Convention 2.5. Unless otherwise stated, $R = \mathbb {C}[x_{1}, \dots , x_{n}]$ and is canonically graded with $x_{i}$ given weight one. Denote the irrelevant ideal $R \cdot (x_{1}, \dots , x_{n})$ of R by $\mathfrak {m}$ . In general, we allow R-modules to be $\mathbb {Z}$ -graded. We grade $\Omega _{R}^{j}$ by giving $dx_{i}$ weight one, and we grade $R[x_{I}^{-1}]$ (the localization of R at each $x_{i}$ for $i \in I \subseteq [n]$ ) by giving $x_{i}^{-1}$ weight one; for a graded R-module M, we grade $M[x_{I}^{-1}]$ similarly. We also grade $\Omega _{R}^{j}(\star \mathscr {A})$ and $\Omega _{R}^{j}(\log \mathscr {A})$ by giving $dx_{i}$ weight one and $\frac {1}{f}$ weight $- \deg (f)$ ; we grade $\operatorname {\mathrm {Der}}_{R}(-\log f)$ by giving $\partial _{i}$ weight $-1$ .

Convention 2.6. In this subsection, $\mathscr {A}$ is always assumed to be central with a homogeneous defining equation f.

Proof. Let $Q_{I} \in R[x_{I}^{-1}]$ and $\eta _{I} \in \Omega _{R}^{j+1}(\log \mathscr {A})$ so that $Q_{I} \eta _{I} \in \Omega _{R}^{j+1}(\log \mathscr {A})[x_{I}^{-1}]$ . Then

$$ \begin{align*} \epsilon(\iota_{E}(Q_{I} \eta_{I})) = \epsilon(Q_{I}( \iota_{E}(\eta_{I}))) &= \sum_{\ell \notin I} (-1)^{\sigma_{I}(\ell)} Q_{I \cup \{\ell\}} \iota_{E}(\eta_{I}) \\ &= \iota_{E}(\sum_{\ell \notin I} (-1)^{\sigma_{I}(\ell)} Q_{I \cup \{\ell\}} \eta_{I}) = \iota_{E}(\epsilon (Q_{I}\eta_{I})).\\[-47pt] \end{align*} $$

Proof. That the squares (2.6) from Proposition 2.9 commute amounts to saying $\nabla _{\omega }$ gives a chain map from $N^{j, \bullet } \to N^{j+1, \bullet }$ for all j. So the induced maps on cohomology

$$\begin{align*}H_{\mathfrak{m}}^{t}(\Omega_{R}^{j}(\log \mathscr{A})) \xrightarrow[]{\nabla_{\omega}} H_{\mathfrak{m}}^{t}(\Omega_{R}^{j+1}(\log \mathscr{A})) \end{align*}$$

are well-defined. Since $\nabla _{\omega }^{2} = 0$ , certainly $(H_{\mathfrak {m}}^{t}(\Omega _{R}^{\bullet }(\log \mathscr {A})), \nabla _{\omega })$ is a complex.

The case of $(H_{\mathfrak {m}}^{t}(\Omega _{R}^{\bullet }(\log \mathscr {A})), \iota _{E})$ is similar, now using Proposition 2.11.

Proof. We first do the case of logarithmic forms. The untwisted statement is due to Naruki according to Lemma 2.1 of [Reference Castro-Jiménez, Narváez-Macarro and Mond9] and is a straightforward computation. Since contraction and exterior differentiation are additive, we need only to confirm that

$$ \begin{align*} \omega \wedge \iota_{E}(\eta) + \iota_{E}(\omega \wedge \eta) &= \omega \wedge \iota_{E}(\eta) + \iota_{E}(\omega) \wedge \eta - \omega \wedge \iota_{E}(\eta) \\ &= \iota_{E}(\omega) \eta = (\sum \lambda_{k} \iota_{E}(d \log f_{k}) )\eta = (\sum \lambda_{k}) \eta. \end{align*} $$

This gives Equation (2.10). As for Equation (2.11), first note that we have confirmed $\iota _{E}$ and $\nabla _{\omega }$ are well-defined maps on local cohomology and are defined by taking representatives. So the second formula follows from the first because: Naruki’s computation for the untwisted case extends to $\Omega _{R}^{j}(\log \mathscr {A})[x_{I}^{-1}]$ ; we can compute the twisted Lie derivative on a homogeneous representative of our homogeneous class of $H_{\mathfrak {m}}^{t}(\Omega _{R}^{j}(\log \mathscr {A}))$ .

Convention 2.14. A quick computation verifies $\iota _{E}(\omega ) = \lambda _{1} + \cdots + \lambda _{d}$ . We use $\iota _{E}(\omega )$ throughout the paper instead of writing this sum.

Proof. Clearly, d is degree preserving map. Since $\omega = \sum \lambda _{k} \frac {df_{k}}{f_{k}}$ is homogeneous of grade zero, $\nabla _{\omega }$ is also a degree-preserving map. That all these complexes decompose into direct sums of their homogeneous parts follows.

As for the quasi-isomorphisms, we first do the case of the logarithmic de Rham complex. Thanks to the direct sum decomposition (which certainly extends to cohomology), it is enough to prove that, for $q \neq - \iota _{E}(\omega )$ , the degree q subcomplex $(\Omega _{R}^{\bullet }(\log \mathscr {A})_{q}, \nabla _{\omega })$ is acyclic. Thanks to Equation (2.10) from Lemma 2.13, and the fact $\iota _{E}$ is itself degree preserving, $\iota _{E}$ defines a chain homotopy between

$$\begin{align*}(\Omega_{R}^{\bullet}(\log \mathscr{A})_{q}, \nabla_{\omega}) \xrightarrow{(q + \iota_{E}(\omega)) \cdot} (\Omega_{R}^{\bullet}(\log \mathscr{A})_{q}, \nabla_{\omega}) \end{align*}$$

and the zero map

$$\begin{align*}(\Omega_{R}^{\bullet}(\log \mathscr{A})_{q}, \nabla_{\omega}) \xrightarrow[]{0 \cdot} (\Omega_{R}^{\bullet}(\log \mathscr{A})_{q}, \nabla_{\omega}). \end{align*}$$

Since $q + \iota _{E}(\omega ) \neq 0$ , the first map is a quasi-isomorphism and so $(\Omega _{R}^{\bullet }(\log \mathscr {A})_{q}, \nabla _{\omega })$ is acyclic. The result for the logarithmic de Rham complex follows.

As for the local cohomology complexes, the same argument works: $\iota _{E}$ defines a chain homotopy between similar maps, now using Equation (2.11) from Lemma 2.13.

Proof. It is enough to verify the claim for $\phi (\eta _{I})$ . We first show that $\phi (\eta _{I}) \in \frac {1}{f}\Gamma (D(x_{I}), \Omega _{X}^{j})$ , which is essentially a statement about the appropriate Laurent series converging absolutely. By Remark 2.16 and the absolute convergence of Laurent expansions, the $-\iota _{E}(\omega )$ -homogeneous piece $\eta _{I, -\iota _{E}(\omega )}$ is itself in $\frac {1}{f}\Gamma (D(x_{I}), \Omega _{X}^{j})$ . And $\phi (\eta _{I}) = \phi (\eta _{I} - \eta _{I,-\iota _{E}(\omega )})$ . To test absolute convergence, we apply the modulus to an evaluation of the formal Laurent series $\phi (\eta _{I} - \eta _{I,-\iota _{E}(\omega )})$ at some point. Under $\phi $ , every (nonzero) homogeneous $\eta _{I, p}$ of $\eta _{I} - \eta _{I, -\iota _{E}(\omega )}$ gets sent to $(1/p+\iota _{E}(\omega )) \eta _{I,p}$ . After applying the modulus, and using the fact that $|1/(p+\iota _{E}(\omega )| \to 0$ as $|p| \to \infty $ , we see that each $\mid (1/p+\iota _{E}(\omega )) \eta _{I,p}\mid $ is smaller than $\mid \eta _{I,p}\mid $ , provided p is sufficiently far from $0$ . Since finite sums of convergent series converge, $\phi (\eta _{I})$ does live in $\frac {1}{f}\Gamma (D(x_{I}), \Omega _{X}^{j})$ .

Now, we confirm that $\phi (\eta _{I}) \in \Gamma (D(x_{I}), \Omega _{X}^{j}(\log f))$ by showing $df/f \wedge \phi (\eta _{I}) \in \frac {1}{f} \Gamma (D(x_{I}), \Omega _{X}^{j+1}).$ Using Remark 2.16 again, $\eta _{I, -\iota _{E}(\omega )}$ is logarithmic, and so $\eta _{I} - \eta _{I, -\iota _{E}(\omega )}$ is also logarithmic. Consequently, $df/f \wedge (\eta _{I} - \eta _{I, -\iota _{E}(\omega )}) \in \frac {1}{f} \Gamma (D(x_{I}, \Omega _{X}^{j+1}).$ But because $df/f$ is homogeneous of weight zero, as formal Laurent series we have the equalities

$$\begin{align*}df/f \wedge \phi(\eta_{I}) = df/f \wedge \phi(\eta_{I} - \eta_{I, -\iota_{E}(\omega)}) = \phi(df/f \wedge (\eta_{I} - \eta_{I, -\iota_{E}(\omega)})). \end{align*}$$

And so the same argument about absolute convergence from the first paragraph, now applied to $\phi (df/f \wedge (\eta _{I} - \eta _{I, -\iota _{E}(\omega )}))$ , shows $df/f \wedge \phi (\eta _{I}) \in \frac {1}{f} \Gamma (D(x_{I}), \Omega _{X}^{j+1})$ as required.

Proof. Suppose $\eta = \bigoplus _{I} \eta _{I} \in \bigoplus _{I} \Gamma (D(x_{I}), \Omega _{X}^{j}(\log f))$ is a t-cycle (so $|I| = t)$ . This means that for each $M \subseteq [n]$ of cardinality $t+1$ we have, in $\Gamma (D(x_{M}), \Omega _{X}^{j}(\log f))$ , the equalities

$$ \begin{align*} \sum_{m \in M} (-1)^{\sigma_{M \setminus \{m\}}(m)} \eta_{M \setminus \{m\}} &= \sum_{m \in M} (-1)^{\sigma_{M \setminus \{m\}}(m)} \sum_{p} \eta_{M \setminus \{m\}, p} \\ & = \sum_{p} \sum_{m \in M} (-1)^{\sigma_{M \setminus \{m\}}(m)} \eta_{M \setminus \{m\}, p} \\ & = 0. \end{align*} $$

(Here, $\eta _{M \setminus \{m\}}$ is naturally regarded in $\Gamma (D(x_{M}),\Omega _{X}^{j}(\log f))$ by restriction.) We see that, for each individual p,

(2.15) $$ \begin{align} \sum_{m \in M} (-1)^{\sigma_{M \setminus \{m\}}(m)} \eta_{M \setminus \{m\}, p} = 0. \end{align} $$

Moreover, the M-component of $\epsilon (\phi (\eta ))$ is

$$ \begin{align*} \sum_{m \in M} (-1)^{\sigma_{M \setminus \{m\}}(m)} \phi(\eta_{M \setminus \{m\}}) &= \sum_{m \in M} (-1)^{\sigma_{M \setminus \{m\}}(m)} \sum_{ p \neq - \iota_{E}(\omega)}\frac{\eta_{M \setminus \{m\}}}{p + \iota_{E}(\omega)} \\ &= \sum_{p \neq - \iota_{E}(\omega)} \sum_{m \in M} (-1)^{\sigma_{M \setminus \{m\}}(m)} \frac{\eta_{M \setminus \{m\}, p}}{p + \iota_{E}(\omega)}. \end{align*} $$

By Equation (2.15), we conclude the above equation equals zero and $\epsilon (\phi (\eta )) = 0$ as well.

The case of boundaries is similar. If $\eta $ is a t-boundary, that means there is a $\zeta $ such that $\epsilon (\zeta ) = \eta $ . Arguing as above, the I-component of $\epsilon (\zeta )$ and $\eta $ agree at each homogeneous piece: $\epsilon (\zeta )_{I, p} = \eta _{I,p}$ for all I of cardinality t and all $p \in \mathbb {Z}$ . As $\phi (\zeta )$ and $\phi (\eta )$ both scale each homogeneous term in a fixed way, $\epsilon (\phi (\zeta )) = \phi (\eta )$ .

Proof. Consider

$$\begin{align*}[\bigoplus_{I} \eta_{I}] \in \ker [H^{t}(C(\{D(x_{i})\}, \Omega_{X}^{j}(\log f))) \xrightarrow[]{\nabla_{\omega}} H^{t}(C(\{D(x_{i})\}, \Omega_{X}^{j}(\log f)))]. \end{align*}$$

We apply $L_{\omega } = \nabla _{\omega }(\iota _{E}) + \iota _{E}(\nabla _{\omega })$ , the twisted Lie derivative, to $[\bigoplus _{I} \eta _{I}] = [\bigoplus _{I} \sum _{p} \eta _{I,p}]$ . Since this is done by applying $L_{\omega }$ to a representative and since we may do so term-by-term thanks to absolute convergence, we may use Lemma 2.13 to compute

$$\begin{align*}[\bigoplus_{I} \sum_{p \in \mathbb{Z}} (p + \iota_{E}(\omega))\eta_{I,p}] = L_{\omega}([\bigoplus_{I} \eta_{I}]) = \nabla_{\omega}(\iota_{E}([\bigoplus_{I} \eta_{I}])) = \nabla_{\omega}([\bigoplus_{I} \iota_{E}(\eta_{I})]). \end{align*}$$

That is, $\bigoplus _{I} \sum _{p \in \mathbb {Z}} (p + \iota _{E}(\omega ))\eta _{I,p} - \nabla _{\omega }(\bigoplus _{I} \iota _{E}(\eta _{I}))$ is a Čech t-boundary. By Proposition 2.21, if we apply $\phi $ to this Čech t-boundary we get another Čech t-boundary. Since $\nabla _{\omega }$ preserves weights, $\phi $ and $\nabla _{\omega }$ commute. So applying $\phi $ to our Čech t-boundary gives the Čech t-boundary

$$\begin{align*}\bigoplus_{I} \sum_{p \neq - \iota_{E}(\omega)} \eta_{I,p} - \nabla_{\omega}(\bigoplus_{I} \phi(\iota_{E}(\eta_{I}))). \end{align*}$$

With some mild rearranging, we have an equality of Čech cohomology classes

$$\begin{align*}[\bigoplus_{I} \sum_{p \in \mathbb{Z}} \eta_{I,p} - \bigoplus_{I} \eta_{I, -\iota_{E}(\omega)}] = [\nabla_{\omega}(\bigoplus_{I} \phi(\iota_{E}(\eta_{I})))]. \end{align*}$$

(Here, we use the fact that $\bigoplus _{I} \sum _{p \in \mathbb {Z}} \eta _{I,p}$ was assumed to be a Čech t-cycle, and so, because the Čech complex preserves weight, each of its homogeneous parts is also a Čech t-cycle.)

This means that we have the following equality of cohomology classes in $H^{j}(H_{\operatorname {\mathrm {\check {C}ech}}}^{t} (\Omega _{X}^{\bullet }(\log \mathscr {A})), \nabla _{\omega })$ :

(2.16) $$ \begin{align} [[\bigoplus_{I} \sum_{p \in \mathbb{Z}} \eta_{I,p}]] = [[\bigoplus_{I} \eta_{I, -\iota_{E}(\omega)}]]. \end{align} $$

Since the map (2.14) explicitly obeys

$$\begin{align*}H_{\operatorname{\mathrm{\check{C}ech}}}^{t}(\Omega_{X}^{j}(\log \mathscr{A})_{-\iota_{E}(\omega)}) \ni [\bigoplus_{I} \eta_{I, -\iota_{E}(\omega)}] \mapsto [\bigoplus_{I} \eta_{I, -\iota_{E}(\omega)}] \in H_{\operatorname{\mathrm{\check{C}ech}}}^{t}(\Omega_{X}^{j}(\log \mathscr{A})), \end{align*}$$

we have, thanks to Equation (2.16), the induced identification

$$ \begin{align*} H^{j}(H_{\operatorname{\mathrm{\check{C}ech}}}^{t}(\Omega_{X}^{\bullet}(\log \mathscr{A})_{-\iota_{E}(\omega)}), \nabla_{\omega}) & \ni [[\bigoplus_{I} \eta_{I,-\iota_{E}(\omega)}]] \\ &\mapsto [[\bigoplus_{I} \eta_{I}]] \in H^{j}(H_{\operatorname{\mathrm{\check{C}ech}}}^{t}(\Omega_{X}^{\bullet}(\log \mathscr{A})), \nabla_{\omega}). \end{align*} $$

The proves the desired cohomological surjections.

Proof. We just do part involving the twisted logarithmic de Rham complexes, as the meromorphic case is similar. Here, we consider $\phi $ as a map from $\Omega _{X,0}^{j}(\log \mathscr {A}) \xrightarrow[]{} \Omega _{X,0}^{j}(\log \mathscr {A})$ with an entirely similar definition as before. Let $\eta $ be a logarithmic j-form that is a $\nabla _{\omega }$ -cycle. One one hand, $L_{\omega }(\eta ) = \nabla _{\omega }(\sum _{p} \eta _{p}) = \sum _{p} (p + \iota _{E}(\omega )) \eta _{p}$ ; on the other hand, $L_{\omega }(\eta ) = \nabla _{\omega }(\iota _{E}(\eta ))$ . Then $\nabla _{\omega }(\phi (\iota _{E}(\eta ))) = \eta - \eta _{-\iota _{E}(\omega )}$ . So the class $[\eta ]$ equals the class $[\eta _{-\iota _{E}(\omega )}]$ .

Convention 2.25. We use the suffix $-(m)$ , with $m \in \mathbb {Z}$ , to denote a twist in the grading. Explicitly, the u-homogeneous part of $R(m)$ is: $R(m)_{u} = R_{u + m}$ . If an isomorphism is written as $M \simeq R(m)$ , then the isomorphism honors the grading. Another convention: we use just $\operatorname {\mathrm {reg}}(M)$ without specifying the ring in question, hoping context rectifies any confusion.

Proof. Throughout, let $f = f_{1} \cdots f_{d}$ be a defining equation for $\mathscr {A}$ and, if not explicitly changed, $n = \operatorname {\mathrm {rank}}(\mathscr {A})$ , $d = \deg (\mathscr {A})$ .

Step 0: We handle the straightforward cases $j=0$ or $j=\operatorname {\mathrm {rank}}(\mathscr {A})$ . All we must show is freeness and then compute regularity. It is well known (and easy to verify using Equation (2.17) below) that $\Omega _{R}^{0}(\log \mathscr {A}) = R$ . Consequently, $\operatorname {\mathrm {reg}}(\Omega _{R}^{0}(\log \mathscr {A})) = 0.$ Now, consider $j=\operatorname {\mathrm {rank}}(\mathscr {A})$ . If $\eta \in \frac {1}{f} \Omega _{R}^{\operatorname {\mathrm {rank}}(\mathscr {A})}$ , then $d(\eta ) = 0 \in \frac {1}{f}\Omega _{R}^{\operatorname {\mathrm {rank}}(\mathscr {A})+1} = 0$ . So $\Omega _{R}^{\operatorname {\mathrm {rank}}(\mathscr {A})}(\log \mathscr {A}) = \frac {1}{f} \Omega _{R}^{\operatorname {\mathrm {rank}}(\mathscr {A})} = R(\deg (\mathscr {A}) - \operatorname {\mathrm {rank}}(\mathscr {A}))$ ; $\operatorname {\mathrm {reg}}(\Omega _{R}^{\operatorname {\mathrm {rank}}(\mathscr {A})}(\log \mathscr {A})) = \operatorname {\mathrm {rank}}(\mathscr {A}) - \deg (\mathscr {A})$ .

Step 1: To deal with $1 \leq j \leq \operatorname {\mathrm {rank}}(\mathscr {A}) - 1$ , we change our target in order to apply Corollary 3.7 of [Reference Derksen and Sidman14] (transcribed as Corollary 2.27). Observe: The logarithmic j-forms can be characterized by $\sum _{\mid I \mid = j} \frac {a_{I}}{f} dx_{I} \in \Omega _{R}^{j}(\log \mathscr {A})$ if and only if $d(\sum _{\mid I \mid = j} \frac {a_{I}}{f} dx_{I}) \in \frac {1}{f} \Omega _{R}^{j+1}$ if and only if $\sum _{\mid I \mid = j} \frac {a_{I}}{f^{2}} d(f) \wedge dx_{I} \in \frac {1}{f} \cdot \Omega _{R}^{j+1}$ if and only if $\sum _{\mid I \mid = j} \frac {a_{I}}{f} d(f) \wedge dx_{I} \in \Omega _{R}^{j+1}$ if and only if $\sum _{\mid I \mid = j} a_{I} d(f) \wedge dx_{I} \in f \cdot \Omega _{R}^{j+1}$ . Since R is a UFD and $f = f_{1} \cdots f_{d}$ is a factorization into irreducibles, we conclude:

(2.17) $$ \begin{align} \sum_{\mid I \mid = j} \frac{a_{I}}{f} dx_{I} \in \Omega_{R}^{j}(\log \mathscr{A}) \iff \sum_{\mid I \mid = j} a_{I} d(f_{k}) \wedge dx_{I} \in f_{k} \cdot \Omega_{R}^{j+1} \text{ for all } f_{k}. \end{align} $$

Define

$$\begin{align*}L^{j}(\mathscr{A}) = \{ \eta \in \Omega_{R}^{j} \mid d(f_{k}) \wedge \eta \in f_{k} \cdot \Omega_{R}^{j+1} \text{ for all } f_{k}\}, \end{align*}$$

and note that

(2.18) $$ \begin{align} L^{j}(\mathscr{A}) = \bigcap_{k} L^{j}(\operatorname{\mathrm{V}}(f_{k})). \end{align} $$

The equivalence (2.17) tells us the map $\sum _{\mid I \mid = j} \frac {a_{I}}{f} dx_{I} \mapsto \sum _{\mid I \mid = j} a_{I} dx_{I}$ gives an isomorphism $\Omega _{R}^{j}(\log \mathscr {A}) \xrightarrow[]{\simeq } L^{j}(\mathscr {A})$ that sends homogeneous elements of degree u to homogeneous elements of degree $u + \deg (\mathscr {A})$ . Therefore,

$$\begin{align*}\Omega_{R}^{j}(\log \mathscr{A})(-\deg(\mathscr{A})) \simeq L^{j}(\mathscr{A}). \end{align*}$$

Consequently, $H_{\mathfrak {m}}^{t}(\Omega _{R}^{j}(\log \mathscr {A}))(-\deg (\mathscr {A})) \simeq H_{\mathfrak {m}}^{t}(L^{j}(\mathscr {A}))$ and

(2.19) $$ \begin{align} \operatorname{\mathrm{reg}}(\Omega_{R}^{j}(\log \mathscr{A})) = \operatorname{\mathrm{reg}}(L^{j}(\mathscr{A})) - d. \end{align} $$

Therefore, our theorem is equivalent to $\operatorname {\mathrm {reg}}(L^{j}(\mathscr {A})) \leq \deg (\mathscr {A})$ , making the following our new objective:

$$\begin{align*}\textbf{The Claim}: \operatorname{\mathrm{reg}}(L^{j}(\mathscr{A})) \leq \deg(\mathscr{A}) \text{ for } 1 \leq j \leq \operatorname{\mathrm{rank}}(\mathscr{A}) - 1. \end{align*}$$

Step 2: We first prove The Claim for $L^{1}(\mathscr {A})$ by induction on the rank and the degree of our arrangement. If $\operatorname {\mathrm {rank}}(\mathscr {A}) = 2$ with $d = \deg (\mathscr {A})$ , then the arrangement is automatically free and by standard considerations, $\Omega _{R}^{1}(\log \mathscr {A}) \simeq R \delta _{1} \oplus R \delta _{2}$ , where $\delta _{1}$ has degree $2-d$ and $\delta _{2}$ has degree $0$ . (Note that the determinant of the Saito matrix corresponding to $\delta _{1}, \delta _{2}$ equals, up to unit, $\frac {1}{f}dx_{1}dx_{2}$ , and one logarithmic form, $\delta _{1}$ , corresponds to the Euler operator, whereas the other, $\delta _{2}$ , corresponds to $(\partial _{1} \bullet f)\partial _{2} - (\partial _{2} \bullet f)\partial _{1}$ .) So $\Omega _{R}^{1}(\log \mathscr {A}) \simeq R(d-2) \oplus R$ and $\operatorname {\mathrm {reg}}( \Omega _{R}^{1}(\log \mathscr {A})) = \max \{ 2-d, 0 \} = 0$ since, by essentiality, $d \geq 2$ . So $\operatorname {\mathrm {reg}}(L^{1}(\mathscr {A})) = d$ .

Assume the rank of $\mathscr {A}$ is at least $3$ , the degree of $\mathscr {A}$ is $d+1$ and the claim holds for all essential, central arrangements of smaller degree or rank. (The base case of a simple normal crossing divisor with rank( $\mathscr {A}$ ) hyperplanes is handled in Example 2.28.) By essentiality, we may find n hyperplanes from $\mathscr {A}$ cutting out the origin.

Case 2.1: There exists n hyperplanes $f_{1}, \dots , f_{n}$ (relabelling as necessary) from $\mathscr {A}$ such that: (1) $\mathfrak {m} = (f_{1}, \dots , f_{n})$ ; (2) deleting any $\operatorname {\mathrm {V}}(f_{i})$ from $\mathscr {A}$ produces an essential arrangement. Let $\mathscr {A}_{i} = \mathscr {A} \setminus \operatorname {\mathrm {V}}(f_{i})$ be the result post deletion. By Equation (2.18), $L^{1}(\mathscr {A}) \subseteq L^{1}(\mathscr {A}_{i}).$ And using Equations (2.17) and (2.18), we confirm that $f_{i} \cdot L^{1}(\mathscr {A}_{i}) \subseteq L^{1}(\mathscr {A})$ . So we use the inductive hypothesis to get $\operatorname {\mathrm {reg}}(L^{1}(\mathscr {A}_{i})) \leq d$ and $\operatorname {\mathrm {reg}}(L^{1}(\mathscr {A}_{i})(1)) \leq d -1.$ As $L^{1}(\mathscr {A}) \subseteq R dx_{1} \oplus \cdots \oplus R dx_{n} = R(-1) \oplus \cdots \oplus R(-1)$ , if we shift degrees and consider $L^{1}(\mathscr {A})(1)$ , $L^{1}(\mathscr {A}_{i})(1) \subseteq R(0) \oplus \cdots \oplus R(0)$ we may invoke Corollary 3.7 of [Reference Derksen and Sidman14] (see Corollary 2.27) to get $\operatorname {\mathrm {reg}}(L^{1}(\mathscr {A})(1)) \leq d$ or equivalently $\operatorname {\mathrm {reg}}(L^{1}(\mathscr {A})) \leq d + 1$ , finishing the case.

Case 2.2: There exists a hyperplane $f_{i}$ such that the deleted arrangement $\mathscr {A} \setminus \operatorname {\mathrm {V}}(f_{i})$ is nonessential. Then, after possibly changing coordinates, $\mathscr {A}$ has a defining equation $x_{n} g$ , where $\operatorname {\mathrm {V}}(g) = \mathscr {A}_{i}$ , $g \in S = \mathbb {C}[x_{1}, \dots , x_{n-1}]$ and $x_{n}$ corresponds to the deleted hyperplane $f_{i}$ . By the Kunneth formula for products, see Remark 2.2,

$$\begin{align*}\Omega_{R}^{1}(\log \mathscr{A}) = (\Omega_{S}^{1}(\log \operatorname{\mathrm{V}}(g)) \otimes_{\mathbb{C}} \mathbb{C}[x_{n}]) \oplus (S \otimes_{\mathbb{C}} \Omega_{\mathbb{C}[x_{n}]}^{1}(\log \operatorname{\mathrm{V}}(x_{n}))) \end{align*}$$

and by the Kunneth formula for local cohomology and regularity; see Remark 2.26,

$$\begin{align*}\operatorname{\mathrm{reg}}(\Omega_{R}^{1}(\log \mathscr{A})) = \max\{ \operatorname{\mathrm{reg}}(\Omega_{S}^{1}(\operatorname{\mathrm{V}}(g))) + \operatorname{\mathrm{reg}}(\mathbb{C}[x_{n}]), \operatorname{\mathrm{reg}}(S) + \operatorname{\mathrm{reg}}(\Omega_{\mathbb{C}[x_{n}]}^{1}(\operatorname{\mathrm{V}}(x_{n}))\}. \end{align*}$$

Because $\Omega _{\mathbb {C}[x_{n}]}^{1}(\operatorname {\mathrm {V}}(x_{n})) = \mathbb {C}[x_{n}]\frac {dx_{n}}{x_{n}} \simeq \mathbb {C}[x_{n}]$ (a graded isomorphism), the only regularity term of the four summands that is potentially nonzero is $\operatorname {\mathrm {reg}}(\Omega _{S}^{1}(\operatorname {\mathrm {V}}(g)))$ . So $\operatorname {\mathrm {reg}}(\Omega _{R}^{1}(\log \mathscr {A})) \leq \max \{\operatorname {\mathrm {reg}}(\Omega _{S}^{1}(\operatorname {\mathrm {V}}(g))), 0\}$ . Since $\mathscr {A}$ is essential of rank at least three, $\operatorname {\mathrm {V}}(g)$ must be essential of rank at least two (and of one less degree), and so we can use the induction hypothesis: $\operatorname {\mathrm {reg}}(L^{1}(\operatorname {\mathrm {V}}(g)) \leq d$ and, by Equation (2.19), $\operatorname {\mathrm {reg}}(\Omega _{S}^{1}(\operatorname {\mathrm {V}}(g))) \leq 0$ . Thus, $\operatorname {\mathrm {reg}}(\Omega _{R}^{1}(\operatorname {\mathrm {V}}(f)) \leq 0$ and $\operatorname {\mathrm {reg}}(L^{1}(\mathscr {A})) = d+1$ , completing this case and terminating the inductive step.

Step 3: Now, we prove The Claim holds for $L^{j}(\mathscr {A})$ , for any $1 \leq j \leq \operatorname {\mathrm {rank}}(\mathscr {A}) - 1$ , by induction on $\deg (\mathscr {A})$ and $\operatorname {\mathrm {rank}}(\mathscr {A})$ , utilizing Step 2. In particular, we may assume $\operatorname {\mathrm {rank}} (\mathscr {A}) \geq 3$ and $j \geq 2$ . The minimal value $\deg \mathscr {A}$ can obtain while remaining essential is $\operatorname {\mathrm {rank}}(\mathscr {A})$ in which case Example 2.28 applies. So assume that the claim holds for any central, essential $\mathscr {B}$ whose degree is less than $\deg (\mathscr {A})$ or whose rank is smaller than $\operatorname {\mathrm {rank}}(\mathscr {A})$ . Now, we do three cases, the first two similar to those from Step 2.

Case 3.1: There exist n hyperplanes $f_{1}, \dots , f_{n}$ (relabelling as necessary) from $\mathscr {A}$ such that: (1) these n hyperplanes generate $\mathfrak {m}$ ; (2) the arrangement $\mathscr {A}_{i} = \mathscr {A} \setminus \operatorname {\mathrm {V}}(f_{i})$ is essential. Since each $\mathscr {A}_{i}$ is promised to be essential, $\operatorname {\mathrm {rank}}(\mathscr {A}_{i}) = \operatorname {\mathrm {rank}}(\mathscr {A})$ and so the $\mathscr {A}_{i}$ fall into our inductive scheme: $\deg (\mathscr {A}_{i}) = \deg (\mathscr {A}) - 1$ and $j \leq \operatorname {\mathrm {rank}}(\mathscr {A}) - 1 = \operatorname {\mathrm {rank}}(\mathscr {A}_{i}) - 1$ . So we know that $\operatorname {\mathrm {reg}}(L^{j}(\mathscr {A}_{i})) \leq \deg (\mathscr {A}_{i}) = \deg (\mathscr {A}) - 1$ . Or, as we will prefer, $\operatorname {\mathrm {reg}}(L^{j}(\mathscr {A}_{i})(j)) \leq \deg (\mathscr {A}) - 1 - j.$ Using Equations (2.17) and (2.18), we again have

$$\begin{align*}f_{i} \cdot L^{j}(\mathscr{A}_{i}) \subseteq L^{j}(\mathscr{A}) \subseteq L^{j}(\mathscr{A}_{i}). \end{align*}$$

Now, $L^{j}(\mathscr {A}) \subseteq \oplus _{\mid I \mid = j} R dx_{I} = \oplus _{\mid I \mid = j} R(-j).$ Shift degrees as in Case 2.1 so that $L^{j}(\mathscr {A})(j)$ , $L^{j}(\mathscr {A}_{i})(j) \subseteq \oplus _{\mid I \mid = j} R$ . Now, we can appeal to Corollary 3.7 of [Reference Derksen and Sidman14] (see Corollary 2.27) to find that $\operatorname {\mathrm {reg}}(L^{j}(\mathscr {A})(j)) \leq \deg (\mathscr {A}) - j$ , that is $\operatorname {\mathrm {reg}}(L^{j}(\mathscr {A})) \leq \deg (\mathscr {A})$ , as required.

Case 3.2: Both: (1) there is a hyperplane $f_{i}$ , where $\operatorname {\mathrm {rank}}(\mathscr {A} \setminus \operatorname {\mathrm {V}}(f_{i})) < \operatorname {\mathrm {rank}}(\mathscr {A})$ ; (2) $j < \operatorname {\mathrm {rank}}(\mathscr {A}) - 1 = n-1$ . Here, we may, changing coordinates as needed, assume $f = x_{n}g$ where $g \in S = \mathbb {C}[x_{1}, \dots , x_{n-1}]$ . Since $\mathscr {A}$ is essential, $\operatorname {\mathrm {V}}(g)$ is itself an essential arrangement whose rank is $\operatorname {\mathrm {rank}}(\mathscr {A}) - 1$ . And $2 \leq j \leq \operatorname {\mathrm {rank}}(\mathscr {A}) - 2 = \operatorname {\mathrm {rank}}(\operatorname {\mathrm {V}}(g)) - 1$ . So $\operatorname {\mathrm {V}}(g)$ falls into the inductive setup and $\operatorname {\mathrm {reg}}(L^{q}(\operatorname {\mathrm {V}}(g))) \leq \deg (\operatorname {\mathrm {V}}(g)) = \deg (\mathscr {A}) - 1$ for all $1 \leq q \leq j$ . We may use the Kunneth formula for products (Remark 2.2) to find

$$\begin{align*}\Omega_{R}^{j}(\log \mathscr{A}) \simeq \Omega_{S}^{j}(\operatorname{\mathrm{V}}(g)) \otimes_{\mathbb{C}} \mathbb{C}[x_{n}] \oplus \Omega_{S}^{j-1}(\operatorname{\mathrm{V}}(g)) \otimes_{\mathbb{C}} \Omega_{\mathbb{C}[x_{n}]}^{1}(\operatorname{\mathrm{V}}(x_{n})) \end{align*}$$

and, using the fact $\Omega _{\mathbb {C}[x_{n}]}^{1}(\operatorname {\mathrm {V}}(x_{n})) \simeq \mathbb {C}[x_{n}]$ (graded), the induced Kunneth formula on regularity (Remark 2.26) gives

$$\begin{align*}\operatorname{\mathrm{reg}}(\Omega_{R}^{j}(\log \mathscr{A})) = \max\{ \operatorname{\mathrm{reg}}(\Omega_{S}^{j}(\operatorname{\mathrm{V}}(g))) + \operatorname{\mathrm{reg}}(\mathbb{C}[x_{n}]), \operatorname{\mathrm{reg}}(\Omega_{S}^{j-1}(\operatorname{\mathrm{V}}(g))) + \operatorname{\mathrm{reg}}(\mathbb{C}[x_{n}])\}. \end{align*}$$

By the inductive setup, $\operatorname {\mathrm {reg}}(\Omega _{S}^{j}(\operatorname {\mathrm {V}}(g))) \leq 0$ and $\operatorname {\mathrm {reg}}(\Omega _{S}^{j-1}(\operatorname {\mathrm {V}}(g))) \leq 0$ . So $\operatorname {\mathrm {reg}}(\Omega _{R}^{j}(\log \mathscr {A})) \leq 0$ and $\operatorname {\mathrm {reg}}(L^{j}(\mathscr {A})) \leq \deg (\mathscr {A})$ .

Case 3.3: j = $\operatorname {\mathrm {rank}}(\mathscr {A}) -1 = n-1.$ Write $d = \deg (\mathscr {A})$ . Here, with $\widehat {dx_{i}} = dx_{1} \wedge \cdots \wedge dx_{i-1} \wedge dx_{i+1} \wedge dx_{n}$ and $dx = dx_{1} \wedge \cdots \wedge dx_{n}$ , the equivalence (2.17) is enriched:

$$ \begin{align*} \sum_{i} \frac{a_{i}}{f} \widehat{dx_{i}} \in \Omega_{R}^{n-1}(\log \mathscr{A}) &\iff \sum_{i} a_{i} d(f) \widehat{dx_{i}} = (\sum_{i} (-1)^{i-1} a_{i} \partial_{i} \bullet f)) dx \in R \cdot f dx \\ & \iff (\sum_{i} (-1)^{i-1} a_{i} \partial_{i}) \bullet f \in R \cdot f \\ & \iff (\sum_{i} (-1)^{i-1} a_{i} \partial_{i}) \in \operatorname{\mathrm{Der}}_{R}(-\log \mathscr{A}). \end{align*} $$

In other words, $\Omega _{R}^{n-1}(\log \mathscr {A}) \ni \sum _{i} a_{i} \widehat {dx_{i}} \mapsto (\sum _{i} (-1)^{i-1} a_{i} \partial _{i}) \in \operatorname {\mathrm {Der}}_{R}(-\log \mathscr {A})$ is an isomorphism sending homogeneous elements of degree u to homogeneous elements of degree $u + d - n$ . (We use the convention that $\partial _{i}$ has degree $-1$ as in [Reference Saito30].) Therefore,

$$\begin{align*}\Omega_{R}^{n-1}(\log \mathscr{A})(n - d) \simeq \operatorname{\mathrm{Der}}_{R}(\log \mathscr{A}) \end{align*}$$

and $\operatorname {\mathrm {reg}}(\Omega _{R}^{n-1}(\log \mathscr {A})) = \operatorname {\mathrm {reg}}(\operatorname {\mathrm {Der}}_{R}(\log \mathscr {A})) + n - d$ . By Proposition 1.3 of [Reference Saito30], $\operatorname {\mathrm {reg}}(\operatorname {\mathrm {Der}}_{R}(\log \mathscr {A})) \leq d - n$ and so $\operatorname {\mathrm {reg}}(\Omega _{R}^{n-1}(\log \mathscr {A})) \leq 0$ . Therefore, $\operatorname {\mathrm {reg}}(L^{n-1}(\mathscr {A})) \leq \deg (\mathscr {A})$ completing this case, the induction and the proof.

Proof. Case 1: $\operatorname {\mathrm {rank}}(\mathscr {A}) \geq 2$ .

By Proposition 2.15, we know that the inclusion of $-\iota _{E}(\omega )$ -homogeneous subcomplex of $(H_{\mathfrak {m}}^{t}(\Omega _{R}^{\bullet }(\log \mathscr {A})), \nabla _{\omega })$ is a quasi-isomorphism:

(2.21)

Since $\iota _{E}(\omega ) = \sum \lambda _{k}$ , it is enough to show that for each t and each j the subspace of $H_{\mathfrak {m}}^{t}(\Omega _{R}^{j}(\log \mathscr {A}))$ degree $-\sum \lambda _{k}$ elements is zero. For then the left-hand side of Equation (2.21) is a complex of zero modules. In an equation, it suffices to show

(2.22) $$ \begin{align} H_{\mathfrak{m}}^{t}(\Omega_{R}^{j}(\log f))_{- \sum \lambda_{k}} = 0. \end{align} $$

First of all, if $\sum \lambda _{k} \notin \mathbb {Z}$ , the criterion (2.22) trivially holds. So we may assume $\sum \lambda _{k} \in \mathbb {Z}$ . And, because the j-logarithmic forms are reflexive (cf. subsection 2.2 of [Reference Denham and Schulze13]), they have depth at least two. By standard facts about local cohomology, $H_{\mathfrak {m}}^{0}(\Omega _{R}^{j}(\log \mathscr {A}))$ and $H_{\mathfrak {m}}^{1}(\Omega _{R}^{j}(\log \mathscr {A}))$ then vanish. Certainly, Equation (2.22) then holds for $t=0,1$ .

We must deal with $t \geq 2$ , where we use the regularity estimate of Theorem 2.29: $\operatorname {\mathrm {reg}}(\Omega _{R}^{j}(\log \mathscr {A})) \leq 0.$ Combining this with the definition of regularity in terms of local cohomology modules yields

$$\begin{align*}\max \deg (H_{\mathfrak{m}}^{t}(\Omega_{R}^{j}(\log \mathscr{A}))) + t \leq 0. \end{align*}$$

Therefore, for all $t \geq 2$ and for all j,

$$\begin{align*}H_{\mathfrak{m}}^{t}(\Omega_{R}^{j}(\log \mathscr{A}))_{\ell} = 0 \text{ whenever } \ell \notin \{-2, - 3, -4, \dots \}. \end{align*}$$

Then our hypothesis (2.20) ensures Equation (2.22) holds for $t \geq 2$ , completing Case 1.

Case 2: $\operatorname {\mathrm {rank}}(\mathscr {A}) = 1$ .

We argue in the same way, except noting that because $\Omega _{R}^{j}(\log \mathscr {A})$ is automatically free and R has depth one, that $H_{\mathfrak {m}}^{0}(\Omega _{R}^{j}(\mathscr {A}))$ vanishes. Thus, we need to show that

$$\begin{align*}H^{1}_{\mathfrak{m}}(\Omega_{R}^{j})_{\ell} = 0 \text{ whenever } \ell \notin \{-1, -2, -3, \dots \} \end{align*}$$

for $j = 0$ and $j = 1$ . Now, use Example 2.28.

Proof. By Lemma 2.22, it suffices to show that for $0 \leq j \leq n$ , the Čech cohomology $H_{\operatorname {\mathrm {\check {C}ech}}}^{t}(\Omega _{X^{\operatorname {\mathrm {an}}}}^{j}(\log \mathscr {A})_{-\iota _{E}(\omega )})$ vanishes whenever $ 1 \leq t \leq n$ . If $-\iota _{E}(\omega ) \notin \mathbb {Z}$ , the claim is immediate: There are no homogeneous (in the Laurent series sense) elements of nonintegral weight. So we may assume $-\iota _{E}(\omega ) \in \mathbb {Z}$ and will proceed by a GAGA argument.

Consider the graded R-module $\Omega _{R}^{j}(\log \mathscr {A})(-\iota _{E}(\omega ))$ . The sheafification $\widetilde {\Omega _{R}^{j}(\log \mathscr {A})(-\iota _{E}(\omega )})$ of this graded module gives a coherent algebraic sheaf of $\mathscr {O}_{\mathbb {P}_{\operatorname {\mathrm {alg}}}^{n-1}}$ -modules on $\mathbb {P}_{\operatorname {\mathrm {alg}}}^{n-1}$ . It is well known that for all $t \geq 1$ we have the following identification of sheaf cohomology with local cohomology:

(2.24) $$ \begin{align} H_{\mathfrak{m}}^{t+1}(\Omega_{R}^{j}(\log \mathscr{A}))_{-\iota_{E}(\omega)} = H^{t}(\widetilde{\Omega_{R}^{j}(\log \mathscr{A})(-\iota_{E}(\omega)})). \end{align} $$

On one hand, the condition (2.23) on the residue of $\omega $ implies that the left-hand size of Equation (2.24) vanishes. Indeed, this follows from Castelnuovo–Mumford regularity estimate of Theorem 2.29 as in Corollary 2.30. On the other hand, let $K^{j, \operatorname {\mathrm {an}}}$ be the analytification of $\widetilde {\Omega _{R}^{j}(\log \mathscr {A})(-\iota _{E}(\omega )})$ . By GAGA, $K^{j, \operatorname {\mathrm {an}}}$ is a coherent analytic sheaf of $\mathscr {O}_{\mathbb {P}^{n-1, \operatorname {\mathrm {an}}}}$ -modules on $\mathbb {P}^{n-1, \operatorname {\mathrm {an}}}$ . Again by GAGA, the sheaf cohomology of $K^{j,\operatorname {\mathrm {an}}}$ equals the right-hand side of Equation (2.24). Altogether this gives $0 = H^{t}(K^{j, \operatorname {\mathrm {an}}})$ for all $t \geq 1$ .

Now, consider the Čech complex $C^{\bullet }(\{D^{+}(x_{i})\}, K^{j, \operatorname {\mathrm {an}}})$ attached to the open cover $\{D^{+}(x_{i})\}_{1 \leq i \leq n}$ of $\mathbb {P}^{n-1, \operatorname {\mathrm {an}}}$ . Since $D^{+}(x_{i})$ and any intersection of the $\{D^{+}(x_{i})\}$ is Stein, and since higher sheaf cohomology (of coherent sheaves) vanishes on Steins, this Čech complex computes the sheaf cohomology $K^{j, \operatorname {\mathrm {an}}}$ . So for all $t \geq 1$ we have

(2.25) $$ \begin{align} 0 = H^{t}(K^{j, \operatorname{\mathrm{an}}}) = H^{t}(C^{\bullet}(\{D^{+}(x_{i})\}, K^{j, \operatorname{\mathrm{an}}})). \end{align} $$

Recall $K^{j, \operatorname {\mathrm {an}}}$ is the analytification of $\widetilde {\Omega _{R}^{j}(\log \mathscr {A})(-\iota _{E}(\omega )})$ . Because membership in $\Omega _{R}^{j}(\log \mathscr {A})[x_{I}^{-1}]$ is determined by the kernel of a map defined wholly in terms of $\partial _{1} \bullet f, \dots , \partial _{n} \bullet f, f$ , all of which are algebraic polynomials, and because analytification is exact, the analytification of $\Gamma (D(x_{I}), \widetilde {\Omega _{R}^{j}(\log \mathscr {A})})$ is $\Gamma (D(x_{I}), \Omega _{X^{\operatorname {\mathrm {an}}}}(\log \mathscr {A}))$ . (This is an affine statement: $D(x_{I}) \subseteq \mathbb {C}^{n}$ .) Because $\Omega _{R}^{j}(\log \mathscr {A})$ is generated by homogeneous terms (as $\mathscr {A}$ is central), we deduce

$$\begin{align*}\Gamma(D^{+}(x_{I}), K^{j, \operatorname{\mathrm{an}}}) = \Gamma(D(x_{I}), \Omega_{X^{\operatorname{\mathrm{an}}}}^{j}(\log \mathscr{A}))_{-\iota_{E}(\omega)}, \end{align*}$$

where by $(-)_{-\iota _{E}(\omega )}$ we mean the homogeneous elements in the sense of Laurent series. This means

$$ \begin{align*} H^{t}(C^{\bullet}(\{D^{+}(x_{i})\}, K^{j, \operatorname{\mathrm{an}}})) &= H^{t}(C^{\bullet}(\{D(x_{i})\}, \Omega_{X^{\operatorname{\mathrm{an}}}}^{j}(\log \mathscr{A}))_{-\iota_{E}(\omega)}) \\ &= H_{\operatorname{\mathrm{\check{C}ech}}}^{t}(\Omega_{X^{\operatorname{\mathrm{an}}}}^{j}(\log \mathscr{A}))_{-\iota_{E}(\omega)}. \end{align*} $$

By Equation (2.25), we conclude $0 = H_{\operatorname {\mathrm {\check {C}ech}}}^{t}(\Omega _{X^{\operatorname {\mathrm {an}}}}^{j}(\log \mathscr {A}))_{-\iota _{E}(\omega )}$ , as required.

Proof. Exactly as in Lemma 2.5 of [Reference Castro-Jiménez, Narváez-Macarro and Mond9].

Proof. It is enough to prove the assertion for V the complement of all the hyperplanes $\operatorname {\mathrm {V}}(f_{j})$ that do not contain the point $\mathfrak {x}$ . Observe:

$$\begin{align*}\omega^{\prime} = \omega - \sum_{\{1 \leq k \leq d \mid \mathfrak{x} \notin \operatorname{\mathrm{V}}(f_{k})\}} \lambda_{j} \frac{df_{k}}{f_{k}} = \omega - \sum_{\{1 \leq k \leq d \mid \mathfrak{x} \notin \operatorname{\mathrm{V}}(f_{k})\}} \lambda_{k} (d \log f_{k}). \end{align*}$$

Rename every such $f_{j}$ not vanishing at $\mathfrak {x}$ as $u_{j}$ . We construct isomorphisms

(3.1) $$ \begin{align} (\Omega_{V}^{\bullet}(\log f), \nabla_{\omega}) \xrightarrow[]{\simeq} (\Omega_{V}^{\bullet}(\log f), \nabla_{\omega - \lambda_{1}(d \log u_{1})}) \xrightarrow[]{\simeq} \cdots \xrightarrow[]{\simeq} (\Omega_{V}^{\bullet}(\log f), \nabla_{\omega^{\prime}}). \end{align} $$

We construct the first map of Equation (3.1). We may assume $\lambda _{1} \neq 0$ . Consider the diagram

Since $u_{1}$ is a unit, $u_{1}^{-\lambda _{1}}$ is a unit as well, making the horizontal maps isomorphisms. A quick computation verifies the diagram commutes. Therefore, multiplication by $u_{1}^{-\lambda _{1}}$ is an invertible chain map, and we have constructed the first isomorphism of Equation (3.1). Iterating the procedure demonstrates (a).

The same construction works for the twisted meromorphic de Rham complex, giving (b). And multiplication by $u_{1}^{\lambda }$ is compatible with including the twisted logarithmic de Rham complex into the meromorphic one.

Proof. Pick local coordinates $(t, x^{\prime })$ for $V = \mathbb {C} \times V^{\prime }$ . And note that $f_{k} = u_{k} f_{k}^{\prime }$ , where $u_{k}$ is a unit. Arguing as in Lemma 3.2, $(\Omega _{V}^{\bullet }(\log f^{\prime }), \nabla _{\omega ^{\prime }}) \xrightarrow[]{\simeq } (\Omega _{V}^{\bullet }(\log f), \nabla _{\omega })$ , where here $f^{\prime }$ and $\omega ^{\prime }$ are viewed as functions and forms over V. (Note that we are removing the summands $\lambda _{k} d \log u_{k}$ from $\omega $ ). Now, weight the local coordinates $(t, x^{\prime })$ by giving t weight one, each $x_{i}^{\prime }$ weight zero, $dx_{i}^{\prime }$ weight zero and $dt$ weight one. Let $\chi = t \partial _{t}$ be a logarithmic derivation on $f^{\prime }$ . Then we can argue as in Lemma 2.23: The twisted Lie derivative $\nabla _{\omega ^{\prime }}(\iota _{\chi }) + \iota _{\chi }(\nabla _{\omega ^{\prime }})$ induces a quasi-isomorphism , where the weight zero subcomplex is defined by the aforementioned weighting. (Note that $\iota _{\chi }(\omega ) = 0$ and the previous claim can be checked stalkwise.) This weight zero subcomplex is $\pi ^{-1}(\Omega _{V^{\prime }}^{\bullet }(\log f^{\prime }), \nabla _{\omega ^{\prime }})$ , giving (a). The proof of (b) is similar.

Proof. While the result is well known, we defer giving details on the derived category identification ‘ $(= \mathbf {R} j_{\star } \operatorname {\mathrm {L}}_{\operatorname {\mathrm {Exp}}(\boldsymbol {\lambda })})$ ’ until Proposition 4.3.

Step 0: Reductions and the inductive scheme.

We proceed stalkwise and induce on $\dim X$ . When $\dim X = 1$ the result is well known and easy to check. So assume $\dim X = n$ and the claim holds for all reduced arrangements in $\mathbb {C}^{p}$ for $p < n$ . Let $\mathfrak {x} \in X$ and E the smallest edge containing $\mathfrak {x}$ . If the subarrangement $f^{\prime }$ consisting of all hyperplanes containing $\mathfrak {x}$ is of rank less than $\dim X$ , then the induction hypothesis applies. For if V is a small Stein containing $\mathfrak {x}$ and

$$\begin{align*}\omega^{\prime} = \sum_{\{1 \leq k \leq d \mid E \subseteq \operatorname{\mathrm{V}}(f_{k})\}} \lambda_{k} \frac{df_{k}}{f_{k}}, \end{align*}$$

then Lemma 3.2 says Equation (3.3) holds at $\mathfrak {x}$ is if and only if holds at $\mathfrak {x}$ . Since $f^{\prime }$ is of rank less than $\dim X$ at $\mathfrak {x}$ it is a product, and we use Lemma 3.3 to invoke the inductive hypothesis. Note that $\omega ^{\prime }$ satisfies its version of Equation (3.2) since $\omega $ satisfies Equation (3.2) itself.

If the subarrangement $f^{\prime }$ at $\mathfrak {x}$ has rank n, we may change coordinates and assume that $\mathfrak {x} = 0$ and $f^{\prime }$ is central and essential. By Lemma 3.2, the quasi-isomorphism (3.3) holds at $\mathfrak {x}$ exactly when the corresponding Twisted Logarithmic Comparison Theorem holds for $f^{\prime }$ and $\omega ^{\prime }$ at $0$ (in these new coordinates).

Therefore, it suffices to prove the following: If $\mathscr {A}$ is a central, essential, reduced hyperplane arrangement such that the Equation (3.3) holds at all points outside of $0$ , then Equation (3.3) holds at $0$ . By Lemma 3.1, we may verify this (equivalently) by checking Equation (3.3) holds at a small open Stein $V \ni 0$ .

Step 1: Setting up the spectral sequences.

For the rest of the proof, we inhabit the spectral sequence infrastructure from the proof of Theorem 1.1 of [Reference Castro-Jiménez, Narváez-Macarro and Mond9] but refine it using the results of our previous sections. Let V be small Stein open about $0$ . Define the Stein opens $V_{i} = V \setminus \{x_{i} = 0\}$ and $V_{i}^{\prime } = V_{i} \setminus (V \cap \mathscr {A})$ . We have Stein open covers $\{V_{i}\}$ and $\{V_{i}^{\prime }\}$ of $V \setminus 0$ and $V \setminus (V \cap \mathscr {A})$ respectively. Consider the following two double complexes with objects:

$$ \begin{align*} K^{p, q} &= \bigoplus_{1 \leq i_{1} \leq \cdots \leq i_{q} \leq n} \Gamma(\bigcap_{i=0}^{q} V_{i_{j}}, \Omega_{X}^{p}(\log \mathscr{A})); \\ \widetilde{K}^{p, q} &= \bigoplus_{1 \leq i_{1} \leq \cdots \leq i_{q} \leq n} \Gamma(\bigcap_{i=0}^{q} V_{i_{j}}^{\prime}, \Omega_{X}^{p}(\star \mathscr{A})). \end{align*} $$

Each complex is positioned in the lattice $\mathbb {Z}^{2}$ so that, for example, $K^{p,q}$ , occurs in row p and column q. $K^{\bullet , \bullet }$ has horizontal differential the Čech differential and vertical differential $\nabla _{\omega }$ ; $\widetilde {K}^{\bullet , \bullet }$ has the same. And restriction commutes with the differentials, giving a map $\rho _{0}^{\bullet , \bullet }: K^{\bullet , \bullet } \to \widetilde {K}^{\bullet , \bullet }$ .

We will consider the resulting four spectral sequences arising from these double complexes using similar notation as in [Reference Castro-Jiménez, Narváez-Macarro and Mond9]. As shorthand, $^{\prime }(-)$ denotes the spectral sequence first taking cohomology vertically and $^{\prime \prime }(-)$ first taking cohomology horizontally. To be precise:

$$ \begin{align*} ^{\prime}E_{1}^{p,q} &= \bigoplus_{1 \leq i_{0} \leq \cdots \leq i_{q} \leq n} H^{p} (\Gamma(\bigcap_{j=0}^{q} V_{i_{j}}, (\Omega_{X}^{\bullet}(\log \mathscr{A}), \nabla_{\omega}))); \\^{\prime \prime}E_{1}^{p,q} &= H^{q}(V \setminus 0, \Omega_{X}^{p}(\log \mathscr{A})); \\^{\prime}\widetilde{E}_{1}^{p,q} &= \bigoplus_{1 \leq i_{0} \leq \cdots \leq i_{q} \leq n} H^{p} (\Gamma(\bigcap_{j=0}^{q} V_{i_{j}}^{\prime} , (\Omega_{X}^{\bullet}(\star \mathscr{A}), \nabla_{\omega}))); \\^{\prime \prime}\widetilde{E}_{1}^{p,q} &= H^{q}(V \setminus (V \cap \mathscr{A}), \Omega_{X}^{p}(\star \mathscr{A})). \end{align*} $$

Step 2: The $^{\prime }(-)$ spectral sequences.

By the inductive setup, Equation (3.3) holds at all points outside the origin. This means that the aformentioned natural map

$$\begin{align*}\rho_{1}^{\bullet, \bullet}: \enspace ^{\prime}E_{1}^{\bullet, \bullet} \xrightarrow[]{} \enspace ^{\prime} \widetilde{E}_{1}^{\bullet, \bullet} \end{align*}$$

is an isomorphism. As $\rho _{0}^{\bullet , \bullet }$ commutes with the differentials, this isomorphism continues on further pages. Since the spectral sequences converge, we have

(3.4) $$ \begin{align} \rho_{\infty}^{\bullet, \bullet}: \enspace ^{\prime}E_{\infty}^{\bullet, \bullet} \xrightarrow[]{\simeq} \enspace ^{\prime}\widetilde{E}_{\infty}^{\bullet, \bullet}. \end{align} $$

Step 3: The $^{\prime \prime }(-)$ spectral sequences.

We first consider $^{\prime \prime }\widetilde {E}^{\bullet , \bullet }$ . Because $V \setminus (V \cap \mathscr {A})$ is Stein and $\Omega _{X}^{p}(\star f) |_{U}$ is coherent, the higher sheaf cohomology $^{\prime \prime }\widetilde {E}_{1}^{p,q}$ vanishes for $q> 0$ . So this page’s nonzero terms lie on the y-axis, the spectral sequence converges on the second page where the nonzero terms still lie on the y-axis, and

(3.5) $$ \begin{align} ^{\prime \prime}\widetilde{E}_{2}^{p,0} = H^{p}(\Gamma(V \setminus (V \cap \mathscr{A}), (\Omega_{X}^{\bullet}(\star \mathscr{A}), \nabla_{\omega}))). \end{align} $$

We claim $^{\prime \prime }E^{\bullet , \bullet }$ has the same vanishing behavior. Combine the long exact sequence of cohomology supported on $\{0\}$

$$\begin{align*}\to H^{q}_{0}(V, \Omega_{X}^{p}(\log \mathscr{A})) \to H^{q}(V, \Omega_{X}^{p}(\log \mathscr{A})) \to H^{q}(V \setminus 0, \Omega_{X}^{p}(\log \mathscr{A})) \to \end{align*}$$

with the higher cohomological vanishing of $H^{q}(V, \Omega _{X}^{p}(\log \mathscr {A}))$ for $q \geq 1$ , to get a natural isomorphism

$$\begin{align*}^{\prime \prime}E_{1}^{p,q} \simeq H_{0}^{q+1}(V,\Omega_{X}^{p}(\log \mathscr{A})) \end{align*}$$

for all $q \geq 1$ . By excision,

$$\begin{align*}H_{0}^{q+1}(V,\Omega_{X}^{p}(\log \mathscr{A})) \simeq H_{0}^{q+1}(X, \Omega_{X}^{p}(\log \mathscr{A})). \end{align*}$$

Again, use the long exact sequence of cohomology supported on $\{0\}$ to get

$$\begin{align*}H^{q}(X \setminus 0, \Omega_{X}^{p}(\log \mathscr{A})) \simeq H_{0}^{q+1}(X, \Omega_{X}^{p}(\log \mathscr{A})) \end{align*}$$

for all $q \geq 1$ . Since $\{D(x_{i})\}_{1 \leq i \leq n}$ is an open cover of $X \setminus 0$ and since every $D(x_{i})$ along with every intersection of elements of the $\{D(x_{i})\}$ is a Stein manifold, the Čech complex $C^{\bullet }(\{D(x_{i})\}, \Omega _{X}^{p}(\log \mathscr {A}))$ computes the sheaf cohomology $H^{\bullet }(X \setminus 0, \Omega _{X}^{p}(\log \mathscr {A}))$ .

Combining these equivalences and using the notation of subsection 2.3 yields

$$\begin{align*}^{\prime \prime}E_{1}^{p,q} \simeq H^{q}(X \setminus 0, \Omega_{X}^{p}(\log \mathscr{A})) \simeq H_{\operatorname{\mathrm{\check{C}ech}}}^{q}(\Omega_{X}^{p}(\log \mathscr{A})) \end{align*}$$

for all $q \geq 1$ . The second page $^{\prime \prime }E_{2}^{\bullet , \bullet }$ arises from the first by taking cohomology vertically with respect to the differential $\nabla _{\omega }$ . So

(3.6) $$ \begin{align} ^{\prime \prime}E_{2}^{p,q} \simeq H^{p}(H_{\operatorname{\mathrm{\check{C}ech}}}^{q}(\Omega_{X}^{\bullet}(\log \mathscr{A})), \nabla_{\omega}) \end{align} $$

for all $q \geq 1.$ Now, our condition on the weights (3.2) considered at the origin become $\sum \lambda _{k} \notin \mathbb {Z}_{\geq \min \{2, \operatorname {\mathrm {rank}}(\mathscr {A})\}}$ . So we may apply Corollary 2.31 to Equation (3.6), deducing $^{\prime \prime }E_{2}^{p,q}$ vanishes for $q \geq 1$ . Therefore, the nonzero terms of the second page $^{\prime \prime }E_{2}^{\bullet , \bullet }$ all lie on the y-axis, the spectral sequence converges on this second page, and

$$\begin{align*}^{\prime \prime}E_{2}^{p, 0} = H^{p}(\Gamma(V \setminus 0, (\Omega_{X}^{\bullet}(\log \mathscr{A}), \nabla_{\omega}))). \end{align*}$$

Since $X = \mathbb {C}^{n}$ with $n \geq 2$ by assumption, the reflexivity of $\Omega _{X}^{j}(\log \mathscr {A})$ (see, for example, Proposition 1.5 [Reference Holland and Mond19]) implies that the global sections of $V \setminus 0$ are the global sections of V, meaning the only nonzero terms of the second page $^{\prime \prime }E_{2}^{\bullet , \bullet }$ are

(3.7) $$ \begin{align} ^{\prime \prime}E_{2}^{p, 0} = H^{p}(\Gamma(V, (\Omega_{X}^{\bullet}(\log \mathscr{A}), \nabla_{\omega}))). \end{align} $$

Step 4: The end.

Because $^{\prime \prime }E_{2}^{\bullet ,\bullet }$ consists of only one column on the y-axis, convergence to $H^{\bullet }\text {Tot}(K^{\bullet ,\bullet })$ happens on the second page. And because there is only one column, the induced filtration on $H^{\bullet }\text {Tot}(K^{\bullet ,\bullet })$ is trivial. Using Equation (3.7), we have

$$\begin{align*}H^{p}(\Gamma(V, (\Omega_{X}^{\bullet}(\log \mathscr{A}), \nabla_{\omega}))) = \enspace ^{\prime \prime}E_{2}^{p, 0} \enspace = H^{p}\text{Tot}(K^{\bullet,\bullet}). \end{align*}$$

Similarly, $^{\prime \prime }\widetilde {E}_{2}^{\bullet ,\bullet }$ has only one column on the y-axis, so convergence to $H^{\bullet } \text {Tot}(\widetilde {K}^{\bullet , \bullet })$ happens on the second page and the isomorphism has no graded data. Using Equation (3.5), we have

$$\begin{align*}H^{p}(\Gamma(V \setminus (V \cap \mathscr{A}), (\Omega_{X}^{\bullet}(\star \mathscr{A}), \nabla_{\omega}))) = \enspace ^{\prime \prime}\widetilde{E}_{2}^{p, 0} \enspace = H^{p}\text{Tot}(\widetilde{K}^{\bullet,\bullet}). \end{align*}$$

On the other hand, the isomorphism $\enspace ^{\prime }E_{\infty }^{\bullet , \bullet } \xrightarrow[]{\simeq } \enspace ^{\prime }\widetilde {E}_{\infty }^{\bullet , \bullet }$ from Equation (3.4) gives (ignoring the extra graded data) the natural isomorphism

$$\begin{align*}H^{p}\text{Tot}(K^{\bullet,\bullet}) \simeq H^{p}\text{Tot}(\widetilde{K}^{\bullet,\bullet}). \end{align*}$$

Putting the three displayed equations of this step together produces

$$\begin{align*}H^{p}(\Gamma(V, (\Omega_{X}^{\bullet}(\log \mathscr{A}), \nabla_{\omega}))) \simeq H^{p}(\Gamma(V \setminus (V \cap \mathscr{A}), (\Omega_{X}^{\bullet}(\star \mathscr{A}), \nabla_{\omega}))), \end{align*}$$

which is what we required.

Proof. Use Theorem 3.4, and set all the weights $\{\lambda _{k}\}$ to zero.

Proof. The quasi-isomorphism induced by inclusion in Theorem 3.4 gives isomorphisms in hypercohomology:

(3.8) $$ \begin{align} \textbf{H}^{t}(\Omega_{X}^{\bullet}(\log \mathscr{A}), \nabla_{\omega}) \xrightarrow[]{\simeq} \textbf{H}^{t}(\Omega_{X}^{\bullet}(\star \mathscr{A}), \nabla_{\omega}). \end{align} $$

Since X and $U = X \setminus \mathscr {A}$ are Stein and $\Omega _{X}^{j}(\log \mathscr {A})$ and $\Omega _{U}^{j}$ are coherent, their higher sheaf cohomology vanishes. So the hypercohomology in Equation (3.8) amounts to taking global sections and then taking cohomology. This gives the promised quasi-isomorphism. As for the claim involving ‘ $(\simeq H^{\bullet }(U, \operatorname {\mathrm {L}}_{\exp (\boldsymbol {\lambda })}))$ ’, it is well known that $\textbf {H}^{t} (\mathbf {R} j_{\star } \operatorname {\mathrm {L}}_{\operatorname {\mathrm {Exp}}(\boldsymbol {\lambda })}) = H^{t}(U, \operatorname {\mathrm {L}}_{\operatorname {\mathrm {Exp}}(\boldsymbol {\lambda })})$ .

Proof. The claim involving $(\simeq H^{\bullet }(U, \operatorname {\mathrm {L}}_{\exp (\boldsymbol {\lambda })}))$ follows from the algebraic de Rham theorems of Deligne and Grothendieck, cf. [Reference Orlik25]. That, when $\mathscr {A}$ is central, the map is a quasi-isomorphism is the first part of Proposition 2.15. As for the finite-dimensional claim: Since R and $\Omega _{R}^{j}(\log \mathscr {A})$ do not have elements of arbitrary large negative degree, the $-\iota _{E}(\omega )$ -homogeneous subspace $\Omega _{R}^{j}(\log \mathscr {A})_{-\iota _{E}(\omega )}$ is finite dimensional.

It remains to validate the purported quasi-isomorphism . Consider the following commutative diagram:

(3.9)

The displayed quasi-isomorphisms are justified by Lemma 2.23 and Proposition 2.15. As for displayed equalities, let $\eta = \sum _{p} \eta _{p} \in \Omega _{X,0}^{j}(\log \mathscr {A})$ be the homogeneous decomposition of $\eta $ as in subsection 2.3, though here this arises from, up to a pole along f, convergent formal power series in $\mathbb {C}[[x_{1}, \dots , x_{n}]]$ as opposed to a Laurent series. If $\eta $ is homogeneous of weight $-\iota _{E}(\omega )$ , then $\eta = \eta _{-\iota _{E}(\omega )}$ , the latter of which is, up to its pole along $\mathscr {A}$ , an algebraic j-form. So $\eta _{-\iota _{E}(\omega )} \in \Omega _{R}^{j}(\log \mathscr {A})_{-\iota _{E}(\omega )}$ . The same story applies for the nonlogarithmic displayed equality. This diagram concludes the proof: Theorem 3.4 implies the topmost horizontal map is a quasi-isomorphism, whence the bottom-most horizontal map is a quasi-isomorphism.

Proof. Brieskorn’s Theorem says the composition of maps in Equation (3.10) is a quasi-isomorphism, so if we show the algebraic Logarithmic Comparison Theorem is true we can conclude the first map in Equation (3.10) is a quasi-isomorphism. From this, the claim $A^{j}(\mathscr {A}) = H^{j}(\Omega _{R}^{\bullet }(\log \mathscr {A}), d)$ is easy: The implicit differential of exterior differentiation on $A^{\bullet }(\mathscr {A})$ is trivial.

It remains to prove the algebraic Logarithmic Comparison Theorem. When $\mathscr {A}$ is central this follows from Theorem 3.8 by setting the weights to zero.

For noncentral $\mathscr {A}$ , first note that we can define a sheaf-theoretic version of the algebraic twisted logarithmic and rational de Rham complexes: $(\Omega _{X_{\operatorname {\mathrm {alg}}}}^{\bullet }(\log \mathscr {A}), \nabla _{\omega })$ and $(\Omega _{X_{\operatorname {\mathrm {alg}}}}^{\bullet }(\star \mathscr {A}), \nabla _{\omega })$ , respectively. These are complexes of $\mathscr {O}_{X_{\operatorname {\mathrm {alg}}}}$ -modules. If we prove

(3.11)

then we can deduce that taking global sections preserves the quasi-isomorphism by arguing as in Corollary 3.7. The proof of Theorem 3.4 gives a proof of Equation (3.11) once you replace every occurrence of ‘Stein’ with ‘affine’. Indeed, note that: Lemma 3.1 holds after replacing ‘Stein’ with ‘affine’; Lemma 3.2 is never invoked since $\omega = 0$ in our case; we may still use Lemma 3.3 since the change of coordinates is linear, and hence algebraic; in Step 3 of the proof, $E_{1}^{p,q} \simeq H_{0}^{q+1}(V, \Omega _{X}^{p}(\log \mathscr {A}))$ which is isomorphic to $H_{\mathfrak {m}}^{q+1}(\Omega _{R}^{p}(\log \mathscr {A}))$ by Exercise III.2.3, III.3.3 [Reference Hartshorne18], and so Corollary 2.30 can be applied to $E_{2}^{p,q}$ directly (i.e., without subsection 2.3 and subsection 2.6).

Proof. We must produce $\boldsymbol {\lambda } \in \operatorname {\mathrm {Exp}}^{-1}(\boldsymbol {\beta })$ such that the combinatorial arithmetical restrictions

$$\begin{align*}\sum_{\{1 \leq k \leq d \mid E \subseteq \operatorname{\mathrm{V}}(f_{k})\}} \lambda_{k} \notin \mathbb{Z}_{\geq \min\{2, \operatorname{\mathrm{rank}}(E)\}} \end{align*}$$

are satisfied at each edge E of the intersection lattice. To do this, pick an arbitrary $\boldsymbol {\lambda } \in \operatorname {\mathrm {Exp}}^{-1}(\boldsymbol {\beta })$ and replace $\boldsymbol {\lambda }$ with $\boldsymbol {\lambda } - \textbf {z}$ , where $\textbf {z} = (z, \dots , z) \in \mathbb {Z}^{d}$ and $z \gg 0$ .

Proof. The vertical maps in the diagram are certainly $\mathscr {O}_{X}$ -isomorphisms, and it is easy to check that action of a derivation $\delta $ commutes with the vertical maps: $\psi (\nabla _{\delta }^{\boldsymbol {\lambda }} (-)) = \delta (\psi (-)).$ This fact simplifies the straightforward calculations mandatory in checking $\nabla ^{\boldsymbol {\lambda }}$ is a well-defined connection (and then certifies the vertical maps are $\mathscr {D}_{X}$ -isomorphisms).