The Jacobian module, the Milnor fiber, and the D-module generated by $f^s$

Abstract

For a germ f on a complex manifold X, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the D-module generated by $f^s$ to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of $f^s$ is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials. In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of 1 / f; we confirm in many cases a corresponding conjecture on the annihilator of $f^s$ but we disprove it in general; we show that the Bernstein–Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of $f^s$ is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.

Appendix: Logarithmic derivations under blow-ups

Notation 6.1

Let X be a smooth scheme over $\mathbb {C}$, with subschemes $C\subseteq Y\subseteq X$. Assume that C is smooth and denote by $\mathscr {I}_C\supseteq \mathscr {I}_Y$ the ideal sheaves. We say that a derivation on X is logarithmic along a subscheme V defined by $\mathscr {I}_V$ if $\delta (\mathscr {I}_V)\subseteq \mathscr {I}_V$.

Let

$$\begin{aligned} \pi :X'\longrightarrow X \end{aligned}$$

be the (smooth) blow-up of X at C and denote $Y'\subseteq X'$ the total transform of Y.

Lemma 6.2

In the context above, $\pi _*(\Omega ^i_{X'}(\log \pi ^*(Y)))=\Omega ^i_X(\log Y)$.

Proof

The statement is local in the base, so we may assume that X is smooth and affine, and represent $\Omega ^i_X(\log Y)$ by their modules of global sections. We may assume further that ${{\mathrm{codim}}}_X(C)\ge 2$ as otherwise $\pi$ is an isomorphism. Set $R=\Gamma (X,\mathscr {O}_X)$ and $I_C=\Gamma (X,\mathscr {I}_C)$.

Pulling back differentials along $\pi$ is a linear faithful functor and there are $\mathscr {O}_X$-module inclusions $\underbrace{\Omega ^i_X(\log Y)}_{=:M}\subseteq \underbrace{\Gamma (X',\Omega ^i_{X'}(\log Y'))}_{=:M'}\subseteq \underbrace{K\otimes _R\Omega ^i_X}_{=:M''}$, K denoting the field of fractions of R.

Let Q be the quotient $M'/M$. Since $M''$, and hence also $M'$, is torsion-free, the associated long exact sequence of local cohomology gives an injection $H^0_{I_C}(Q)\hookrightarrow H^1_{I_C}(M)$. Now M is a second syzygy, say the kernel of the map $F_1\longrightarrow F_0$ between free modules. Then if L is the image of this map, long exact sequences again show that $H^1_{I_C}(M)=H^0_{I_C}(L)$ which vanishes as L sits in a free module and $I_C$ has height at least two. We have shown that $H^0_{I_C}(Q)=0$ but since Q is supported in C this forces $Q=0$. $\square$

We now consider lifting logarithmic derivations along a blow-up. While logarithmic derivations and logarithmic $(n-1)$-forms can be identified locally, globally $\Omega ^{n-1}_X(\log Y)$ is really ${{\mathrm{Der}}}_X(-\log Y)\otimes \mathscr {O}_X(Y)\otimes \Omega ^n_Y$ and so exhibits different behavior: the push-forward of the logarithmic derivations along $\pi ^*(Y)$ is, in general, not all of the logarithmic derivations along Y .

Theorem 6.3

In the setting of Notation 6.1,

• 1. an element of ${{\mathrm{Der}}}_X$ lifts to $X'$ if and only if it is logarithmic along $\mathscr {I}_C$; we have $\pi _*({{\mathrm{Der}}}_{X'})={{\mathrm{Der}}}_X(-\log \mathscr {I}_C)$;

• 2. if Y is C-saturated, $\mathscr {I}_Y:_{\mathscr {O}_X}\mathscr {I}_C=\mathscr {I}_Y$, then

  $$\begin{aligned} \pi _*({{\mathrm{Der}}}_{X'}(-\log \pi ^*(Y)))={{\mathrm{Der}}}_X(-\log C)\cap {{\mathrm{Der}}}_X(-\log Y). \end{aligned}$$

Proof

Again, we assume that X is smooth and affine, and that ${{\mathrm{codim}}}_X(C)\ge 2$. Set $R=\Gamma (X,\mathscr {O}_X)$ and write $C={{\mathrm{Var}}}(I_C)$.

By shrinking X we can assume that $k={{\mathrm{codim}}}_X(C)$ and $I_C=(f_1,\ldots ,f_k)R\subseteq R$ where the $f_i$ form a regular sequence in any order. Let $\delta \in {{\mathrm{Der}}}_X$ be logarithmic along C. Since $\delta \bullet (I_C)\subseteq I_C$, it induces a derivation $\tilde{\delta }$ of t-degree zero on the Rees ring

$$\begin{aligned} \mathscr {R}=\mathscr {R}(R,I_C)=\bigoplus _{i\in \mathbb {N}} (I_Ct)^i. \end{aligned}$$

Derivations of t-degree zero induce derivations on each homogeneous localization $\mathscr {R}[(tf_i)^{-1}]$, stabilize the degree-zero part, and then obviously agree on overlaps. Since $X'={{\mathrm{Proj}}}(\mathscr {R})$, $\tilde{\delta }$ induces a global derivation $\delta '=:\pi ^*(\delta )$ on $\mathscr {O}_{X'}$. The exceptional divisor $E_\pi$ is defined by the ideal sheaf $I_Ct\mathscr {R}$; as $\tilde{\delta }\bullet (I_Ct\mathscr {R})\subseteq I_Ct\mathscr {R}$, $\delta '$ is logarithmic along $E_\pi$. The R-morphism $\delta \mapsto \pi ^*(\delta )$ from ${{\mathrm{Der}}}_X$ to ${{\mathrm{Der}}}_{X'}(-\log E_\pi )$ is injective, since X and $X'$ are smooth and agree outside C.

Now suppose additionally that $\delta \in {{\mathrm{Der}}}_X(-\log Y)$; then $\tilde{\delta }$ preserves the extension of $I_Y$ to $\mathscr {R}$, and hence $\delta '$ is logarithmic along the total transform of Y. Since $I_C$ is prime and $I_Y:_R I_C=I_Y$, a dense open set of each component of the proper transform $Y'$ is outside $E_\pi$. It follows that $\delta '$ is also logarithmic along $Y'$ and so ${{\mathrm{Der}}}_X(-\log Y)\cap {{\mathrm{Der}}}_X(-\log C)$ is a submodule of $\Gamma (X',{{\mathrm{Der}}}_{X'}(-Y'))$ and of $\Gamma (X',{{\mathrm{Der}}}_{X'}(-\pi ^*(Y)))$.

Conversely, let $\delta '$ be a global derivation on $X'$ and let $U_i$ be the standard open set ${{\mathrm{Spec}}}(\mathscr {R}_i)$ in $X'$ where $\mathscr {R}_i=(R[tf_1,\ldots ,tf_k,(tf_i^{-1})])_0=R[f_1/f_i,\ldots ,f_k/f_i]$. By definition, $\delta '$ induces a derivation on each $\mathscr {R}_i$ which we also denote $\delta '$.

Since R is a domain, $R\subseteq \mathscr {R}_i \subseteq R[1/f_i]$ and so $\delta '\bullet (R)\subseteq \bigcap _i\mathscr {R}_i\subseteq \bigcap _i R[1/f_i]$. The latter is the ideal transform of R with respect to $I_C$, and since the depth of $I_C$ on R is at least 2 we have $(\bigcap R[1/f_i])/R=H^1_I(R)=0$. Hence $\delta '$ is a derivation on R which we denote $\delta$. We must show that it is logarithmic along the center of the blow-up. Note that $\pi ^*(\delta )=\delta '$ again.

Lemma 6.4

Let R be a domain, $f_1,\ldots ,f_k,g$ a regular sequence in R, and $\delta '$ a derivation on both R and $R[f_1/g,\ldots ,f_k/g]$. Let I be the R-ideal generated by $f_1,\ldots ,f_k$. Then $\delta '\bullet (I)\subseteq I+Rg$.

Proof

By hypothesis, $\delta '\bullet (f_i/g)=\delta '\bullet (f_i)/g-f_i\delta '\bullet (g)/g^2$ can be written as $\sum _{j=0}^rP_j(f_1/g,\ldots ,f_k/g)$ for suitable homogeneous polynomials $P_0,\ldots ,P_r\in R[x_1,\ldots ,x_k]$, $P_j$ being of degree j. Choose such presentation with r minimal.

We show first that one can assume $r\le 2$. Indeed, if $r>2$ clear denominators to see that $P_r(f_1,\ldots ,f_k)\in Rg$. By Lemma 6.5 below, $P_r(f_1,\ldots ,f_k)=Q_r(f_1,\ldots ,f_k)$ for a suitable homogeneous polynomial $Q_r(x)=g\sum _M q_Mx^M\in gR[x_1,\ldots ,x_k]$ using multi-index notation $x^M=\prod _ix_i^{m_i}$ with $|M|=\deg (P_r)=r$. Then, abbreviating $f_1,\ldots ,f_k$ to f, and $f_1/g,\ldots ,f_k/g$ to f / g,

$$\begin{aligned} P_r(f/g)=\sum _{M}q_M\cdot g\cdot (f/g)^M =\sum _{M}q_M\cdot g^{1-|M|}\cdot f^M. \end{aligned}$$

For each multi-index M, let i(M) be some index with $M_i>0$ and denote $e_i$ the unit vector in direction i. Then $\sum _{M}q_Mg^{1-|M|}f^M=\sum _{M}q_Mf_{i(M)}g^{1-|M|}f^{M-e_{i(M)}}$ can be viewed as evaluation of $\sum _{M}q_Mf_{i(M)}x^{M-e_{i(M)}}$ at f / g. The latter is a polynomial of degree $r-1$. And so the presentation $\sum _{j=0}^rP_i(f_1/g,\ldots ,f_k/g)$ for $\delta '(f_i/g)$ didn’t use minimal r.

Hence, we may assume that $r\le 2$. In that case, we obtain, clearing denominators,

$$\begin{aligned} \delta '\bullet (f_i)g-\delta '\bullet (g)f_i=P_0(f)g^2+P_1(f)g+P_2(f). \end{aligned}$$

In particular, $\delta '\bullet (f_i)g-P_0(f)g^2=P_1(f)g+P_2(f)+\delta '\bullet (g)f_i\in I$. Since g is regular on R / I, $\delta '\bullet (f_i)-P_0(f)g\in I$ and so $\delta '\bullet (f_i)\in R(f,g)$. This finishes the proof of Lemma 6.4 $\square$

Now return to the proof of the Theorem. Lemma 6.4 implies, looking at $U_i$, that $\delta \bullet (f_i)\in I$ for each i. Hence, $\delta$ is logarithmic along $I_C$, and therefore $\delta '$ is logarithmic along $E_\pi$.

If $\delta '$ is logarithmic along $Y'$ then $\delta '$ preserves each $I_Y\cdot \mathscr {R}_i$, and hence $\delta$ sends $I_Y$ into $\bigcap (I_Y\cdot \mathscr {R}_i)$. However, $\bigcap (I_Y\cdot \mathscr {R}_i)/I_Y=H^1_{I_C}(I_Y)$. Since C is cut out by a regular sequence of length at least 2 in R, $H^1_{I_C}(I_Y)=H^0_{I_C}(R/I_Y)$ and this last module vanishes since by hypothesis R is $\mathscr {I}_C$-torsion-free. It follows that $\delta$ is logarithmic along Y and the theorem follows. $\square$

Lemma 6.5

Let R be a domain, and let $f_1,\ldots ,f_m,g_1,\ldots ,g_{m'}$ be a regular sequence in every order on R. Write $f=f_1,\ldots ,f_m$ and $g=g_1,\ldots ,g_{m'}$.

Suppose $P(x)\in R[x_1,\ldots ,x_{m}]$ is homogeneous of degree k and satisfies $P(f)\in Rg$. Then there exists $Q\in R[x_1,\ldots ,x_m]$, homogeneous of degree k, with $P(f)=Q(f)$ and $Q(x)\in gR[x]$.

Proof

• 1. If $m=1$ then we have $rf^k\in Rg$ and so regularity implies that the lemma holds in that case.

• 2. If P is linear, $\sum r_if_i\in Rg$ and so $r_m\in R(g,f_{<m})$ can be written as $\sum b_jg_j+\sum _{i<m}a_if_i$. Rewriting, we obtain $P(f)=\sum _{<m} r_if_i+r_mf_m=\sum _{<m}r_if_i+\sum _{<m} f_ma_if_i+ \sum f_mb_jg_j=\sum _{i<m}(f_ma_i+r_i)f_i+\sum f_mb_jg_j\in R(f_{<m,g})$. By induction on m, there is a linear polynomial $Q'(x_1\ldots ,x_{m-1})$ over R with coefficients in Rg which when evaluated at $f_{<m}$ yields $\sum _{i<m}(f_ma_i+r_i)f_i$. But then $P(f)=Q'(f_{<m})+f_m\sum b_jg_j$ and the lemma follows in this case.

• 3. The general case. We use induction on m. Consider the stratification of the monomials in m variables of degree k into the following subsets: $S_m=\{x_m^k\}$, $S_j=\{\text {monomials in } x_{\ge j }\}{\smallsetminus } S_{j+1}$ for $0<j<n$. So $S_j$ is the set of monomials in $R[x_j,\ldots ,x_m]$ divisible $x_j$. We are going to construct a sequence of polynomial identities

  $$\begin{aligned} (j):\qquad P^{(j)}(x)=\underbrace{\sum _{\ell<j}P^{(j)}_\ell (x)}_ {P^{(j)}_{<j}(x)}+P^{(j)}_j(x)+\underbrace{\sum _{\ell>j}P^{(j)}_\ell (x)}_ {P^{(j)}_{>j}(x)} \end{aligned}$$

  such that the following hold: $P^{(j)}_\ell (x)$ is a sum of monomials indexed by $S_\ell$; $P^{(j)}_{>j}(x)\in gR[x_{j+1},\ldots ,x_m]$; $P^{(j)}(f)=P(f)$ for all j. For $j=m$, take for $P^{(m)}_m(x)$ the part of P(x) indexed by $S_m$, and put $P^{(m)}_{>m}(x)=0$ and $P^{(m)}_{<m}(x)=P(x)-P^{(m)}_m(x)$. For any j, $P^{(j)}_{<j}(x)$ is automatically in the ideal generated by $x_{<j}$, so evaluation at $x=f$ gives $P^{(j)}_j(f)=P(f)-P^{(j)}_{<j}(f)-P^{(j)}_{>j}(f)\in R(g,f_{<j})$. Given identity (j) we now construct identity $(j-1)$ with the corresponding properties. As $x_j$ divides $P^{(j)}_j(x)$, $P^{(j)}_j(f)/f_j\in R$. Then $R(g,f_{<j})\ni P^{(j)}_j(f)=f_j\cdot ( P^{(j)}_j(f)/f_j)$ implies by regularity that $P^{(j)}_j(f)/f_j\in R(g,f_{<j})$. As the polynomial $P^{(j)}_j(x)/x_j\in R[x_{\ge j}]$ is homogeneous of degree $k-1$, the inductive hypothesis asserts the existence of a homogeneous polynomial of degree $k-1$ in $x_j,\ldots ,x_m$ with coefficients in $R(g,f_{<j})$ and which evaluates at f to $P^{(j)}_j(f)/f_j$. Explicitly, $P^{(j)}_j(f)$ is the value at $x=f$ of the sum of one polynomial $P^{(j)}_{j,g}(x)\in gR[x]$ and another polynomial in $f_{<j}R[x]$, both homogeneous of degree $k-1$. The latter polynomial can be changed (without affecting its value at $x\leadsto f$) into a polynomial $P^{(j)}_{j,<}(x)\in x_{<j}R[x]$ which is then supported in $S_{<j}$. Moving terms, if necessary, from $P^{(j)}_{j,g}(x)$ to $P^{(j)}_{j,<}(x)$ one may assume that $P^{(j)}_{j,g}(x)\in R[x_{>j-1}]$. Now update identity (j) as follows: set $P^{(j-1)}_{<j-1}(x)$ to be the terms in $P^{(j)}_{<j}(x)+P^{(j)}_{j,<}(x)$ supported in $S_{<j-1}$; let $P^{(j-1)}_{j-1}(x)$ be the other terms in this sum; set $P^{(j-1)}_{>j-1}(x)= P^{(j)}_{>j}(x)+P^{(j)}_{j,g}(x)$. The stipulated conditions then hold for the new display. It follows that from identity (m) above we can proceed to identity (0). However, then $P^{(0)}(x)=P^{(0)}_{>0}(x)\in gR[x]$. $\square$

We now record the existence of an embedded resolution of singularities that is particularly well adapted to computing with logarithmic vector fields.

Proposition 6.6

For every divisor Y on X, there is an algorithm for construction of an embedded resolution of singularities that is a composition of blow-ups such that at each step the center of the blow-up is smooth and a union of logarithmic strata of the total transform of Y under the previous blow-ups.

Proof

By [50], there is an embedded resolution of singularities $\pi :X'\longrightarrow X$ that is a sequence of blow-ups $\pi =\pi _k\circ \cdots \circ \pi _1$ with smooth centers that furthermore is functorial: for any analytic isomorphism $\iota :X\longrightarrow X$ there is an analytic isomorphism $\tilde{\iota } :X'\longrightarrow X'$ such that $\pi \circ \tilde{\iota }=\iota \circ \pi$.

Take $\delta \in {{\mathrm{Der}}}_X(-\log Y)$ and choose ${\mathfrak x}_0\in X$ with $\delta ({\mathfrak x}_0)\ne 0$. By the Picard–Lindelöf theorem, there is a foliation of integrating curves $\gamma _{\mathfrak x}(t)$ for $\delta$ at all ${\mathfrak x}$ near ${\mathfrak x}_0$: $\gamma _{\mathfrak x}(0)={\mathfrak x}$ and $\frac{{\mathrm d}}{{\mathrm d}t}(\gamma _{\mathfrak x}(t))$ is $\delta$ evaluated at $\gamma _{\mathfrak x}(t)$ for small t. It follows that the assignment $\Phi ({\mathfrak x},t):{\mathfrak x}\mapsto \gamma _{\mathfrak x}(t)$ is well-defined for ${\mathfrak x}$ sufficiently near ${\mathfrak x}_0$, and for all $t\ll 1$. Since $\delta$ is analytic and $\gamma _{\mathfrak x}(t)$ is defined via an integral, $\Phi$ is analytic.

As $\delta$ is logarithmic along Y, ${\mathfrak x}\in Y$ implies $\gamma _{\mathfrak x}(t)\in Y$. Hence, $\Phi (-,t)$ is a local automorphism of the pair (X, Y) near ${\mathfrak x}\in Y$ for $t\ll 1$. If $\pi _1:X'_1\longrightarrow X$ is the first blow-up in the resolution, then by functoriality $\Phi (-,t)$ lifts to an analytic family of local automorphisms of $(X'_1, \pi ^*Y)$. This is only possible if the center $C_1$ of the blow-up is stable under $\delta$. This being true for all logarithmic vector fields along Y, $C_1$ must be a union of logarithmic strata of Y. The proposition follows by iterating the argument. $\square$

Corollary 6.7

If $Y\subseteq X$ is a divisor in a smooth $\mathbb {C}$-scheme then there is a resolution of singularities $\pi :X'\longrightarrow X$ such that $\pi _*({{\mathrm{Der}}}_{X'}(-\log \pi ^*(Y)))={{\mathrm{Der}}}_X(-\log Y)$ and $\pi _*(\Omega ^i_{X'}(\log \pi ^*(Y)))=\Omega ^i_X(\log Y)$.

Proof

For any divisor $Y_i$ and any smooth center $C_i\subseteq Y_i$ of codimension 2 or more on smooth $X_i$ the ideal quotient $\mathscr {I}_{Y_i} :_{\mathscr {O}_{X_i}} \mathscr {I}_{C_i}$ equals $\mathscr {I}_{Y_i}$. Thus Theorem 6.3 applies in each step of the resolution obtained in Proposition 6.6. The final claim follows from Lemma 6.2. $\square$

Not every logarithmic stratum will appear as a center in a resolution:

Example 6.8

Let $Y={{\mathrm{Var}}}(xy(x+y)(x+ty))\subseteq \mathbb {C}^3$. Then every point of the t-axis is a logarithmic stratum. In any reasonable resolution of the pair $(\mathbb {C}^3,Y)$, of all points on the t-axis, the only zero-dimensional canonical Whitney strata that feature as blow-up centers are (0, 0, 0) and (0, 0, 1).