - 1) \frac{\phi(\eta) \bullet f}{f} \biggr) \biggr) + \biggl( \frac{dx}{f} \otimes \biggl(\sum_{1 \leq i \leq n} (-1)^{i-1} (\partial_{i} \bullet a_{i}) \biggr) \biggr) \nonumber \\ &= \frac{dx}{f} \otimes - \biggl( \tau \biggl( \phi(\eta) - (s-1) \biggl(\frac{\phi(\eta) \bullet f}{f}\biggr) \biggr) \biggr) \end{align}

Here, the second ‘ $=$ ’ follows from the definition of $\phi$ in (4.5), and the third ‘ $=$ ’ is the transposition identity Lemma 3.9.

Since $\mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1}$ is generated by $\delta - (s-1) (\delta \bullet f) / f $ as $\delta$ varies over all elements of $\mathrm{Der}_{X}({-}\!\log f)$ , under the natural right $\mathscr{D}_{X}[s]$ -isomorphisms

$$U_{f}^{n} = \omega_{X}(D) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \simeq \mathscr{D}_{X}[s],$$

we can use (4.6) to deduce right $\mathscr{D}_{X}[s]$ -isomorphisms

\[ \frac{U_{f}^{n}}{\mathrm{im} [ \nabla_{s}: U_{f}^{n-1} \to U_{f}^{n} ] } \simeq \frac{\mathscr{D}_{X}[s]}{\tau(\mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1} )} \simeq \biggl( \frac{\mathscr{D}_{X}[s]}{\mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1}} \biggr)^{\!\!r}. \]

Our eventual aim is to compute certain $\mathrm{Ext}_{\mathscr{D}_{X}[s]}^{\bullet}(- , \mathscr{D}_{X}[s])$ -modules. Hence the behavior of $U_{f}^{\bullet}$ under duality is salient. We record the following observation here, since it does not require any assumptions on f except reducedness.

Proposition 4.4. Let f be reduced. Then applying $\mathrm{Hom}_{\mathscr{D}_{X}[s]}(-, \mathscr{D}_{X}[s])$ to $U_{f}^{\bullet}$ gives the following complex of left $\mathscr{D}_{X}[s]$ -modules:

(4.7) \begin{align} \mathrm{Hom}_{\mathscr{D}_{X}[s]}(U_{f}^{\bullet}, \mathscr{D}_{X}[s]) \simeq 0 &\xleftarrow[]{} \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(\Omega_{X}^{0}(\log f), \mathscr{O}_{X}) \xleftarrow[]{} \cdots \nonumber\\ &\xleftarrow[]{}\mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(\Omega_{X}^{p}(\log f), \mathscr{O}_{X}) \cdots \nonumber \\ &\xleftarrow[]{} \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(\Omega_{X}^{n}(\log f), \mathscr{O}_{X}). \end{align}

Moreover, the leftmost nontrivial cohomology of the above is given by

(4.8) \begin{equation} H^{0}(\mathrm{Hom}_{\mathscr{D}_{X}[s]}(U_{f}^{\bullet}, \mathscr{D}_{X}[s])) \simeq \frac{\mathscr{D}_{X}[s]}{\mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)}f^{-s}}.\end{equation}

Proof. That $\mathrm{Hom}_{\mathscr{D}_{X}[s]}(U_{f}^{\bullet}, \mathscr{D}_{X}[s])$ is a complex of left $\mathscr{D}_{X}[s]$ -modules is standard. As for computing each element of the complex, we have

\begin{align*} \mathrm{Hom}_{\mathscr{D}_{X}[s]}(\Omega_{X}^{p}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s], \mathscr{D}_{X}[s]) &\simeq \mathrm{Hom}_{\mathscr{O}_{X}}(\Omega_{X}^{p}(\log f), \mathrm{Hom}_{\mathscr{D}_{X}[s]}(\mathscr{D}_{X}[s], \mathscr{D}_{X}[s])) \\ &\simeq \mathrm{Hom}_{\mathscr{O}_{X}}(\Omega_{X}^{p}(\log f), \mathscr{D}_{X}[s]) \\ &\simeq \mathrm{Hom}_{\mathscr{O}_{X}}(\Omega_{X}^{p}(\log f), \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathscr{O}_{X}) \\ &\simeq \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(\Omega_{X}^{p}(\log f), \mathscr{O}_{X}), \nonumber \end{align*}

where the left $\mathscr{D}_{X}[s]$ -module structure is the obvious one. (Note that the first ‘ $\simeq$ ‘ is tensor-hom adjunction, and the fourth ‘ $\simeq$ ’ is a consequence of the flatness of $\mathscr{D}_{X}[s]$ over $\mathscr{O}_{X}$ .)

As for (4.8) and computing $H^{0}$ , we must compute the cokernel of

(4.9) \begin{equation} \mathrm{Hom}_{\mathscr{D}_{X}[s]}(\Omega_{X}^{0}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]) \xleftarrow[]{} \mathrm{Hom}_{\mathscr{D}_{X}[s]}(\Omega_{X}^{1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]). \end{equation}

In the original complex $U_{f}^{\bullet}$ we know that

(4.10) \begin{align} \Omega_{X}^{0}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] &\ni 1 \otimes 1 \nonumber\\ &\mapsto \biggl( \frac{df}{f} \otimes s\biggr) + \biggl(\sum_{i} dx_{i} \otimes \partial_{i}\biggr) \nonumber \\ &\in \Omega_{X}^{1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]. \end{align}

Using the perfect pairing between logarithmic derivations and log 1-forms (see Remark 2.14(f)),

\begin{align*} P \otimes \delta \in \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Der}_{X}({-}\!\log f) &\simeq \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(\Omega_{X}^{1}(\log f), \mathscr{O}_{X}) \\ &\simeq \mathrm{Hom}_{\mathscr{D}_{X}[s]}(\Omega_{X}^{1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s], \mathscr{D}_{X}[s]) \end{align*}

acts on the image of the element 1 inside $\Omega_{X}^{1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]$ as computed in (4.10) via

\begin{align*} (P \otimes \delta ) \biggl( \biggl( \frac{df}{f} \otimes s\biggr) + \biggl(\sum_{i} dx_{i} \otimes \partial_{i}\biggr) \biggr) &= P \biggl( \biggl( \iota_{\delta}\biggl(\frac{df}{f}\biggr) s\biggr) + \sum_{i} \iota_{\delta}(dx_{i}) \partial_{i} \biggr) \\[-3pt] &= P \biggl( s \frac{\delta \bullet f}{f} + \delta\biggr) \\ &= P \biggl( \delta - \biggl(-s \frac{\delta \bullet f}{f}\biggr) \biggr), \end{align*}

where we use $\delta$ to denote the same element in $\mathrm{Der}_{X} \subseteq \mathscr{D}_{X}[s]$ . It follows that the cokernel of (4.9) is exactly the left $\mathscr{D}_{X}[s]$ -module $\mathscr{D}_{X}[s] / \mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{-s}.$

4.2 Conditions for a resolution

We now require additional assumptions on f. When $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ is reduced and locally quasi-homogeneous it will turn out that all but the terminal cohomology $U_{f}^{\bullet}$ to vanish, i.e., it will force $U_{f}^{\bullet}$ to be a resolution of $(\mathscr{D}_{X}[s] / \mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1})^{r}$ (cf. Proposition 4.3). However, we can get the same result with weaker assumptions. Hence we have the following definition.

Definition 4.6. We say that a divisor $D \subseteq X$ is locally tame at $\mathfrak{x} \in X$ when, for any defining equation germ f at $\mathfrak{x}$ , we have

\[ \mathrm{pdim}_{\mathscr{O}_{X,\mathfrak{x}}} \Omega_{X, \mathfrak{x}}^{p} (\log f) \leq p. \]

We say D is tame when it is tame locally everywhere.

Here is the promised resolution result for $U_{f}^{\bullet}$ .

Proposition 4.8 Suppose that $f \in R = \mathbb{C}[x_{1}, \dots, x_{n}]$ defines a tame, Saito holonomic divisor. Then $U_{f}^{\bullet}$ resolves $(\mathscr{D}_{X}[s] / \mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1})^{r}$ , i.e., the augmented complex

\[ U_{f}^{\bullet} \to \biggl( \frac{\mathscr{D}_{X}[s]}{\mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1}} \biggr)^{\!\!r} \]

is acyclic. If in addition f is locally quasi-homogeneous, then $U_{f}^{\bullet}$ resolves $\mathscr{D}_{X}[s]f^{s-1}$ . In particular, $U_{f}^{\bullet}$ resolves $\mathscr{D}_{X}[s]f^{s-1}$ when $f \in \mathbb{C}[x_{1}, x_{2}, x_{3}]$ is locally quasi-homogeneous.

Proof. First of all, the ‘in particular’ claim is well known: such a divisor is tame by Remark 4.7; such a divisor also has a finite logarithmic stratification (see Remark 2.10(c)). The claim of the penultimate sentence follows by [Reference WaltherWal17, Theorem 3.26] which gives, in this setting, the natural isomorphism

\[ \frac{\mathscr{D}_{X}[s]}{\mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1}} \xrightarrow[]{\simeq} \mathscr{D}_{X}[s] f^{s-1}. \]

All that is left is to justify the second sentence. Each $U_{f}^{p}$ can be given a decreasing filtration by s-degree in such a way as to make $U_{f}^{\bullet}$ a filtered complex. Moreover, the resulting associated graded complex $\mathrm{gr}\ U_{f}^{\bullet}$ has (locally) objects and maps

\begin{align*} \Omega_{X}^{p}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] &\ni \eta \otimes P(s) \\ &\mapsto (d(\eta) \otimes P(s) ) + \Bigl( \sum_{i} dx_{i} \wedge \eta \otimes \partial_{i} P(s) \Bigr) \\ &\in \Omega_{X}^{p+1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]. \end{align*}

(To be explicit, first define a decreasing filtration on $U_f^p$ by s-degree: $G^{t,p} (U_f^p) = \Omega_X^p(\log f) \otimes_{\mathscr{O}_X} \mathscr{D}_{X} \otimes_{\mathbb{C}} \mathbb{C}[s]_{\geq t}$ . Note that $G^{t+1, p} (U_f^p) \subseteq G^{t, p}(U_f^p)$ . Now set

\begin{equation*} G^t(U_f^\bullet) = [ \cdots \xrightarrow[]{} G^{t, p-1}(U_f^p) \to G^{t, p}(U_f^p) \to G^{t,p+1}(U_f^{p+1}) \to \cdots] \end{equation*}

with differentials induced by those of $U_f^\bullet$ . The aforementioned decreasing filtration on $U_f^\bullet$ is $\cdots \subseteq G^{t+1}(U_f^\bullet) \subseteq G^t(U_f^\bullet) \subseteq \cdots $ .) By standard spectral sequence considerations, it suffices to show that $H^{k} (\mathrm{gr}\ U_{f}^{\bullet}) = 0$ for all $k \neq n$ .

We sketch a proof of this fact exploiting the local finiteness of the logarithmic strata, i.e., Saito holonomicity. We refer the reader to [Reference WaltherWal17, Theorem 3.9] or [Reference BathBat21, Proposition 3.11] for more details about Saito holonomic induction; our proof here has similarities to the arguments cited.

First assume $\dim X = 1$ . That $H^k(\mathrm{gr}\ U_f^\bullet) = 0$ for all $k \neq n$ (i.e., for $k=0)$ can be checked by hand since $\mathrm{gr}\ U_f^0$ is cyclically generated by $1 \otimes 1$ (recall that $\Omega_X^0(\log f) = \Omega_X^0$ ). Now assume that $\dim X = n$ and the claim holds on smaller-dimensional spaces. Fix $\mathfrak{x} \in X$ . If $\mathfrak{x}$ belongs to a positive-dimensional logarithmic stratum, the Saito holonomic assumption guarantees a local analytic isomorphism $(X, D, \mathfrak{x}) \simeq (\mathbb{C}, \mathbb{C}, 0) \times (X^{\prime}, D^{\prime}, \mathfrak{x}^{\prime})$ where $D \subseteq X$ is the divisor germ of f at $\mathfrak{x}$ , $\dim X^{\prime} < X$ , and $D^{\prime} \subseteq X^{\prime}$ is the divisor germ of some $f^{\prime}$ at $\mathfrak{x}^{\prime} \in X^{\prime}$ . This local analytic isomorphism arises exactly as in Remark 2.10(c), since we have assumed there is a nonvanishing logarithmic vector field at $\mathfrak{x}$ . Remark 2.10(c) says that Saito holonomicity is preserved along such products. And there is a Künneth formula for logarithmic forms in such situations (see, for example, [Reference Cohen, Denham, Falk and VarchenkoCDFV11, Lemma 3.10]), meaning tameness is also preserved under such products. So $(X, D^\prime, \mathfrak{x}^\prime)$ falls into our inductive setup. Since this Künneth formula is compatible with our complex $\mathrm{gr}\ U_f^\bullet$ , the induction hypothesis implies the desired cohomological vanishing at $\mathfrak{x}$ .

We deduce that, for all $k \neq n$ , any nonvanishing $H^k(\mathrm{gr}\ U_f^\bullet)$ has zero-dimensional support. We conclude $H^k(\mathrm{gr}\ U_f^\bullet)$ vanishes for all $k \neq 0$ by invoking [Reference BathBat21, Lemma 3.6] (see also [Reference BathBat21, Remark 3.7]), which is a sort of noncommutative variant of the Peskine–Szpiro acyclicity lemma.

4.3 The double complex

In this subsection we finally restrict to $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ reduced and locally quasi-homogeneous. We will construct a double complex $K_{f}^{\bullet, \bullet}$ out of $U_{f}$ ; we will use this double complex to start a computation of $\mathrm{Ext}_{\mathscr{D}_{X}[s]}^{3}((\mathscr{D}_{X}[s] f^{s})^{r}, \mathscr{D}_{X}[s])$ . Actually, using a standard spectral sequence approach, we will uncover what map of $\mathscr{D}_{X}[s]$ -modules we need to understand to perform this computation (see Lemma 4.15). So the Ext-module computation will have to wait until the next section, wherein we analyze this map.

To set up the double complex’s foundations, recall (Proposition 2.18, Definition 2.19) our fixed free $\mathscr{O}_{X}$ -resolutions

\[Q_{\bullet}^{2} = 0 \to Q_{1}^{2} \to Q_{0}^{2} \to \Omega_{X}^{2}(\log f) \to 0,\]

\[Q_{\bullet}^{1} = 0 \to Q_{1}^{1} \to Q_{0}^{2} \to \Omega_{X}^{1}(\log f) \to 0.\]

As $\mathscr{D}_{X}[s]$ is a flat $\mathscr{O}_{X}$ -module, $Q_{\bullet}^{2} \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]$ and $Q_{\bullet}^{1} \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]$ are free, right $\mathscr{D}_{X}[s]$ -resolutions of $\Omega_{X}^{2}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]$ and $\Omega_{X}^{1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]$ , respectively.

By general facts about projective modules, we can lift the differentials of the all but terminally acyclic $U_{f}^{\bullet}$ (Proposition 4.8) along the data of our preferred $\mathscr{D}_{X}[s]$ -resolutions, obtaining a double complex. Thus we are entitled to make the following definition.

Definition 4.11 . Let $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be locally quasi-homogeneous. Consider a double complex $K_{f}^{\bullet, \bullet}$ of free right $\mathscr{D}_{X}[s]$ -modules and maps

(4.11)

such that:

1. (i) $K^{p,q} = 0$ for all $p, q \neq 0, 1$ ;

2. (ii) $K^{p, q} = Q_{q}^{p} \otimes_{\mathscr{O}_X} \mathscr{D}_X[s]$ ;

3. (iii) the vertical differentials are $\nu_{q}^{p}: K^{p, q} \to K^{p, q-1}$ ;

4. (iv) the horizontal differentials are $\sigma_{q}^{p} : K^{p,q} \to K^{p+1, q}$ ;

5. (v) the augmentation of $K^{1,\bullet}$ is/induces the acyclic complex

   \[ 0 \to Q_{1}^{1} \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{\nu_{1}^{1}} Q_{0}^{1} \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{} \Omega_{X}^{1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{} 0; \]

6. (vi) the augmentation of $K^{2, \bullet}$ is/induces the acyclic complex

   \[ 0 \to Q_{1}^{2} \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{\nu_{1}^{2}} Q_{0}^{2} \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{} \Omega_{X}^{2}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{} 0; \]

7. (vii) all the maps of (4.11) are right $\mathscr{D}_{X}[s]$ -linear and all the squares of (4.11) commute.

Finally, use $\mathrm{Tot}(K_{f}^{\bullet, \bullet})$ to denote the total complex of $K^{\bullet, \bullet}$ .

Definition 4.11 is useful to us because of the following proposition.

Proof . Certainly $\mathrm{Tot}(K_{f}^{\bullet, \bullet})$ is a complex of free (right) $\mathscr{D}_{X}[s]$ -modules. We compute the cohomology of $\mathrm{Tot}(K_{f}^{\bullet, \bullet})$ by using the spectral sequence that first computes cohomology vertically. By the construction, the first page of the spectral sequence has only nonzero entries on the x-axis and this row is exactly $U_{f}^{\bullet}$ . The second page of the spectral sequence is $H^{\bullet}(U_{f}^{\bullet})$ , as positioned on the x-axis. By Proposition 4.8 all these cohomologies vanish except for

\[ H^{n}(U_{f}^{\bullet}) \simeq \frac{\mathscr{D}_{X}[s]}{\mathrm{ann}_{\mathscr{D}_{X}[s]}^{(1)} f^{s-1}} \xrightarrow[]{\simeq} \mathscr{D}_{X}[s] f^{s-1}. \]

For our computation of $\mathrm{Ext}_{\mathscr{D}_{X}[s]}^{3}((\mathscr{D}_{X}[s])^{r}, \mathscr{D}_{X}[s])$ we will Hom the total complex $\mathrm{Tot}(K_{f}^{\bullet, \bullet})$ into $\mathscr{D}_{X}[s]$ . This gives a double complex vulnerable to spectral sequence tactics. The $\infty$ -page of our spectral sequence will describe the associated graded pieces of a filtration on $\mathrm{Ext}_{\mathscr{D}_{X}[s]}^{\bullet}((\mathscr{D}_{X}[s])^{r}, \mathscr{D}_{X}[s])$ and it is from this that our first main result, Theorem 6.4, follows.

Understanding this spectral sequence will require a fair amount of work. The opening salvo, which makes our required labors explicit, requires reducedness and is as follows.

Lemma 4.15. Let $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be reduced and locally quasi-homogeneous. Let $E_{v}^{\bullet, \bullet}$ denote the vth page of the spectral sequence of the double complex of left $\mathscr{D}_{X}[s]$ -modules $\mathrm{Hom}_{\mathscr{D}_{X}[s]}(\mathrm{Tot}(K_{f}^{\bullet, \bullet}), \mathscr{D}_{X}[s])$ where one computes cohomology first vertically and then horizontally. Then the second page of the spectral sequence is

(4.12)

where all entries outside of (4.12) are zero, and the $E_{2}^{1,1}$ and $E_{2}^{2,1}$ entries refer to the cokernel and kernel, respectively, of the natural map

(4.13) \begin{equation} \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{X}^{2}(\log f), \mathscr{O}_{X}) \xrightarrow[]{\mathrm{Hom}_{\mathscr{D}_{X}[s]}(\sigma_{1}^{1}, \mathscr{D}_{X}[s])} \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{X}^{1}(\log f), \mathscr{O}_{X}). \end{equation}

Proof. First consider the complex $\mathrm{Hom}_{\mathscr{D}_{X}[s]}(\mathrm{Tot}(K_{f}), \mathscr{D}_{X}[s])$ , which can be viewed as the total complex of a double complex. The pth column is of the form $\mathrm{Hom}_{\mathscr{D}_{X}[s]}(Q_{\bullet}^{p} \otimes_{\mathscr{O}_X} \mathscr{D}_{X}[s], \mathscr{D}_{X}[s])$ . By arguing as in the first display equation of (4.8), we may rewrite this complex of left $\mathscr{D}_{X}[s]$ -modules as $\mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(Q_{\bullet}^{p}, \mathscr{O}_{X})$ . Since $\mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}}-$ is flat and hence exact, applying this functor commutes with taking cohomology. By the construction of $Q_{\bullet}^{p}$ we deduce the only nonzero entries on the first page $E_{1}^{\bullet, \bullet}$ of the spectral sequence are

(4.14)

To compute $E_{2}^{\bullet, \bullet}$ we compute cohomology of the rows of the first page. Proposition 4.4 computes the terminal cohomology of row $E_{1}^{0, \bullet}$ : this gives $E_{2}^{0,0}$ as described in (4.12). As for row $E_{1}^{1, \bullet}$ we are tasked with computing cohomology of the complex

\begin{align*} 0 \xleftarrow[]{} \mathscr{D}_{X}[s] &\otimes_{\mathscr{O}_{X}} \mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{X}^{1}(\log f), \mathscr{O}_{X}) \\ &\xleftarrow[]{\mathrm{Hom}_{\mathscr{D}_{X}[s]}(\sigma_{1}^{1}, \mathscr{D}_{X}[s])} \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{X}^{2}(\log f), \mathscr{O}_{X}) \xleftarrow[]{} 0, \nonumber \end{align*}

which agrees with the description of $E_{2}^{1,1}$ and $E_{2}^{2,1}$ in (4.12), (4.13).

Lemma 4.15 makes it clear we must understand the aforementioned map (4.13). It will turn out that this map is injective (see the proof of Theorem 6.4). Therefore $E_{2}^{1,1}$ and $E_{2}^{0,0}$ of (4.12) will be the associated graded pieces of a filtration on $\mathrm{Ext}_{\mathscr{D}_{X}[s]}^{3}((\mathscr{D}_{X}[s] f^{s-1})^{r}, \mathscr{D}_{X}[s])$ , which in turn describe this Ext-module in terms of a short exact sequence.

5.1 Lift of an algebraic map

Throughout this subsection, $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ is quasi-homogeneous with weighted homogeneity E.

We start our construction of a suitable replacement for the lift $\sigma_{1}^{1}$ with an algebraic analog. We will use the algebraic version to build the analytic version we eventually require.

While we have defined $U_{f}^{\bullet}$ in the analytic category, we can also define it in the algebraic category. In particular, this will give a map after taking global sections. As prophesied, in this subsection we focus on lifting one such map along free resolutions of the domain and codomain.

To set this up, let $\mathbb{A}_{3}$ be the Weyl algebra in three variables, i.e., the global sections of the algebraic sheaf of differential operators over $\mathbb{A}_{\mathbb{C}}^{3}$ ; let $\mathbb{A}_{3}[s]$ be a polynomial ring extension. We will compute explicit lifts of the right $\mathbb{A}_{3}[s]$ -linear map

(5.2) \begin{equation} \nabla_{s}: \Omega_{R}^{1}(\log f) \otimes_{R} \mathbb{A}_{3}[s] \to \Omega_{R}^{2}(\log f) \otimes_{R} \mathbb{A}_{3}[s]\end{equation}

along explicit free resolutions of the domain and codomain.

Recall our fixed graded minimal free resolution of $\Omega_{R}^{2}(\log_{0} f)$ ,

\[F_{\bullet} = 0 \to F_{1} \xrightarrow[]{\beta_{1}} F_{0} \xrightarrow[]{\beta_{0}} \Omega_{R}^{2}(\log_{0} f) \to 0,\]

and the induced minimal graded free resolutions (cf. Proposition 2.16, Definition 2.17)

\begin{eqnarray*}P_{\bullet}^{2} &=& 0 \to P_{1}^{2} = F_{1}\xrightarrow[]{(\beta_{1},0)} P_{0}^{2} = F_{0} \oplus R\Bigl({-}d +\sum w_{i}\Bigr) \xrightarrow[]{(\beta_{0}, \iota_{E} dx/f)}\Omega_{R}^{2}(\log f) \to 0,\\P_{\bullet}^{1} &=& 0 \to P_{1}^{1} = F_{1} \xrightarrow[]{(\beta_{1}, 0)} P_{0}^{0} = F_{0} \oplus R \xrightarrow[]{(\iota_{E} \beta_{0}, \cdot df/f)} \Omega_{R}^{1}(\log f) \to 0.\end{eqnarray*}

Since $\mathbb{A}_{3}[s]$ is flat over R we see that $P_{\bullet}^{2} \otimes \otimes_{R} \mathbb{A}_{3}[s]$ and $P_{\bullet}^{1} \otimes_{R} \mathbb{A}_{3}[s]$ are free resolutions of $\Omega_{R}^{2}(\log f) \otimes_{R} \mathbb{A}_{3}[s]$ and $\Omega_{R}^{1}(\log f) \otimes_{R} \mathbb{A}_{3}[s]$ , respectively. We want to construct right $\mathbb{A}_{3}[s]$ -maps $\epsilon_{0}$ and $\epsilon_{1}$ so that the following diagram commutes.

(5.3)

(The left column is $P_{\bullet}^{1} \otimes_{R} \mathbb{A}_{3}[s]$ written explicitly, the right is similar.) While the promised lifts of (5.2) exist by general facts about projective resolutions, we will construct lifts so that $\epsilon_{1}$ is essentially given by left multiplication with $\mathrm{deg}_{\mathbf{w}}(f)s + q + E$ where q is degree data depending on the input of $\epsilon_{1}$ . The rest of this subsection is devoted to setting up and proving the following proposition.

We lay the foundation for the proof of Proposition 5.1. Return to the original map $\nabla_{s}$ and recall the direct sum decomposition of $\Omega_{R}^{p}(\log f) = \Omega_{R}^{p}(\log_{0} f) \oplus \iota_{E} \Omega_{R}^{p+1}(\log_{0} f)$ . Our strategy depends on the behavior of $\nabla_{s}$ on members of each direct summand, which in turn hinges on a sort of Lie derivative trick. We denote by $L_{E}$ the Lie derivative with respect to the Euler derivation E. It is easy to check that our grading conventions on $\Omega_{R}^{p}(\log f)$ guarantee that $L_{E}(\eta) = \mathrm{deg}_{\mathbf{w}}(\eta) \eta$ , provided $\eta$ is homogeneous. If $\eta \in \Omega_{R}^{p}(\log f)$ is homogeneous, by basic properties of Lie derivatives and contractions, then

(5.4)

(In the expression $\iota_E (\nabla_s(\eta \otimes P))$ of the final line, we are implicitly letting the interior derivative act on all terms of the complex $\Omega_R^\bullet(\log f) \otimes_R \mathbb{A}_3[s]$ , by linearly extending its action across $\mathbb{A}_3[s]$ .) Note that if $\eta \in \Omega_R^2(\log_0 f)$ , then in the final line of (5.4) the first summand lives in the direct summand $\Omega_{R}^{2}(\log_{0} f) \otimes_R \mathbb{A}_3[s]$ whereas the second summand in the direct summand $\iota_{E}\Omega_{R}^{3}(\log f) \otimes_R \mathbb{A}_3[s]$ (see Remark 2.14(b)).

We now return to the lifting problem where we compute $\epsilon_{1}$ in two steps. We first restrict the domain of $\Omega_{R}^{2}(\log f) \otimes_{R} \mathbb{A}_{3}[s] \xrightarrow[]{\nabla_{s}} \Omega_{R}^{3}(\log f) \otimes_{R} \mathbb{A}_{3}[s]$ by considering the induced map $\iota_{E}\Omega_{R}^{2}(\log_{0} f) \otimes_{R} \mathbb{A}_{3}[s] \xrightarrow[]{\nabla_{s}} \Omega_{R}^{2}(\log f) \otimes_{R} \mathbb{A}_{3}$ (cf. Remark 2.14(b)). We compute certain lifts $\mu_{0}, \mu_{1}$ of the induced map $\nabla_{s}$ as depicted here.

(5.5)

Proposition 5.2. The map $\mu_{0} : F_{0} \otimes_{R} \mathbb{A}_{3}[s] \to (F_{0} \otimes_{R} \mathbb{A}_{3}[s]) \oplus R(n-d) \otimes_{R} \mathbb{A}_{3}[s]$ , which, for $a \in F_{0}$ homogeneous, is given by

\[ \mu_{0}(a \otimes A) = ( a \otimes (\mathrm{deg}_{\mathbf{w}}(f)s + \mathrm{deg}_{\mathbf{w}}(a) + E)A ) - ( (\iota_{E} dx/f \otimes \mathrm{id})^{-1} ( (\iota_{E} \circ \nabla_{s})(\beta_{0}(a) \otimes A) ) ), \]

is well defined, right $\mathbb{A}_{3}[s]$ -linear, and makes the lower square of (5.5) commute.

Proof. First note that $(\iota_{E} dx/f \otimes \mathrm{id}) : R(n-d) \otimes_{R} \mathbb{A}_{3}[s] \to \iota_{E} \Omega_{R}^{3}(\log f) \otimes_{R} \mathbb{A}_{3}[s]$ is an isomorphism, so its inverse uniquely selects an element. To confirm well-definedness, since $\nabla_{s}$ and $\iota_{E}$ are well-defined right- $\mathbb{A}_{3}[s]$ maps, we must check the assignment

(5.6) \begin{equation} a \otimes A \mapsto a \otimes (\mathrm{deg}_{\mathbf{w}}(f)s + \mathrm{deg}_{\mathbf{w}}(a) + E) A \end{equation}

is well defined. In other words, that if $a \in F_{0}$ is homogeneous and $r \in R$ homogeneous, then (5.6) has the same output on $ar \otimes A$ and $a \otimes rA$ . This reduces to the commutator identity $rE = Er - [E,r] = Er - E \bullet r = Er - \mathrm{deg}_{\mathbf{w}}(r)$ and the obvious fact that $\mathrm{deg}_{\mathbf{w}}(ra) = \mathrm{deg}_{\mathbf{w}}(a) + \mathrm{deg}_{\mathbf{w}}(r)$ .

We now confirm $\mu_{0}$ makes the lower square of (5.5) commute. Let $a \in F_{0}$ be homogeneous. On the one hand,

\begin{align*} &\big( \nabla_{s} \circ (\iota_{E} \beta_{0} \otimes \mathrm{id}) \big) (a \otimes A) \\ &\quad = \nabla_{s} \big( \iota_{E} \beta_{0} (a) \otimes A \big) \\ &\quad = \big( \beta_{0}(a) \otimes (\mathrm{deg}_{\mathbf{w}}(f)s + \mathrm{deg}_{\mathbf{w}}(\beta_{0}(a)) + E) A \big) - \big( \iota_{E} (\nabla_{s} (\beta_{0}(a) \otimes A)) \big) \\ &\quad = \big( \beta_{0}(a) \otimes (\mathrm{deg}_{\mathbf{w}}(f)s + \mathrm{deg}_{\mathbf{w}}(a) + E) A \big) - \big( \iota_{E} (\nabla_{s} (\beta_{0}(a) \otimes A)) \big). \end{align*}

Here, the second ‘ $=$ ’ is (5.4), and the third ‘ $=$ ’ uses the fact that $\beta_{0}$ is degree-preserving. On the other hand,

\begin{align*} (\beta_{0} \otimes \mathrm{id} , \iota_{E} df/f \otimes \mathrm{id}) \big( \mu_{0}(a \otimes A) \big) &= \big( \beta_{0}(a) \otimes (\mathrm{deg}_{\mathbf{w}}(f)s + \mathrm{deg}_{\mathbf{w}}(a) + E)A \big) \\ &\quad - (\iota_{E} \circ \nabla_{s}) (\beta_{0}(a) \otimes A).\\[-32pt] \end{align*}

Proposition 5.3. The map $\mu_{1}: F_{1} \otimes_{R} \mathbb{A}_{3}[s] \to F_{1} \otimes_{R} \mathbb{A}_{3}[s]$ which, for $b \in F_{1}$ homogeneous, is given by

\[ \mu_{1}(b \otimes B) = b \otimes (\mathrm{deg}_{\mathbf{w}}(f)s + \mathrm{deg}_{\mathbf{w}}(b) + E)B, \]

is well defined and right $\mathbb{A}_{3}[s]$ -linear. Moreover, with $\mu_{0}$ as defined in Proposition 5.2, $\mu_{1}$ makes the upper square of (5.5) commute.

Proof. The well-definedness of $\mu_{1}$ follows exactly from the commutator identity used in checking the well-definedness of $\mu_{0}$ (cf. Proposition 5.2). To check the upper square of (5.5) commutes, let us compute. On the one hand,

\begin{align*} \big( \mu_{0} \circ (\beta_{1} \otimes \mathrm{id}) \big) (b \otimes B) &= \mu_{0} \big( \beta_{1}(b) \otimes B) \\ &= \beta_{1}(b) \otimes (\mathrm{deg}_{\mathbf{w}}(f)s + \mathrm{deg}_{\mathbf{w}}(\beta_{1}(b)) + E)B \\ &= \beta_{1}(b) \otimes (\mathrm{deg}_{\mathbf{w}}(f)s + \mathrm{deg}_{\mathbf{w}}(b) + E)B. \end{align*}

Here, the second ‘ $=$ ’ uses the fact that $(\beta_{0} \circ \beta_{1})(b) = 0$ , and the third ‘ $=$ ’ uses the fact that $\beta_{1}$ preserves degree. On the other hand,

\begin{align*} \big((\beta_{1} \otimes \mathrm{id}, 0) \circ \mu_{1} \big) (b \otimes B) &= (\beta_{1} \otimes \mathrm{id}, 0) \big( b \otimes (\mathrm{deg}_{\mathbf{w}}(f) s + \mathrm{deg}_{\mathbf{w}}(b) + E)B \big) \\ &= \beta_{1}(b) \otimes (\mathrm{deg}_{\mathbf{w}}(f) s + \mathrm{deg}_{\mathbf{w}}(b) + E)B.\\[-36pt] \end{align*}

We can now return to the original problem of constructing $\epsilon_{0}, \epsilon_{1}$ and filling in (5.3). Restrict the domain of $\nabla_{s}$ in a different way by considering $\nabla_{s}: \Omega_{R}^{1}(\log_{0} f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \to \Omega_{R}^{2}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]$ (cf. Remark 2.14(b)). By basic facts about projective resolutions, there exists a right $\mathbb{A}_{3}[s]$ -linear maps $\nu_{0}$ so that the following diagram commutes.

(5.7)

(By Remark 2.14(d) the left column in (5.7) is a free resolution; by Proposition 2.16 and Definition 2.17 the same holds for the right column.)

We are ready to prove the subsection’s main result.

Proof of Proposition 5.1. First define

\[ \epsilon_{0} : (F_{0} \otimes_{R} \mathbb{A}_{3}[s]) \oplus (R \otimes_{R} \mathbb{A}_{3}[s]) \to (F_{0} \otimes_{R} \mathbb{A}_{3}[s]) \oplus (R(n-d) \otimes_{R} \mathbb{A}_{3}[s]) \]

by $\epsilon_{0} = \mu_{0} \oplus \nu_{0}$ , where $\mu_{0}$ is as defined in Proposition 5.2 and $\nu_{0}$ is as promised by (5.7). As the summands of $\epsilon_{0}$ are right $\mathbb{A}_{3}[s]$ -linear, so is $\epsilon_{0}$ ; as the summands of $\epsilon_{0}$ make the lowest squares of (5.5) and (5.7) commute, this choice of $\epsilon_{0}$ makes the lowest square of (5.3) commute.

Now define

\[ \epsilon_{1} : F_{1} \otimes_{R} \mathbb{A}_{3}[s] \to F_{1} \otimes_{R} \mathbb{A}_{3}[s] \]

by $\epsilon_{1} = \mu_{1}$ . Proposition (5.3) shows that $\mu_{1}$ is right $\mathbb{A}_{3}[s]$ -linear and that the upper square of (5.5) commutes. By the definition of $\epsilon_{0}$ , the commutativity of said square is equivalent to the commutativity of the upper square of (5.3). We have checked (a). That (b) holds is by definition of $\epsilon_{1} = \mu_{1}$ (see Proposition 5.3).

5.2 Lift of an analytic map and its behavior on Ext

In this subsection $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ is quasi-homogeneous with weighted homogeneity E.

We return to the analytic setting. We use the previous subsection to compute a lift of the map

(5.8) \begin{equation} \nabla_{s} : \Omega_{X}^{1}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s] \xrightarrow[]{} \Omega_{X}^{2}(\log f) \otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]\end{equation}

along explicit free resolutions of the domain and codomain.

Again, we have fixed a minimal graded free resolution of $\Omega_{R}^{2}(\log f)$ ,

\[F_{\bullet} = 0 \to F_{1} = \bigoplus_{i} R(b_{i}) \xrightarrow[]{\beta_{1}} F_{0} = \bigoplus_{j} R(a_{j}) \xrightarrow[]{\beta_{0}} \Omega_{R}^{2}(\log_{0} f) \to 0,\]

which induces (cf. Proposition 2.16, Definition 2.17, Proposition 2.18, Definition 2.19) free $\mathscr{O}_{X}$ -resolutions

\[Q_{\bullet}^{2} = 0 \to Q_{1}^{2} \to Q_{0}^{2} \to \Omega_{X}^{2}(\log f) \to 0,\]

\[Q_{\bullet}^{1} = 0 \to Q_{1}^{1} \to Q_{0}^{2} \to \Omega_{X}^{1}(\log f) \to 0.\]

As we saw earlier, tensoring on the right with $\otimes_{\mathscr{O}_{X}} \mathscr{D}_{X}[s]$ preserves the property of ‘being a resolution’ thanks to flatness. So we know there are lifts of (5.8) along free resolutions so that the following diagram commutes.

(5.9)

More specifically, we have the following proposition.

Proposition 5.4. In the setup of this section, recall that $F_{1} = \bigoplus_{i} R(b_{i})$ , and that $Q_{1}^{2} = Q_{1}^{1}$ are the (algebraic) sheafification and analytification of $F_{1}$ . Let $\mathbf{e}_{t}$ be the tth indicator vector of $Q_{1}^{2} = Q_{1}^{1}$ . Then we may construct right $\mathscr{D}_{X}[s]$ -maps $\theta_{1}, \theta_{0}$ making (5.9) commute, where

(5.10) \begin{equation} \theta_{1} (\mathbf{e}_{t} \otimes 1) = \mathbf{e}_{t} \otimes (\mathrm{deg}_{\mathbf{w}}(f)s - b_{t} + E). \end{equation}

To do this, set $\theta_{j}$ to be the map induced by the $\epsilon_{j}$ constructed in the previous subsection making (5.3) commute. Indeed, $P_{j}^{1} \otimes_{R} \mathbb{A}_{3}[s]$ is a free $\mathbb{A}_{3}[s]$ -module and so $\epsilon_{j}$ can be defined by its behavior on a basis, reducing it to left matrix multiplication. Then each $\theta_{j}$ is given by the same map. (The matrices defining $\epsilon_{j}$ are globally defined with (homogeneous) polynomial entries.) That the squares of (5.9) commute amounts to checking a matrix identity, which we know is true by Proposition 5.1. Finally, $\epsilon_{1}$ sends $\mathbf{e}_{t} \otimes 1$ , where $\mathbf{e}_{t}$ is the tth indicator vector of $F_{1}$ to $\mathbf{e}_{t} \otimes (\mathrm{deg}_{\mathbf{w}}(f) s - b_{t} + E)$ , since $\mathbf{e}_{t} \in F_{1} = \bigoplus_{i} R(b_{i})$ has degree $-b_{i}$ . So the matrix defining $\epsilon_{1}$ is a diagonal matrix with (t,t)th entry $(\mathrm{deg}_{\mathbf{w}}(f)s - b_{t} + E)$ and the same matrix will define $\theta_{1}$ .

This confirms the upper square of (5.9) commutes and confirms (5.10). We will be done once we show the bottom square of (5.9) commutes. But since f is a polynomial, $\Omega_{X}^{1}(\log f)$ is (locally everywhere) generated by polynomials. So the commutativity of the lower square of (5.9) follows from the commutativity of the lower square of (5.5) and the construction of $\theta_{0}$ .

As our eventual aim is to understand the behavior of lifts of (5.8) after applying $\mathrm{Hom}_{\mathscr{D}_{X}[s]}(- , \mathscr{D}_{X}[s])$ we need the following proposition.

Proposition 5.6. In the setup of Proposition 5.4, applying $\mathrm{Hom}_{\mathscr{D}_{X}[s]}(- , \mathscr{D}_{X}[s])$ to (5.9) gives the map of left $\mathscr{D}_{X}[s]$ -modules

(5.11) \begin{equation} \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(Q_{1}^{1}, \mathscr{O}_{X}) \xleftarrow[]{\mathrm{Hom}_{\mathscr{D}_{X}[s]}(\theta_{1}, \mathscr{D}_{X}[s])} \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(Q_{1}^{2}, \mathscr{O}_{X}). \end{equation}

If $\mathbf{e}_{t}$ is the tth indicator vector of $\mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(Q_{1}^{2}, \mathscr{O}_{X}) = \mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Hom}_{\mathscr{O}_{X}}(Q_{1}^{1}, \mathscr{O}_{X})$ then

(5.12) \begin{equation} \mathrm{Hom}_{\mathscr{D}_{X}[s]}(\theta_{1}, \mathscr{D}_{X}[s]) \left( 1 \otimes \mathbf{e}_{t} \right) = (\mathrm{deg}_{\mathbf{w}}(f) s - b_{t} + E) \otimes \mathbf{e}_{t}. \end{equation}

The preceding result immediately gives an induced map on $\mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Ext}$ -modules. Since these Ext-modules vanish everywhere except, possibly, at the origin (Proposition 2.20), it suffices to describe the map at zero.

Proposition 5.7. Stay in the setup of Propositions 5.4 and 5.6. Let $\Delta_{f}$ be a finite homogeneous generating set of $\mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R)$ as a $\mathbb{C}$ -vector space; for $u \in \Delta_{f}$ , let $\mathrm{deg}_{\mathbf{w}}(u)$ denotes its degree. Then $\Delta_{f}$ is a finite homogeneous generating set of both $\mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X,0}^{1}(\log f), \mathscr{O}_{X,0})$ and $\mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\log f, \mathscr{O}_{X,0})$ and the induced map

(5.13) \begin{align} \mathscr{D}_{X,0}[s] &\otimes_{\mathscr{O}_{X,0}} \mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X,0}^{1}(\log f), \mathscr{O}_{X,0}) \nonumber\\ &\xleftarrow[]{\mathrm{Hom}_{\mathscr{D}_{X,0}[s]}(\theta_{1}, \mathscr{D}_{X,0}[s])} \mathscr{D}_{X,0}[s] \otimes_{\mathscr{O}_{X,0}} \mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X,0}^{2}(\log f), \mathscr{O}_{X,0}) \end{align}

is given, and completely determined, by

(5.14) \begin{equation} \mathrm{Hom}_{\mathscr{D}_{X,0}[s]}(\theta_{1}, \mathscr{D}_{X,0}[s]) ( 1 \otimes u ) = (\mathrm{deg}_{\mathbf{w}}(f)s - \mathrm{deg}_{\mathbf{w}}(u) + E) \otimes u \end{equation}

for all $u \in \Delta_{f}$ .

Proof. Thanks to Proposition 2.20, all we must prove is (5.14). (As $\Delta_{f}$ will generate both Ext-modules, this formula would completely determine the map (5.13).) Take $u \in \Delta_{f}$ viewed inside

\[ \mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X, 0}^{2}(\log f), \mathscr{O}_{X,0}) \simeq H^{1} (\mathrm{Hom}_{\mathscr{O}_{X,0}}(Q_{\bullet}^{2}, \mathscr{O}_{X,0})). \]

Then we can find weighted homogeneous polynomials $p_{t,k}$ such that $u = \overline{\sum_{t,k} p_{t,k} \mathbf{e}_{t}}$ where $\mathbf{e}_{t}$ is the tth indicator vector of $\mathrm{Hom}_{\mathscr{O}_{X,0}}(Q_{1}^{2}, \mathscr{O}_{X})$ and $\overline{(-)}$ denotes the element in the corresponding quotient computing $H^{1} (\mathrm{Hom}_{\mathscr{O}_{X,0}}(Q_{\bullet}^{2}, \mathscr{O}_{X,0}))$ . Then

\begin{align*} \mathrm{Hom}_{\mathscr{D}_{X,0}[s]}(\theta_{1}, \mathscr{D}_{X,0}[s]) ( 1 \otimes u ) &= \mathrm{Hom}_{\mathscr{D}_{X,0}[s]}(\theta_{1}, \mathscr{D}_{X,0}[s]) \biggl( \sum_{t,k} p_{t,k} \otimes \overline{\mathbf{e}_{t}} \biggr) \\ &= \sum_{t,k} \big( p_{t,k}(\mathrm{deg}_{\mathbf{w}}(f)s - b_{t} + E) \big) \otimes \overline{\mathbf{e}_{t}} \\ &= \sum_{t,k} \big( \mathrm{deg}_{\mathbf{w}}(f)s - b_{t} - \mathrm{deg}_{\mathbf{w}}(p_{t,k}) + E \big) p_{t,k} \otimes \overline{\mathbf{e}_{t}} \\ &= (\mathrm{deg}_{\mathbf{w}}(f)s - \mathrm{deg}_{\mathbf{w}}(n) + E) \otimes u. \end{align*}

(The second ‘ $=$ ’ uses Proposition 5.6; the third ‘ $=$ ’ uses $[E, p_{t,k}] = \mathrm{deg}_{\mathbf{w}}(p_{t,k}) p_{t,k}$ ; and the fourth ‘ $=$ ’ uses $\mathrm{deg}_{\mathbf{w}}(n) = \mathrm{deg}_{\mathbf{w}}(p_{t,k}) + \mathrm{deg}_{\mathbf{w}}(\mathbf{e}_{t}) = \mathrm{deg}_{\mathbf{w}}(p_{t,k}) + b_{t}$ for all valid k, where in this parenthetical remark we are thinking of $\mathbf{e}_{t}$ as the indicator vector in the algebraic setting so we can compute weighted degrees.)

6.1 A study in cokernel

We study the cokernel of the map in Proposition 5.7. (As the domain and codomain vanish at other stalks (Proposition 2.20), the description of Proposition 5.7 can be trivially extended to the sheaf level.) Recall this map naturally arises from our specific choice of a nice lift $\theta_{1}$ of $U_{f}^{1} \to U_{f}^{2}$ along our specific free resolutions of $U_{f}^{1}$ and $U_{f}^{2}$ .

Let us bestow a name on our cokernel.

Definition 6.1. For a quasi-homogeneous $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ we denote by $L_{f}$ the left $\mathscr{D}_{X}[s]$ -module that is the cokernel

(6.1) \begin{align} L_{f} = \text{cokernel}: [ \mathrm{Hom}_{\mathscr{D}_{X}[s]}(\theta_{1}, \mathscr{D}_{X}[s]): &\mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{X}^{2}(\log f), \mathscr{O}_{X}) \to \nonumber\\ &\mathscr{D}_{X}[s] \otimes_{\mathscr{O}_{X}} \mathrm{Ext}_{\mathscr{O}_{X}}^{1}(\Omega_{X}^{1}(\log f), \mathscr{O}_{X}) ] \end{align}

Recall that $\theta_{1}$ is as constructed in Proposition 5.4 and that the map appearing in (6.1) was explicitly studied in Propositions 5.6 and 5.7.

We now show that $L_{f,0}$ has a b-function which we are able to compute exactly. Our formula for $B(L_{f,0})$ is often used in later applications.

Proposition 6.3. Let $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$ be quasi-homogeneous with weighted homogeneity $E = \sum w_{i} x_{i} \partial_{i}$ . Then $L_{f}$ vanishes at all stalks $\mathfrak{x} \neq 0$ , and at 0 the stalk $L_{f,0}$ has a b-function. Explicitly,

\[ B(L_{f,0}) = \prod_{t \in \mathrm{deg}_{\mathbf{w}}(\mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R))} \left( s - \frac{t + \sum w_{i}}{\mathrm{deg}_{\mathbf{w}}(f)} \right) \]

and we take $B (L_{f,0}) = 1$ if $\mathrm{deg}_{\mathbf{w}}(\mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R)) = \emptyset$ .

Proof. We may assume that $L_{f,0} \neq 0$ as otherwise there is nothing to prove. (The case $L_{f,0} = 0$ is the ‘and we take’ part of the proposition.) Select a $\mathbb{C}$ -basis $u_{1}, \dots, u_{r}$ of $\mathrm{Ext}_{R}^{1}(\Omega_{R}^{1}(\log f), R)$ of homogeneous elements and regard them as a $\mathbb{C}$ -basis of $\mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X,0}^{1}(\log f), \mathscr{O}_{X,0})$ as we have been entitled to do. Then we may order the $u_{k}$ so that

(6.2) \begin{equation} \mathfrak{m}_{0} \cdot u_{\ell} \in \sum_{k \geq \ell+1} \mathscr{O}_{X,0} \cdot u_{k} \quad \text{and}\quad u_{\ell} \notin \sum_{k \geq \ell+1} \mathscr{O}_{X,0} \cdot u_{k} \text{ for all } u_{\ell}. \end{equation}

By the ordering (6.2), $\mathrm{deg}_{\mathbf{w}}(u_{k}) \leq \mathrm{deg}_{\mathbf{w}}(u_{k+1})$ for all k. Let $m_{1}, \dots, m_{q}$ denote the indices at which $\mathrm{deg}_{\mathbf{w}}(u_{k})$ jumps, i.e.,

(6.3) \begin{equation} \mathrm{deg}_{\mathbf{w}}(u_{k}) \leq \mathrm{deg}_{\mathbf{w}}(u_{m_{\ell}}) \enspace \forall k \leq \ell \quad\text{and}\quad \mathrm{deg}_{\mathbf{w}}(u_{k}) > \mathrm{deg}_{\mathbf{w}}(u_{m_{\ell}}) \enspace \forall k > \ell. \end{equation}

This lets us define a decreasing filtration $L_{f} = N_{0} \supsetneq N_{1} \supsetneq \cdots \supsetneq N_{q} = 0$ of $L_{f}$ where

(6.4) \begin{equation} N_{\ell} = \sum_{k > m_{\ell}} \mathscr{D}_{X,0}[s] \cdot \overline{1 \otimes u_{k}} \subseteq L_{f}. \end{equation}

(We take $m_{0} \ll 0$ so that $N_{0} = L_{f}$ is the initial term of the filtration.) We first make and prove the following claim.

Proof of Claim 1. Fix such an $\ell$ and select $u_{j} \in \{u_{1}, \dots, u_{r}\}$ such that $\mathrm{deg}_{\mathbf{w}}(u_{j}) = m_{\ell}$ . To prove the claim it suffices to prove that

(6.5) \begin{equation} 0 \neq \overline{1 \otimes u_{j}} \in N_{\ell - 1} / N_{\ell}. \end{equation}

If (6.5) were false, then by the definition of $L_{f}$ we could write $1 \otimes u_{\ell} \in \mathscr{D}_{X,0}[s] \otimes_{\mathscr{O}_{X,0}} \mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X,0}^{1}(\log f), \mathscr{O}_{X,0})$ as

(6.6) \begin{align} 1 \otimes u_{\ell} &= \bigg(\sum_{k} A_{k}(\mathrm{deg}_{\mathbf{w}}(f) s - \mathrm{deg}_{\mathbf{w}}(u_{k}) + E)(1 \otimes u_{k}) \bigg) \nonumber\\ &\quad + \bigg( \sum_{\{j \mid \mathrm{deg}_{\mathbf{w}}(u_{j}) > m_{\ell}\}} A_{j}^{\prime} (1 \otimes u_{j}) \bigg) \nonumber \\ &= \bigg( \sum_{\{k \mid \mathrm{deg}_{\mathbf{w}}(u_{k}) \leq m_{\ell}\}} A_{k}(\mathrm{deg}_{\mathbf{w}}(f) s - \mathrm{deg}_{\mathbf{w}}(u_{k}) + E)(1 \otimes u_{k}) \bigg) \nonumber \\ &\quad + \bigg( \sum_{\{j \mid \mathrm{deg}_{\mathbf{w}}(u_{j}) > m_{\ell}\}} B_{j} (1 \otimes u_{j}) \bigg), \end{align}

for suitable $A_{k}, A_{j}^{\prime},$ and $B_{j} \in \mathscr{D}_{X,0}[s]$ . Write $A_{k} = \sum_{a} p_{a}(\partial, s) \alpha_{a}(x)$ where $p_{a} \in \mathbb{C}[\partial, s]$ and $\alpha_{a}(x) \in \mathscr{O}_{X,0}$ ; write $B_{j} = \sum_{b} q_{b}(\partial, s) \beta_{b}(s)$ with the corresponding conventions. Since $\mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X,0}(\log f), \mathscr{O}_{X,0})$ is killed by high enough powers of $\mathfrak{m}_{0}$ we may assume that each $\alpha_{a}(x)$ and each $\beta_{b}(x)$ are actually polynomials as opposed to just convergent power series. To wit, write $A_{k} = \sum_{m} p_{m,k}(\partial, s) g_{m,k}$ where $p_{m,j}(\partial, s) \in \mathbb{C}[\partial, s]$ and $g_{m,j} \in R_{m}$ ; write $B_{j} = \sum_{m} q_{m,k}(\partial_{s}) h_{m,k}$ similarly. (Here $R_{m}$ refers the w-grading of R.) Using that

\begin{align*} g_{m,k}E = E g_{m,k} - [E, g_{m,k}] = E g_{m,k} - m g_{m,k} &= \bigg(\sum_{i} \partial_{i} w_{i} x_{i} - w_{i}\bigg) g_{m,k} - m g_{m,k} \\ &= \bigg( {-} m - \sum_{i} w_{i} + \sum_{i} \partial_{i} w_{i} x_{i} \bigg) g_{m,k} \end{align*}

and that our assumption (6.2) implies that $g_{m,k} u_{k} \in \sum_{v > k} \mathscr{O}_{X,0} \cdot u_{v}$ whenever $m > 0$ (along with a similar statement for $h_{m,j} u_{j}$ ), we deduce that if (6.6) holds then we may write $1 \otimes u_{\ell} \in \mathscr{D}_{X,0}[s] \otimes_{\mathscr{O}_{X,0}} \mathrm{Ext}_{\mathscr{O}_{X,0}}^{1}(\Omega_{X,0}^{1}(\log f), \mathscr{O}_{X,0})$ as

(6.7) \begin{align} 1 \otimes u_{\ell} &= \sum_{\{k \mid \mathrm{deg}_{\mathbf{w}}(u_{k}) \leq m_{\ell}\}} p_{k}(\partial, s) \Bigl(\mathrm{deg}_{\mathbf{w}}(f)s + m_{k} + \sum \partial_{i} w_{i} x_{i} \Bigr) (1 \otimes u_{k}) \nonumber\\ &\quad = \sum_{\{j \mid \mathrm{deg}_{\mathbf{w}}(u_{j}) > m_{\ell}\}} q_{j}(\partial, s) (1 \otimes u_{j}) \end{align}

where $p_{k}(\partial, s), q_{j}(\partial, s) \in \mathbb{C}[\partial,s]$ and $m_{k} \in \mathbb{C}$ for each k.