Minimal exponents of hyperplane sections: a conjecture of Teissier

Introduction

Let $X$ be a smooth complex algebraic variety and $f\in\mathcal{O}_{X}(X)$ nonzero, defining a hypersurface $Y$. For a point $P\in Y$, the minimal exponent $\widetilde{\alpha}_{P}(f)$ can be defined as the negative of the largest root of the reduced Bernstein-Sato polynomial of $f$ at $P$. This is a very interesting invariant of singularities that refines the log canonical threshold $\operatorname{lct}_{P}(f)$; more precisely, by a result of Lichtin and Kollár, we have ${\rm lct}_{P}(f)=\min\{\widetilde{\alpha}_{P}(f),1\}$ (see [Kollar, Section 10]). Moreover, by a result of Saito [Saito-B, Theorem 0.4] the hypersurface $Y$ has rational singularities at $P$ if and only if $\widetilde{\alpha}_{P}(f)>1$. In the setting where $Y$ has an isolated singularity at $P$, the minimal exponent can be described via asymptotic expansion of integrals along vanishing cycles, see [Malgrange] and [Malgrange2]. In this incarnation, it has been extensively studied in [AGZV] and is also known as the Arnold exponent of $f$ at $P$.

In this article we are interested in the behavior of the minimal exponent under restriction to a smooth hypersurface $H$ in $X$, containing $P$. When $Y$ has an isolated singularity at $P$, Teissier introduced and studied in [Teissier2] the invariant $\theta_{P}(f)$ defined as

where $E$ runs over the prime divisors over $X$ with center at $P$, $\mathfrak{m}_{P}$ is the ideal defining $P$ in $X$, and $J_{f}$ is the Jacobian ideal of $f$. Using the description of the integral closure of an ideal in terms of divisorial valuations (see [Lazarsfeld, Chapter 9.6.A]) one can see that the maximum in (1) is achieved by a divisor on the normalized blow-up of $X$ along $\mathfrak{m}_{P}\cdot J_{f}$; moreover, $\theta_{P}(f)$ is the minimum of the positive rational numbers $\tfrac{r}{s}$ with the property that $\mathfrak{m}_{P}^{r}$ is contained in the integral closure $\overline{J_{f}^{s}}$ of $J_{f}^{s}$. The following is our main result, giving a positive answer to a conjecture of Teissier [Teissier3].

Suppose that $n=\dim(X)\geq 2$ and the hypersurface defined by $f$ in $X$ has an isolated singularity at $P$. If $H$ is a smooth hypersurface in $X$, with $f|_{H}\neq 0$ and $P\in H$, then

By successively applying the theorem for general hyperplane sections (which automatically have isolated singularities), we obtain the following:

If the hypersurface $Y$ defined by $f$ in $X$ has an isolated singularity at $P$ and if $H_{1},\ldots,H_{n-1}$ are general smooth hypersurfaces in $X$, containing $P$, then

The inequality in Theorem 1.1 was proved by Loeser [Loeser], with $\theta_{P}(f)$ replaced by its round-up $\lceil\theta_{P}(f)\rceil$ (that is, the smallest integer that is $\geq\theta_{P}(f)$). Assuming that $f\in{\mathbf{C}}[x_{1},\ldots,x_{n}]$, $P=0$, and $H$ is the hyperplane given by $x_{n}=0$, the argument in [Loeser] made use of the family of hypersurfaces

where $\theta=\lceil\theta_{P}(f)\rceil$. Note that $\widetilde{\alpha}_{0}(h_{1})=\widetilde{\alpha}_{0}(f)$ and $\widetilde{\alpha}_{0}(h_{0})=\widetilde{\alpha}_{0}(f|_{H})+\frac{1}{\theta+1}$ by the Thom-Sebastiani property of Arnold exponents (see for example [Malgrange, Example (6.8)]). The definition of $\theta_{0}(f)$ implies that the Milnor number of $h_{t}$ at is constant in a neighborhood $V$ of $1$, hence by a result of Varchenko [Varchenko2] it follows that $\widetilde{\alpha}_{0}(h_{t})$ is constant on $V$. Finally, by the semicontinuity of the Arnold exponent [Steenbrink, Theorem 2.11], it follows that $\widetilde{\alpha}_{0}(h_{t})\geq\widetilde{\alpha}_{0}(h_{0})$ for $t\in V$, and we conclude that $\widetilde{\alpha}_{0}(f)\geq\widetilde{\alpha}_{0}(f|_{H})+\frac{1}{\theta+1}$.

This argument was modified in [EM] to prove the weaker version of Theorem 1 for log canonical thresholds. The idea was to consider the same family $(h_{t})_{t\in{\mathbf{C}}}$, with $\theta=\theta_{P}(f)$. In order to make sense of this, one pulls-back this expression by the finite cover given by $\pi(x_{1},\ldots,x_{n})=(x_{1},\ldots,x_{n-1},x_{n}^{d})$, where $d$ is a divisible enough positive integer. Recall that the log canonical threshold is characterized by the triviality of certain invariants associated to $f$, the multiplier ideals, see [Lazarsfeld, Chapter 9]; due to the presence of the finite cover $\pi$, the argument in [EM] relied on considering whether the equation $x_{n}^{d-1}$ (defining the relative canonical divisor of $\pi$) lies in a suitable multiplier ideal $h_{t}\circ\pi$, by making use of various properties of multiplier ideals.

In this note we follow the same approach. Since we deal with the minimal exponent, we need to make use of more refined invariants, the Hodge ideals $I_{p}(f^{\lambda})$ introduced and studied in [MP1]. The definition of these invariants (as well as the proofs of their basic properties) makes use of Saito’s theory of mixed Hodge modules [Saito-MHM]. It was shown in [Saito-MLCT] and [MP2] that in the same way that triviality of multiplier ideals characterizes log canonical thresholds, triviality of Hodge ideals characterizes minimal exponents. In order to extend the approach in [EM] to the setting of Hodge ideals, we need two new properties of these invariants, that are of independent interest.

Let $\pi\colon Y\to X$ be a finite surjective morphism between smooth varieties and let $K_{Y/X}$ be the effective divisor on $Y$ defined by the determinant of the Jacobian matrix of $\pi$. If $0\neq f\in\mathcal{O}_{X}(X)$ and $g=\pi^{*}(f)$ both define reduced divisors, then for every $h\in\mathcal{O}_{X}(X)$, every nonnegative integer $p$, and every $\lambda\in{\mathbf{Q}}_{>0}$, if $\pi^{*}(h)\cdot\mathcal{O}_{Y}(-K_{Y/X})\subseteq I_{p}(g^{\lambda})$, then $h\in I_{p}(f^{\lambda})$.

For a more general statement, which does not assume that $f$ and $g$ define reduced divisors, see Theorem 3.5 below. We also give a partial converse of this result in the case of a Galois cover (see Theorem 3.6). At least for such covers, we thus have an extension of the formula in [Lazarsfeld, Theorem 9.5.42] relating multiplier ideals under finite maps.

The next result is concerned with certain Hodge ideals associated to families of hypersurfaces with constant Milnor number. Let $\varphi\colon{\mathcal{X}}\to T$ be a smooth morphism of smooth complex algebraic varieties (in particular, $T$ is connected), and let $s\colon T\to{\mathcal{X}}$ be a section of $\varphi$. Let $f\in\mathcal{O}_{\mathcal{X}}(\mathcal{X})$ be such that $f\circ s=0$ and for every $t\in T$, the restriction $f_{t}$ to ${\mathcal{X}}_{t}=\varphi^{-1}(t)$ is nonzero. We assume that for every $t\in T$, the hypersurface defined by $f_{t}$ in ${\mathcal{X}}_{t}$ is reduced, has at most one singular point at $s(t)$, and furthermore, that the Milnor number of this hypersurface at $s(t)$ is independent of $t\in T$ (note that the condition to be reduced is a consequence of isolated singularities as long as the relative dimension is at least 2). In this case, a result of Varchenko [Varchenko2] says that the spectrum of $f_{t}$ at $s(t)$ is independent of $t\in T$; in particular, the minimal exponent $\widetilde{\alpha}_{s(t)}(f_{t})$ is independent of $t\in T$.

With the above notation, if $\alpha=\widetilde{\alpha}_{s(t)}(f_{t})$ for $t\in T$, then for every nonnegative integer $p$ and every $\gamma\in{\mathbf{Q}}\cap(0,1]$ with $p+\gamma\leq\alpha+1$, the subscheme of ${\mathcal{X}}$ defined by $I_{p}(f^{\gamma})$ is finite and flat over $T$ (possibly empty). Moreover, for every $t\in T$, we have

We deduce this result from the above-mentioned result of Varchenko on the constancy of the spectrum and a result due to Jung, Kim, Yoon, and Saito [Saito_et_al] that allows us to relate the Hodge ideals satisfying the condition in the theorem with the Hodge filtration in Steenbrink’s mixed Hodge structure on the cohomology of the Milnor fiber. Finally, for the proof of Theorem 1.1 we also need the Restriction Theorem for Hodge ideals from [MP1], as well as a version of the Thom-Sebastiani property for certain Hodge ideals. Regarding the latter property, in our setting it is enough to use a Thom-Sebastiani type result for some related ideals, Saito’s microlocal multiplier ideals; this property was proved by Maxim, Saito, and Schürmann in [MSS].

Our main result gives a lower bound for the difference between the minimal exponent of $f$ and the minimal exponent of a restriction of $f$ to a smooth hypersurface, in terms of Teissier’s invariant $\theta_{P}(f)$. Our last result is an upper bound for the same difference in terms of multiplicity, when we restrict to a general hypersurface. We note that in this result we do not require isolated singularities.

Let $X$ be a smooth complex algebraic variety with ${\rm dim}(X)\geq 2$, $f\in\mathcal{O}_{X}(X)$ nonzero, and $P\in X$ such that $f(P)=0$. If $H$ is a general hypersurface in $X$ containing $P$, then

The case when $f$ has an isolated singularity at $P$ is a consequence of a stronger bound proved by Loeser in [Loeser]. We deduce the general case from this one by making use of a result from [MP2]. As a consequence of Theorem 1.5 is that we get conditions, in terms of the minimal exponent $\widetilde{\alpha}_{P}(f)$, that guarantee that successive general hyperplane sections through $P$ have rational singularities (see Corollary 6.1). Another application concerns a characterization of singular points with maximal minimal exponent (see Corollary 6.3).

The paper is structured as follows. In the next section we review briefly the Hodge ideals and the microlocal multiplier ideals, the connection between them, and the corresponding characterization of minimal exponents. In Section 3 we discuss the behavior of Hodge ideals under finite maps and prove Theorem 1.3, as well as its partial converse in the case of Galois finite covers. In Section 4 we recall the relevant result from [Saito_et_al] and use it to relate certain jumping numbers for Hodge ideals to the spectrum. In particular, we prove Theorem 1.4. We combine these results to give the proof of Theorem 1.1 in Section 5. The last section of the paper is devoted to the proof of the bound in the opposite direction in Theorem 1.5 above and of the above-mentioned applications.

We are grateful to Eva Elduque for many discussions related to this project. We would like to thank Sebastián Olano and Jakub Witaszek for a conversation that led to Corollary 6.3. We are also indebted to the anonymous referees for the careful reading of the paper and for many useful comments.

Hodge ideals and minimal exponents

In this section we review some basic facts about Hodge ideals and their connection with minimal exponents and microlocal multiplier ideals, following [MP1] and [MP2]. Let $X$ be a smooth $n$-dimensional complex algebraic variety and $f\in\mathcal{O}_{X}(X)$ a nonzero regular function. We denote by $\mathcal{D}_{X}$ the sheaf of differential operators on $X$.

For every positive rational number $\alpha$, we have a left $\mathcal{D}_{X}$-module

This is a free module of rank 1 over the sheaf $\mathcal{O}_{X}[1/f]$, generated by the element $f^{-\alpha}$, with differential operators acting in the expected way: if $D$ is a derivation on $\mathcal{O}_{X}$, then

Since $\mathcal{M}(f^{-\alpha})$ is a filtered direct summand of a mixed Hodge module in the sense of [Saito-MHM], it carries a canonical filtration $F_{\bullet}\mathcal{M}(f^{-\alpha})$, compatible with the filtration on $\mathcal{D}_{X}$ given by order of differential operators. The Hodge ideals \big{(}I_{p}(f^{\alpha})\big{)}_{p\geq 0} describe this filtration.

In what follows we will only be interested in the case when $f$ defines a reduced divisor $D$. In this case, the Hodge ideals are given by

(we note that $I_{p}(f^{\alpha})$ was denoted by $I_{p}(\alpha D)$ in [MP1]).

It is sometimes convenient to also consider the right $\mathcal{D}_{X}$-module corresponding to $\mathcal{M}(f^{-\alpha})$. Recall that there is an equivalence of categories between left and right $\mathcal{D}_{X}$-modules such that if $\mathcal{M}$ is a left $\mathcal{D}_{X}$-module, the $\mathcal{O}_{X}$-module underlying the right $\mathcal{D}_{X}$-module corresponding to $\mathcal{M}$ is $\omega_{X}\otimes_{\mathcal{O}_{X}}\mathcal{M}$; see [HTT, Section 1.2]. We denote by $\mathcal{M}_{r}(f^{-\alpha})$ the right $\mathcal{D}_{X}$-module corresponding to $\mathcal{M}(f^{-\alpha})$. The filtration on $\mathcal{M}(f^{-\alpha})$ induces a filtration on $\mathcal{M}_{r}(f^{-\alpha})$, with the convention

We next recall the $V$-filtration associated to $f$. Let $\iota\colon X\to X\times{\mathbf{A}}^{1}$ be the graph embedding given by \iota(x)=\big{(}x,f(x)\big{)}. We denote the standard coordinate on ${\mathbf{A}}^{1}$ by $t$. The $\mathcal{D}$-module theoretic push-forward $B_{f}:=\iota_{+}\mathcal{O}_{X}$ of $\mathcal{O}_{X}$ can be described as

(see [HTT, Example 1.3.5]). We thus have an $\mathcal{O}_{X}$-basis of $B_{f}$ given by $\partial_{t}^{j}\delta$, for $j\geq 0$, where we put $\delta=1\otimes 1\in B_{f}$. The action of $t$ on the elements of this basis is given by

Every $V^{\gamma}B_{f}$ is a coherent module over $\mathcal{D}_{X}[t,\partial_{t}t]$.

$t\cdot V^{\gamma}B_{f}\subseteq V^{\gamma+1}B_{f}$ for all $\gamma\in{\mathbf{Q}}$, with equality if $\gamma>0$.

$\partial_{t}\cdot V^{\gamma}B_{f}\subseteq V^{\gamma-1}B_{f}$ for all $\gamma\in{\mathbf{Q}}$.

For every $\gamma\in{\mathbf{Q}}$, the operator $\partial_{t}t-\gamma$ on ${\rm Gr}_{V}^{\gamma}:=V^{\gamma}B_{f}/V^{>\gamma}B_{f}$ is nilpotent, where $V^{>\gamma}B_{f}=\bigcup_{\gamma^{\prime}>\gamma}V^{\gamma^{\prime}}B_{f}$.

It follows from property iv) above that if $\alpha\neq 1$, then $t\partial_{t}$ is invertible on ${\rm Gr}_{V}^{\alpha}$; in particular, $\partial_{t}\colon{\rm Gr}_{V}^{\alpha}\to{\rm Gr}_{V}^{\alpha-1}$ is injective. This implies that if $u\in B_{f}$ is such that $\partial_{t}u\in V^{>0}B_{f}$, then $u\in V^{>1}B_{f}$.

In particular, the $V$-filtration on $B_{f}$ induces a $V$-filtration $(V^{\gamma}\mathcal{O}_{X})_{\gamma\in{\mathbf{Q}}}$ on $\mathcal{O}_{X}$ via the inclusion $\mathcal{O}_{X}\hookrightarrow B_{f}$ that maps $h$ to $h\delta$. Saito introduced in [Saito_microlocal] a microlocal version of the $V$-filtration. This in turn induces the microlocal $V$-filtration $(\widetilde{V}^{\gamma}\mathcal{O}_{X})_{\gamma\in{\mathbf{Q}}}$ on $\mathcal{O}_{X}$, that we can describe via the usual $V$-filtration, as follows. For $\gamma\leq 0$, we have $\widetilde{V}^{\gamma}\mathcal{O}_{X}=\mathcal{O}_{X}$. If $\gamma>0$, write $\gamma=p+\alpha$, for an integer $p$ and $\alpha\in(0,1]$ (hence $p=\lceil\alpha\rceil-1$). With this notation, $\widetilde{V}^{\gamma}\mathcal{O}_{X}$ consists of those regular functions $h\in\mathcal{O}_{X}$, with the property that there are regular functions $h_{0},\ldots,h_{p-1}\in\mathcal{O}_{X}$ such that

Whenever the function $f$ is not clear from the context, we write $\widetilde{V}^{\gamma}\mathcal{O}_{X}(f)$ for $\widetilde{V}^{\gamma}\mathcal{O}_{X}$.

A basic fact is that $\delta\in V^{>0}B_{f}$. For example, this follows from Sabbah’s description of the $V$-filtration in terms of $b$-functions (see [Sabbah]) and the fact, due to Kashiwara [Kashiwara], that all the roots of the $b$-function of $f$ are negative rational numbers.

The microlocal $V$-filtration on $\mathcal{O}_{X}$ is a rational, decreasing, exhaustive, left-continuous and discrete filtration by coherent ideals. With the above definition, the only assertion that is not clear is that $\widetilde{V}^{\gamma_{1}}\mathcal{O}_{X}\subseteq\widetilde{V}^{\gamma_{2}}\mathcal{O}_{X}$ if $\gamma_{1}>\gamma_{2}$. In order to check this, we can easily reduce to the case when $\gamma_{2}=p$ is a positive integer and $\gamma_{1}=p+\alpha$, for a rational number $\alpha\in(0,1]$. In order to prove the inclusion, suppose that $h\in\widetilde{V}^{p+\alpha}\mathcal{O}_{X}$, hence there are $h_{0},\ldots,h_{p-1}\in\mathcal{O}_{X}$ such that

Since $h_{0}\delta\in V^{>0}B_{f}$ by Remark 2.2, it follows that if

then $\partial_{t}u\in V^{>0}B_{f}$, hence $u\in V^{>1}B_{f}\subseteq V^{1}B_{f}$ by Remark 2.1. This implies that $h\in\widetilde{V}^{p}\mathcal{O}_{X}$.

The microlocal multiplier ideals of $f$ are defined by

(see [Saito-MLCT] or [MSS]). The shift in the definition is convenient since it implies that for $\gamma<1$, the microlocal multiplier ideal $\widetilde{\mathcal{J}}(f^{\gamma})$ coincides with the usual multiplier ideal $\mathcal{J}(f^{\gamma})$ (this is a consequence of a theorem of Budur and Saito [BS] relating multiplier ideals to the $V$-filtration on $\mathcal{O}_{X}$). In what follows we will not shift by $\epsilon$ since, as we will see shortly, this indexing matches the one for Hodge ideals, but we will still refer to the elements of the filtration $(\widetilde{V}^{\gamma}\mathcal{O}_{X})_{\gamma\in{\mathbf{Q}}}$ as microlocal multiplier ideals.

The following is the main result relating Hodge ideals and microlocal multiplier ideals. It was proved in [Saito-MLCT, Theorem 1] for $\gamma\in{\mathbf{Z}}_{>0}$ and in [MP2, Theorem A’] in general.

If $f$ defines a reduced divisor, then for every $\gamma=p+\alpha$, with $p\in{\mathbf{Z}}_{\geq 0}$ and $\alpha\in{\mathbf{Q}}\cap(0,1]$, we have

In particular, it follows from the theorem that given $P\in X$ with $f(P)=0$, we have $I_{p}(f^{\alpha})=\mathcal{O}_{X}$ around $P$ if and only if $\widetilde{V}^{\gamma}\mathcal{O}_{X}=\mathcal{O}_{X}$ around $P$. It was shown by Saito in [Saito-MLCT] that if $\widetilde{\alpha}_{P}(f)$ is the minimal exponent of $f$ at $P$, then

(see also [MP2, Remark 6.13]). As a consequence, we get the fact that if $f$ defines a reduced divisor, then

We note that the definition of the minimal exponent that is used in the above results is in terms of the Bernstein-Sato polynomial of $f$. We will only need the above characterization and thus do not recall the precise definition. For more details and basic properties of the minimal exponent that follow from the above characterization in terms of Hodge ideals, we refer to [MP2]. The fact that for isolated singularities the minimal exponent coincides with the Arnold exponent follows from [Malgrange2]. We will discuss in more detail the case of isolated singularities in Section 4.

For us it will be important that Hodge ideals are equal to microlocal multiplier ideals also in an interval of length 1 starting with the minimal exponent. More precisely, we have the following result. Recall that the Jacobian ideal $J_{f}$ of $f$ is defined as follows: if $x_{1},\ldots,x_{n}$ are algebraic coordinates in an open subset $U$ of $X$, then $J_{f}|_{U}$ is generated by $\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial f}{\partial x_{n}}$ (this definition is independent of the choice of coordinates and thus by gluing the local definitions we get a coherent ideal sheaf of $\mathcal{O}_{X}$).

Let $f$ be a nonzero regular function on the smooth complex algebraic variety $X$, defining a reduced divisor, and $P\in X$ such that $f(P)=0$. Suppose that $\gamma$ is a positive rational number and we write $\gamma=p+\alpha$, with $p=\lceil\gamma\rceil-1$. If $\gamma\leq\widetilde{\alpha}_{P}(f)+1$, then

in a neighborhood of $P$; moreover, in such a neighborhood these ideals contain $(f)+J_{f}$.

Suppose first that $\gamma\leq 1$. In this case $p=0$ and both ideals $I_{0}(f^{\gamma})$ and $\widetilde{V}^{\gamma}\mathcal{O}_{X}$ are equal to the multiplier ideal $\mathcal{J}(f^{\gamma-\epsilon})$ for $0<\epsilon\ll 1$ (for the Hodge ideal this follows from [MP1, Proposition 9.1], while for the microlocal multiplier ideal this follows from the result of Budur and Saito [BS] relating multiplier ideals and the $V$-filtration). These ideals contain $f$ since $(f)=\mathcal{J}(f)\subseteq\mathcal{J}(f^{\gamma-\epsilon})$ for every $\gamma\leq 1$. The fact that $J_{f}$ is contained in $\mathcal{J}(f^{1-\epsilon})$ for $\epsilon>0$ (which, in turn, is contained in $\mathcal{J}(f^{\gamma-\epsilon})$) is proved in [ELSV, Theorem 4.2].

Suppose now that $\gamma>1$, hence $p\geq 1$, and that $\gamma\leq\widetilde{\alpha}_{P}(f)+1$. In this case we have

in a neighborhood of $P$ by (2) and (3). After possibly replacing $X$ by this neighborhood of $P$, we may and will assume that these equalities hold on $X$. If we show that $f\in I_{p}(f^{\alpha})\cap\widetilde{V}^{\gamma}\mathcal{O}_{X}$, then the equality of the ideals in the proposition follows from Theorem 2.5. Then the last assertion follows as well if we show that $J_{f}\subseteq I_{p}(f^{\alpha})$.

The fact that $f\in I_{p}(f^{\alpha})$ has already been noticed in [MP2, Corollary 5.5]. The point is that since $I_{p-1}(f^{\alpha})=\mathcal{O}_{X}$, we have

The fact that $F_{p-1}\mathcal{M}(f^{-\alpha})\subseteq F_{p}\mathcal{M}(f^{-\alpha})$ and the definition of $I_{p}(f^{\alpha})$ then give $f\in I_{p}(f^{\alpha})$. Moreover, since we have $F_{1}\mathcal{D}_{X}\cdot F_{p-1}\mathcal{M}(f^{-\alpha})\subseteq F_{p}\mathcal{M}(f^{-\alpha})$, we see that if $x_{1},\ldots,x_{n}$ are local algebraic coordinates on $X$, then

hence $J_{f}\subseteq I_{p}(f^{\alpha})$.

In order to complete the proof, it is enough to show that $f\in\widetilde{V}^{\gamma}\mathcal{O}_{X}$. Since $\widetilde{V}^{\gamma-1}\mathcal{O}_{X}=\mathcal{O}_{X}$, it follows that we have in $V^{\alpha}B_{f}$ an element of the form

In this case we have in $V^{\alpha}B_{f}$ also the element

hence $f\in\widetilde{V}^{\gamma}\mathcal{O}_{X}$. ∎

Hodge ideals under finite maps

In this section we consider the behavior of Hodge ideals under finite surjective morphisms. Let us fix such a morphism $\pi\colon Y\to X$ between smooth complex $n$-dimensional algebraic varieties. Since we deal with push-forward of $\mathcal{D}$-modules, in this section we typically consider right $\mathcal{D}$-modules.

Recall that $\pi^{*}(\mathcal{D}_{X})$ has a canonical structure of \big{(}\mathcal{D}_{Y},\pi^{-1}(\mathcal{D}_{X})\big{)}-bimodule; as such, it is denoted by $\mathcal{D}_{Y\to X}$. As for every proper morphism, we have an induced push-forward morphism between the corresponding derived categories of (quasi-coherent) right $\mathcal{D}$-modules given by $R\pi_{*}(-\otimes^{L}_{\mathcal{D}_{Y}}\mathcal{D}_{Y\to X})$, see [HTT, Chapters 1.3 and 2.5]. The case of finite maps is easier: first, $\pi_{*}$ is exact on quasi-coherent $\mathcal{O}_{Y}$-modules. Second, $\mathcal{D}_{Y\to X}$ is a flat left $\mathcal{D}_{Y}$-module (see [Bjork, Theorem 2.11.10] or [FiniteMaps, Proposition 2.10]). This implies that the functor between derived categories is induced by the exact functor $\pi_{+}=\pi_{*}(-\otimes_{\mathcal{D}_{Y}}\mathcal{D}_{Y\to X})$ between the corresponding Abelian categories of right $\mathcal{D}$-modules. Note that if $\pi$ is étale, then $\mathcal{D}_{Y}=\pi^{*}(\mathcal{D}_{X})$ and the functor $\pi_{+}$ is equal to $\pi_{*}$ on right $\mathcal{D}_{Y}$-modules.

We have a morphism of left $\mathcal{D}_{Y}$-modules $\mathcal{D}_{Y}\to\mathcal{D}_{Y\to X}$ that maps $1$ to $\pi^{*}(1)$. This induces a canonical morphism of $\mathcal{O}_{X}$-modules $\pi_{*}(\mathcal{M})\to\pi_{+}(\mathcal{M})$, which is an isomorphism if $\pi$ is étale.

We also write $i_{+}$ for the push-forward functor between the Abelian categories of $\mathcal{D}$-modules when $i\colon U\hookrightarrow X$ is the open immersion corresponding to the complement of an effective divisor in $X$. We thus have $i_{+}\mathcal{M}=i_{*}\mathcal{M}$ for every right $\mathcal{D}_{U}$-module $\mathcal{M}$. It is straightforward to see (and well-known) that if $j\colon V\hookrightarrow Y$ is the open immersion with $V=\pi^{-1}(U)$ and $\varphi\colon V\to U$ is the induced morphism, then we have a canonical isomorphism of functors $\pi_{+}\circ j_{+}\simeq i_{+}\circ\varphi_{+}$.

If $\pi\colon Y\to X$ is a finite surjective morphism between smooth complex algebraic varieties and if $\mathcal{M}$ is a right $\mathcal{D}_{Y}$-module having no torsion as an $\mathcal{O}_{Y}$-module, then $\pi_{+}(\mathcal{M})$ has no torsion as an $\mathcal{O}_{X}$-module.

The assertion is local on $X$, hence we may assume that $X$ (and thus also $Y$) is affine. By generic smoothness, we can find a nonzero $h\in\mathcal{O}_{X}(X)$ such that $\pi$ is étale over the open subset $U=(h\neq 0)$. Let $V=\pi^{-1}(U)$ and $j\colon V\hookrightarrow Y$ and $i\colon U\hookrightarrow X$ be the corresponding open immersions. Note that we have a canonical morphism $\mathcal{M}\to j_{+}(\mathcal{M}|_{V})$, which is injective since $\mathcal{M}$ has no torsion as an $\mathcal{O}_{Y}$-module. By taking the direct image, we get an injective morphism

where $\varphi\colon V\to U$ is the induced morphism; therefore it is enough to show that the right-hand side has no torsion. Note that as an $\mathcal{O}_{X}$-module, $i_{+}\varphi_{+}(\mathcal{M}|_{V})$ is simply $i_{*}\varphi_{*}(\mathcal{M}|_{V})$, since $\varphi$ is étale. Since $\mathcal{M}|_{V}$ is an $\mathcal{O}_{V}$-module without torsion, it follows that $i_{*}\varphi_{*}(\mathcal{M}|_{V})$ is an $\mathcal{O}_{X}$-module without torsion. This completes the proof. ∎

In what follows, we will make use of Saito’s theory of pure and mixed Hodge modules, for which we refer to [Saito-MHP] and [Saito-MHM]. Recall that a mixed Hodge module has an underlying filtered (right) $\mathcal{D}$-module. For example, on a smooth $n$-dimensional variety $X$, we have the pure Hodge module ${\mathbf{Q}}_{X}^{H}[n]$, whose underlying $\mathcal{D}_{X}$-module is $\omega_{X}$ (the right $\mathcal{D}_{X}$-module corresponding to $\mathcal{O}_{X}$), with the filtration given by $F_{p-n}\omega_{X}=\omega_{X}$ for $p\geq 0$ and $F_{p-n}\omega_{X}=0$, otherwise.

Suppose now that $\varphi\colon W\to X$ is a morphism between smooth varieties which is either finite and surjective or an open immersion, given by the complement of an effective divisor. If $(\mathcal{M},F_{\bullet})$ is a filtered $\mathcal{D}_{W}$-module on $W$ that underlies a mixed Hodge module $M$, then we have by [Saito-MHM] a mixed Hodge module that we will denote by $\varphi_{+}M$ and whose underlying filtered $\mathcal{D}_{X}$-module we will denote by $\varphi_{+}(\mathcal{M},F_{\bullet})$. The corresponding $\mathcal{D}_{X}$-module is just $\varphi_{+}(\mathcal{M})$, but the description of the filtration is rather subtle. When $W$ is the complement of the hypersurface defined by $f\in\mathcal{O}_{X}(X)$ and $M={\mathbf{Q}}_{W}^{H}[n]$, we get the filtration on $\mathcal{O}_{X}[1/f]$ described by the Hodge ideals of $f$. One easy case is that when $\varphi$ is finite and étale, in which case $F_{p}(\varphi_{+}\mathcal{M})=\varphi_{*}(F_{p}\mathcal{M})$ for all $p$.

Given a finite surjective morphism $\pi\colon Y\to X$ of smooth $n$-dimensional algebraic varieties, we have a canonical morphism of mixed Hodge modules

that commutes with restriction to open subsets of $X$. At the level of $\mathcal{D}_{X}$-modules, this is given by the composition

where the first morphism maps a form $\eta\in\omega_{X}$ to its pull-back $\pi^{*}(\eta)\in\omega_{Y}$ and the second morphism is the one in Remark 3.1.

The next lemma provides the ingredient to relate Hodge ideals under finite maps. Suppose that $\pi\colon Y\to X$ is a finite surjective morphism between $n$-dimensional smooth varieties, $f\in\mathcal{O}_{X}(X)$ is nonzero, and $g=f\circ\pi\in\mathcal{O}_{Y}(Y)$. Recall that associated to $f$ and $g$ we have the filtered right $\mathcal{D}$-modules $\mathcal{M}_{r}(f^{-\alpha})$ and $\mathcal{M}_{r}(g^{-\alpha})$ on $X$ and $Y$, respectively. Note that for every $\alpha\in{\mathbf{Q}}_{>0}$, we have a canonical morphism of $\mathcal{O}_{X}$-modules

that maps $u=f^{-\alpha}\frac{\eta}{f^{m}}$, with $\eta\in\omega_{Y}$, to $\pi^{*}(u):=g^{-\alpha}\frac{\pi^{*}(\eta)}{g^{m}}$.

With the above notation, for every $\alpha\in{\mathbf{Q}}_{>0}$, the map $\tau$ given by the composition

where the first map is the one in (5) and the second map is the one in Remark 3.1, is an injective strict morphism of filtered $\mathcal{D}_{X}$-modules.

For the proof, we will need to make use of the definition of the filtrations on $\mathcal{M}_{r}(f^{-\alpha})$ and $\mathcal{M}_{r}(g^{-\alpha})$ in [MP1, Section 2]. Let $X^{\prime}=X\smallsetminus V(f)$ and $Y^{\prime}=Y\smallsetminus V(g)$. Choose an integer $m\geq 2$ such that $m\alpha\in{\mathbf{Z}}$ and let

so that we have a diagram with Cartesian squares

with $i$ and $j$ open immersions and $p$ and $q$ finite étale morphisms. We have by (4) a canonical morphism of mixed Hodge modules

that induces after applying $i_{+}p_{+}$ and taking the underlying filtered $\mathcal{D}_{X}$-modules, a morphism of filtered $\mathcal{D}_{X}$-modules

By taking the suitable eigenspace with respect to the ${\mathbf{Z}}/m{\mathbf{Z}}$ action on both sides, we obtain, for $k=1$, a morphism of filtered $\mathcal{D}_{X}$-modules

Checking that it is given by the composition in the statement is an easy exercise. Strictness follows from the fact that for every morphism of mixed Hodge modules, the underlying morphism of filtered $\mathcal{D}$-modules is strict. The fact that $\tau$ is injective is clear if $\pi$ is étale; the general case follows by restricting to an open subset $U$ of $X$ such that $\pi$ is étale over $U$ and using the fact that as an $\mathcal{O}_{X}$-module, $\mathcal{M}_{r}(f^{-\alpha})$ has no torsion. ∎

In order to describe the filtration on $\pi_{+}\mathcal{M}_{r}(g^{-\alpha})$, we take the usual approach, by factoring $\pi$ as $q\circ\rho$, where $\rho\colon Y\to Y\times X$ is the graph embedding given by \rho(y)=\big{(}y,\pi(y)\big{)} and $q\colon Y\times X\to X$ is the projection onto the second component. Let us assume that we have algebraic coordinates $x_{1},\ldots,x_{n}$ defined on $X$ (we can always reduce to this case by taking a suitable cover of $X$). Let $\pi_{i}=\pi^{*}(x_{i})\in\mathcal{O}_{Y}(Y)$. If $\mathcal{M}$ is a right $\mathcal{D}_{Y}$-module, then the $\mathcal{D}$-module push-forward via $\rho$ is easy to compute: we have an isomorphism

where on the right-hand side a function $h\in\mathcal{O}_{X}$ acts via

and a derivation $D\in{\rm Der}_{{\mathbf{C}}}(\mathcal{O}_{Y})$ acts by

where we use the multi-index notation and $e_{1},\ldots,e_{n}$ is the standard basis of ${\mathbf{Z}}^{n}$. Moreover, if $(\mathcal{M},F_{\bullet})$ is a filtered $\mathcal{D}_{Y}$-module, then via the isomorphism (6) we have

where for $\beta=(\beta_{1},\ldots,\beta_{n})$, we put $|\beta|=\sum_{i}\beta_{i}$.

On the other hand, the $\mathcal{D}$-module push-forward via $q$ is computed by the relative Spencer complex. Given a right $\mathcal{D}_{Y\times X}$-module $\mathcal{N}$, the relative Spencer complex of $\mathcal{N}$ is the complex

where $\mathcal{T}_{Y}$ is the tangent sheaf of $Y$ and $p\colon Y\times X\to Y$ is the projection onto the first component. The map $d_{1}$, which is the only one we will need, is given by right multiplication.

If $\mathcal{M}$ is a $\mathcal{D}_{Y}$-module, then we have a canonical isomorphism

Moreover, if $(\mathcal{M},F_{\bullet})$ is the filtered $\mathcal{D}_{Y}$-module underlying a mixed Hodge module, then via this isomorphism, the filtration on $\pi_{+}\mathcal{M}$ is the quotient filtration induced by the filtration on $\rho_{+}\mathcal{M}$ described in (8).

If we are in the setting of Lemma 3.3, then via the isomorphism

the morphism $\tau$ maps $u\in\mathcal{M}_{r}(f^{-\alpha})$ to the class of $\pi^{*}(u)\otimes 1\in\rho_{+}\mathcal{M}_{r}(g^{-\alpha})$. Indeed, since as an $\mathcal{O}_{X}$-module $\pi_{+}\mathcal{M}_{r}(g^{-\alpha})$ has no torsion by Lemma 3.2, in order to check the assertion we may restrict to on open subset $U$ of $X$ such that $\pi$ is étale over $U$; in this case the assertion follows via an easy computation.

Let $\pi\colon Y\to X$ be a finite surjective morphism between smooth $n$-dimensional complex algebraic varieties. If $f\in\mathcal{O}_{X}(X)$ is nonzero and $g=\pi^{*}(f)\in\mathcal{O}_{Y}(Y)$, then for every $\alpha\in{\mathbf{Q}}_{>0}$ and $k\in{\mathbf{Z}}$, we have the following inclusion

Note that if we assume that $f$ and $g$ define reduced divisors, then by passing from right to left $\mathcal{D}$-modules and using the definition of Hodge ideals, we obtain the assertion in Theorem 1.3.

After taking a suitable open cover of $X$, we may assume that we have algebraic coordinates defined on $X$. If $u\in\mathcal{M}_{r}(f^{-\alpha})$ is such that $\pi^{*}(u)\in F_{k}\mathcal{M}_{r}(g^{-\alpha})$, then it follows from the above description of the filtration on $\pi_{+}\mathcal{M}_{r}(g^{-\alpha})$ that the class of $\pi^{*}(u)\otimes 1$ in {\rm coker}\big{(}\rho_{+}\mathcal{M}_{r}(g^{-\alpha})\otimes p^{*}(\mathcal{T}_{Y})\to\rho_{+}\mathcal{M}_{r}(g^{-\alpha})\big{)} lies in F_{k}\big{(}\pi_{+}\mathcal{M}_{r}(g^{-\alpha})\big{)}. Since this is equal to $\tau(u)$ and $\tau$ is strict by Lemma 3.3, we conclude that $u\in F_{k}\mathcal{M}_{r}(f^{-\alpha})$. ∎

Our next goal is to understand how far the inclusion in Theorem 3.5 is from being an equality. We will do this under the extra assumption that $\pi$ is Galois. This result will not be used in the following sections.

Let $\pi\colon Y\to X$ be a finite surjective morphism between smooth complex algebraic varieties. Recall that $\pi$ is Galois if the corresponding field extension ${\mathbf{C}}(X)\hookrightarrow{\mathbf{C}}(Y)$ is normal (hence Galois, since we are in characteristic ). Suppose that this is the case and let $G$ be the Galois group of this field extension. Since $Y$ is the integral closure of $X$ in ${\mathbf{C}}(Y)$, we have an induced action of $G$ on $Y$ and $\pi$ is the quotient morphism with respect to this action.

Let $\pi\colon Y\to X$ be a Galois finite surjective morphism between smooth $n$-dimensional complex algebraic varieties. Suppose that $f\in\mathcal{O}_{X}(X)$ is nonzero and let $g=\pi^{*}(f)\in\mathcal{O}_{Y}(Y)$. For every $\alpha\in{\mathbf{Q}}_{>0}$ and $k\in{\mathbf{Z}}$, we put

We then have for every $k\in{\mathbf{Z}}$

After taking a suitable open cover of $X$, we may and will assume that $X$ (hence also $Y$) is affine and we have algebraic coordinates $x_{1},\ldots,x_{n}$ defined on $X$. We may thus use the description of the filtration on $\pi_{+}\mathcal{M}_{r}(g^{-\alpha})$ that we have discussed.

Note that the action of $G$ on $Y$ induces a $G$-action on $\mathcal{O}_{Y}(Y)$. Moreover, it also induces a $G$-action on $\Gamma(Y,\omega_{Y})$ that makes $\omega_{Y}$ a $G$-equivariant sheafSince $G$ is a finite group and $Y$ is affine, this simply means that the scalar multiplication $\mathcal{O}_{Y}(Y)\times\Gamma(Y,\omega_{Y})\to\Gamma(Y,\omega_{Y})$ is compatible with the $G$-actions.. Moreover, its subspace of invariant sections is precisely the image of the map

given by the pull-back of differential forms (see [Brion, Theorem 1]). Since $g$ is a $G$-invariant section of $\mathcal{O}_{Y}$, it follows that the sheaf $\mathcal{M}_{r}(g^{-\alpha})$ has an induced structure of $G$-equivariant sheaf; moreover, its subspace of $G$-invariant sections is the image of the pull-back map

We also have an induced $G$-action on \Gamma(X,\pi_{+}\mathcal{M}_{r}(g^{-\alpha})\big{)}. In order to see this, note that we have a natural $G$-action on \Gamma\big{(}Y\times X,\rho_{+}\mathcal{M}_{r}(g^{-\alpha})\big{)}=\Gamma\big{(}Y,\mathcal{M}_{r}(g^{-\alpha})\big{)}\otimes_{{\mathbf{C}}}{\mathbf{C}}[\partial_{x_{1}},\ldots,\partial_{x_{n}}], where $G$ acts trivially on ${\mathbf{C}}[\partial_{x_{1}},\ldots,\partial_{x_{n}}]$. The $G$-action on $Y$ induces an action of $G$ on $\Gamma(Y,\mathcal{T}_{Y})$ such that the multiplication map $\rho_{+}\mathcal{M}_{r}(g^{-\alpha})\otimes\mathcal{T}_{Y}\to\rho_{+}\mathcal{M}_{r}(g^{-\alpha})$ is compatible with the $G$-actions. We thus get an induced $G$-action on the cokernel of this map and thus on \Gamma(X,\pi_{+}\mathcal{M}_{r}(g^{-\alpha})\big{)} via the isomorphism (9). It also follows from the description of the canonical morphism

in Remark 3.4 that the image of $\tau$ lands in the subspace of $G$-invariant sections. A key point is that due to the fact that $g$ is $G$-invariant, the filtration on \Gamma\big{(}Y,\mathcal{M}_{r}(g^{-\alpha})\big{)} is preserved by the $G$-action and therefore so is the filtration on \Gamma\big{(}X,\pi_{+}\mathcal{M}_{r}(g^{-\alpha})\big{)}.

Given a complex vector space $V$ with a $G$-action, we consider the linear map $A=A_{V}\colon V\to V$ given by $A(v)=\frac{1}{|G|}\sum_{g\in G}gv$. Note that $A(v)$ lies in the subspace $V^{G}$ of invariant elements for every $v\in V$ and $A(v)=v$ if $v\in V^{G}$.

The inclusion “$\supseteq$” in (10) is clear: it follows from Theorem 3.5 that for every $j$ we have $F^{\prime}_{j}\mathcal{M}_{r}(f^{-\alpha})\subseteq F_{j}\mathcal{M}_{r}(f^{-\alpha})$ and thus

We now prove the reverse inclusion. Given a global section $u$ of $F_{k}\mathcal{M}_{r}(f^{-\alpha})$, it follows from Lemma 3.3 that $\tau(u)$ is a global section of F_{k}\big{(}\pi_{+}\mathcal{M}_{r}(g^{-\alpha})\big{)}. This means that we can find global sections $w_{\beta}$ of $F_{k-|\beta|}\mathcal{M}_{r}(g^{-\alpha})$ for $\beta\in{\mathbf{Z}}_{\geq 0}^{n}$, with $w_{\beta}=0$ for all but finitely many $\beta$, such that

If we put w^{\prime}_{\beta}=A(w_{\beta})\in\Gamma(Y,F_{k-|\beta|}\mathcal{M}_{r}(g^{-\alpha})\big{)}^{G}, we see that

Since each $w^{\prime}_{\beta}$ is $G$-invariant, it follows that we can write $w^{\prime}_{\beta}=\pi^{*}(u_{\beta})$ for some global section $u_{\beta}$ of $F^{\prime}_{k-|\beta|}\mathcal{M}_{r}(f^{-\alpha})$. We now claim that for every $\beta$, $\pi^{*}(u_{\beta})\otimes\partial_{x}^{\beta}$ and $\pi^{*}(u_{\beta}\partial_{x}^{\beta})\otimes 1$ have the same image in \Gamma\big{(}X,\pi_{+}\mathcal{M}_{r}(g^{-\alpha})\big{)}. If this is the case, it follows that $\tau(u)=\tau\left(\sum_{\beta}u_{\beta}\partial_{x}^{\beta}\right)$. Since $\tau$ is injective by Lemma 3.3, we conclude that $u\in\sum_{i\geq 0}F^{\prime}_{k-i}\mathcal{M}_{r}(f^{-\alpha})\cdot F_{i}\mathcal{D}_{X}$.

In order to prove the above claim, since $\pi_{+}\mathcal{M}_{r}(g^{-\alpha})$ is an $\mathcal{O}_{X}$-module with no torsion by Lemma 3.2, it is enough to prove it on some nonempty open subset of $X$. We thus may and will assume that $\pi$ is étale. In this case, if $\pi_{i}=x_{i}\circ\pi$, then $\pi_{1},\ldots,\pi_{n}$ give an algebraic system of coordinates on $Y$ and we have a corresponding system of derivations $\partial_{\pi_{1}},\ldots,\partial_{\pi_{n}}$. Of course, arguing by induction on $|\beta|$, it is enough to show that for every global section $\eta$ of $\mathcal{M}_{r}(f^{-\alpha})$, the elements $\pi^{*}(\eta\partial_{x_{i}})\otimes 1$ and $\pi^{*}(\eta)\otimes\partial_{x_{i}}$ in $\rho_{+}\mathcal{M}_{r}(g^{-\alpha})$ have the same image in $\pi_{+}\mathcal{M}_{r}(g^{-\alpha})$. This follows from the fact that by (7), the map $\rho_{+}\mathcal{M}_{r}(g^{-\alpha})\otimes\mathcal{T}_{Y}\to\rho_{+}\mathcal{M}_{r}(g^{-\alpha})$ maps $\pi^{*}(\eta)\otimes\partial_{\pi_{i}}$ to

This completes the proof of (10) and the last assertion in the theorem is an immediate consequence. ∎

We now give a consequence of the results in Theorems 1.3 and 3.6 to minimal exponents.

Let $\pi\colon Y\to X$ be a finite surjective morphism between smooth complex algebraic varieties and let $K_{Y/X}$ be the effective divisor on $Y$ locally defined by the determinant of a Jacobian matrix of $\pi$. If $0\neq f\in\mathcal{O}_{X}(X)$ and $g=f\circ\pi$ both define reduced divisors, then the following hold for every $P$ in $X$ with $f(P)=0$, every nonnegative integer $p$, and every $\alpha\in{\mathbf{Q}}\cap(0,1]$:

If $\mathcal{O}_{Y}(-K_{Y/X})\subseteq I_{p}(g^{\alpha})$ in a neighborhood of the fiber $\pi^{-1}(P)$, then $\widetilde{\alpha}_{P}(f)\geq p+\alpha$.

If $\pi$ is Galois and the hypersurface defined by $f$ is not smooth at $P$, then the converse of i) holds: if $\widetilde{\alpha}_{P}(f)\geq p+\alpha$, then $\mathcal{O}_{Y}(-K_{Y/X})\subseteq I_{p}(g^{\alpha})$ in a neighborhood of the fiber $\pi^{-1}(P)$.

Note that a similar assertion holds for log canonical thresholds of arbitrary regular functions in the setting of a finite surjective morphism between smooth varieties, as above: we have ${\rm lct}_{P}(f)>\lambda$ if and only if $\mathcal{O}_{Y}(-K_{Y/X})$ is contained in the multiplier ideal $\mathcal{J}(g^{\lambda})$ in a neighborhood of the fiber $\pi^{-1}(P)$. This follows from the fact that ${\rm lct}_{P}(f)>\lambda$ if and only if $\mathcal{J}(f^{\lambda})=\mathcal{O}_{X}$ in a neighborhood of $P$ and the theorem relating the multiplier ideals of $f$ and $g$ (see [Lazarsfeld, Theorem 9.5.42]).