Introduction

Let X be a connected complex projective manifold of dimension \(n\ge 1\). Given any base-point free linear system \(\Lambda \) on X, it follows from the classical Bertini theorem [9] that a general hyperplane H of \(\Lambda \) is smooth. Let \(\varphi \) be a quasi-plurisubharmonic (quasi-psh) function on X. For a general member \(H\in \Lambda \), the multiplier ideal sheaf \({\mathcal {I}}(\varphi |_H)\) makes sense. It is natural to wonder if

$$\begin{aligned} {\mathcal {I}}(\varphi |_H)={\mathcal {I}}(\varphi )|_H \end{aligned}$$
(1.1)

holds for general H. It is well-known that the \({\mathcal {I}}(\varphi |_H)\subseteq {\mathcal {I}}(\varphi )|_H\) direction always holds for a general H, as a consequence of the Ohsawa–Takegoshi \(L^2\)-extension theorem. Conversely, it is easy to construct examples such that the set \({\mathcal {B}}\) of \(H\in \Lambda \) where the equality fails is not contained in any proper Zariski closed subset of \(\Lambda \). A natural question arises: is the set \({\mathcal {B}}\) small in a suitable sense? This kind of problem was first studied by Fujino and Matsumura, see [4, 5]. They proved that the complement of \({\mathcal {B}}\) is dense with respect to the complex topology of \(\Lambda \) (regarded as a projective space). More recently, Meng and Zhou [11] proved that the complement of \({\mathcal {B}}\) has zero Lebesgue measure. In this paper, we prove the following refinement:

Theorem 1.1

There is a pluripolar set \(\Sigma \subseteq \Lambda \) such that for all \(H\in \Lambda {\setminus } \Sigma \), H is smooth and (1.1) holds.

This result affirmatively answers a problem of Boucksom, see [5, Question 1.2]. From the point of view of pluripotential theory, this theorem is quite natural: a small set in pluripotential theory just means a pluripolar set. As shown in [4, Example 3.12], the exceptional set is not contained in a countable union of proper Zariski closed subsets in general, so Theorem 1.1 seems to be the optimal result. We also prove a more general analytic Bertini type result for fibrations Corollary 2.9.

Let us mention a key advantage of Theorem 1.1: our theorem can be applied to a countable family of quasi-psh functions at the same time, see Corollary 2.10. This corollary makes it possible to perform induction on the dimension when studying psh singularities.

Analytic Bertini theorem

In this section, varieties or algebraic varieties mean reduced separated schemes of finite type over \({\mathbb {C}}\).

Definition 2.1

Let Y be a complex projective manifold. A subset \(A\subseteq Y\) is

  1. (1)

    co-pluripolar if \(Y{\setminus } A\) is pluripolar. When \(\dim Y=1\), we also say \(A\subseteq Y\) is co-polar.

  2. (2)

    co-meager if \(Y{\setminus } A\) is contained in a countable union of proper Zariski closed sets.

We say a condition in \(y\in Y\) is satisfied quasi-everywhere if there is a co-pluripolar subset \(Y_0\subseteq Y\) such that the condition is satisfied for \(y\in Y_0\).

Clearly, a co-meager set is co-pluripolar. Both classes are preserved by countable intersections.

Lemma 2.2

Let \(\pi {:}\,Y\rightarrow X\) be a smooth morphism of smooth algebraic varieties. Let \(\varphi \) be a quasi-plurisubharmonic function on X, then

$$\begin{aligned} \pi ^*{\mathcal {I}}(\varphi )= {\mathcal {I}}(\pi ^*\varphi ). \end{aligned}$$
(2.1)

Here \({\mathcal {I}}(\varphi )\) denotes the multiplier ideal sheaf of \(\varphi \) in the sense of Nadel. Observe that as \(\pi \) is flat, \(\pi ^*{\mathcal {I}}(\varphi )\) is a subsheaf of \({\mathcal {O}}_Y\), so in (2.1) equality makes sense, the two sheaves are actually equal, not just isomorphic.

Proof

As pointed out by the referee, \(\pi ^*{\mathcal {I}}(\varphi )\supseteq {\mathcal {I}}(\pi ^*\varphi )\) is proved in [1, Proposition 14.3]. So it suffices to prove the reverse inclusion.

By decomposing \(\pi \) into the composition of an étale morphism and a projection locally, it suffices to deal with the two cases separately. Fix a local section f of \({\mathcal {I}}(\varphi )\).

Assume that \(\pi {:}\,X\times {\mathbb {C}}^n\rightarrow X\) is the natural projection. Fix a volume form \(\mathrm {d}V\) on X. Take the product volume form \(\mathrm {d}V\otimes \mathrm {d}\lambda \) on \(X\times {\mathbb {C}}^n\), where \(\mathrm {d}\lambda \) denotes the Lebesgue measure. It follows from Fubini theorem that \(|\pi ^*f|^2e^{-\pi ^*\varphi }\) is locally integrable with respect to \(\mathrm {d}V\otimes \mathrm {d}\lambda \).

Now assume that \(\pi \) is étale. The change of variable formula shows that \(|\pi ^*f|^2e^{-\pi ^*\varphi }\) is locally integrable. \(\square \)

In Lemma 2.2, we do not really need the algebraic structures on X and Y. For general complex manifolds, it suffices to apply the co-area formula.

We recall the notion of positive metrics on a torsion-free coherent sheaf.

Definition 2.3

Let X be a smooth complex algebraic variety. Let \({\mathcal {F}}\) be a torsion-free (algebraic) coherent sheaf on X. Let \(Z\subseteq X\) be the smallest Zariski closed set such that \({\mathcal {F}}|_{X{\setminus } Z}={\mathcal {O}}_{X{\setminus } Z}(F)\) for some vector bundle F on \(X{\setminus } Z\). A singular Hermitian metric (resp. positive singular Hermitian metric) on \({\mathcal {F}}\) is a singular Hermitian metric (resp. Griffiths positively curved singular Hermitian metric) on F in the sense of [13].

Theorem 2.4

Let X be a connected projective manifold of dimension \(n\ge 1\). Let \(\varphi \) be a quasi-plurisubharmonic function on X. Let \(p{:}\,X\rightarrow {\mathbb {P}}^N\) be a morphism (\(N\ge 1\)). Define

$$\begin{aligned} {\mathcal {G}}:=\left\{ \,H\in |{\mathcal {O}}_{{\mathbb {P}}^N}(1)|{:}\,H':=H\cap X \text { is smooth and } {\mathcal {I}}(\varphi |_{H'})={\mathcal {I}}(\varphi )|_{H'}\,\right\} . \end{aligned}$$

Then \({\mathcal {G}}\subseteq |{\mathcal {O}}_{{\mathbb {P}}^N}(1)|\) is co-pluripolar.

Remark 2.5

Here and in the sequel, we slightly abuse the notation by writing \(H\cap X\) for \(p^{-1}H\), the scheme-theoretic inverse image of H. In other words, \(H\cap X:=H\times _{{\mathbb {P}}^N} X\).

By definition, any \(H\in |{\mathcal {O}}_{{\mathbb {P}}^N}(1)|\) such that \(p^{-1}H=\emptyset \) lies in \({\mathcal {G}}\).

We briefly sketch the argument. We need to show that for quasi-every \(H\in |{\mathcal {O}}_{{\mathbb {P}}^N}(1)|\), the restriction formula (1.1) holds. A standard argument in algebraic geometry allows us to reduce the proof of (1.1) to proving the corresponding equality on global sections, after tensoring with a sufficiently ample line bundle L. In this case, we group the pairs consisting of \(H\in \Lambda \) and points on \(H\cap X\) as a single fibration \(\pi _1{:}\,U\rightarrow \Lambda \). The theory of positivity of direct images allows us to construct a coherent sheaf \({\mathcal {F}}\) on \(\Lambda \) endowed with a positive metric \(h_{{\mathcal {H}}}\) out of \(\pi _1\) and \(\varphi \). By the construction of \(h_{{\mathcal {H}}}\), the locus where the restriction formula (1.1) fails is contained in the singular locus of \(h_{{\mathcal {H}}}\), which finishes the proof.

Proof

Take an ample line bundle L with a smooth Hermitian metric h such that \(c_1(L,h)+\mathrm {dd}^{\mathrm {c}}\varphi \ge 0\), where \(c_1(L,h)\) is the first Chern form of (Lh), namely the curvature form of h. Let \({\mathcal {L}}\) be the invertible sheaf corresponding to L. We introduce \(\Lambda :=|{\mathcal {O}}_{{\mathbb {P}}^N}(1)|\) to simplify our notations.

Step 2.6

We prove that the following set is co-pluripolar:

$$\begin{aligned} \begin{array}{c} {\mathcal {G}}_{{\mathcal {L}}} :=\{H\in \Lambda : H\cap X \text { is smooth and } H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi |_{H\cap X}))\\ = H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi )|_{H\cap X})\}. \end{array} \end{aligned}$$

Here \(\omega _{H\cap X}\) denotes the dualizing sheaf of \(H\cap X\).

Let \(U\subseteq \Lambda \times X\) be the closed subvariety whose \({\mathbb {C}}\)-points correspond to pairs \((H,x)\in \Lambda \times X\) with \(p(x)\in H\). Let \(\pi _1{:}\,U\rightarrow \Lambda \) be the natural projection. We may assume that \(\pi _1\) is surjective, as otherwise there is nothing to prove.

Observe that U is a local complete intersection scheme by Krulls Hauptidealsatz and a fortiori a Cohen–Macaulay scheme. It follows from miracle flatness [10, Theorem 23.1] that the natural projection \(\pi _2{:}\,U\rightarrow X\) is flat. As the fibers of \(\pi _2\) over closed points of X are isomorphic to \({\mathbb {P}}^{N-1}\), it follows that \(\pi _2\) is smooth. Thus, U is smooth as well.

In the following, we will construct pluripolar sets \(\Sigma _1\subseteq \Sigma _2 \subseteq \Sigma _3\subseteq \Sigma _4\subseteq \Lambda \) such that the behaviour of \(\pi _1\) is improved successively on the complement of \(\Sigma _i\).

Step 2.6.1 The usual Bertini theorem shows that there is a proper Zariski closed set \(\Sigma _1\subseteq \Lambda \) such that \(\pi _1\) has smooth fibres outside \(\Sigma _1\). This is slightly more general than the version that one finds in [7], see [9, Thèorème 6.3] for a proof.

Step 2.6.2 By Kollár’s torsion-free theorem [4, Theorem C],

$$\begin{aligned} {\mathcal {F}}^i:=R^i\pi _{1*}\left( \omega _{U/\Lambda }\otimes \pi _2^*{\mathcal {L}}\otimes {\mathcal {I}}(\pi _2^*\varphi ) \right) \end{aligned}$$

is torsion-free for all i. Here \(\omega _{U/\Lambda }\) denotes the relative dualizing sheaf of the morphism \(U\rightarrow \Lambda \). Thus, there is a proper Zariski closed set \(\Sigma _2\subseteq \Lambda \) such that

  1. (1)

    \(\Sigma _2\supseteq \Sigma _1\).

  2. (2)

    The \({\mathcal {F}}^i\)’s are locally free outside \(\Sigma _2\).

  3. (3)

    \(\omega _{U/\Lambda }\otimes \pi _2^*{\mathcal {L}}\otimes {\mathcal {I}}(\pi _2^*\varphi )\) is \(\pi _1\)-flat on \(\pi _1^{-1}(\Lambda {\setminus } \Sigma _2)\) [3, Thèoréme 6.9.1].

We write \({\mathcal {F}}={\mathcal {F}}^0\). By cohomology and base change [7, Theorem III.12.11], for any \(H\in \Lambda {\setminus } \Sigma _2\), the fibre \({\mathcal {F}}|_H\) of \({\mathcal {F}}\) is given by

$$\begin{aligned} {\mathcal {F}}|_H= H^0\left( \pi _{1,H},\omega _{U/\Lambda }|_{\pi _{1,H}}\otimes \pi _2^*{\mathcal {L}}|_{\pi _{1,H}}\otimes {\mathcal {I}}(\pi _2^*\varphi )|_{\pi _{1,H}} \right) . \end{aligned}$$

Here \(\pi _{1,H}\) denotes the fibre of \(\pi _1\) at H.

Step 2.6.3 In order to proceed, we need to make use of the Hodge metric \(h_{{\mathcal {H}}}\) on \({\mathcal {F}}\) defined in [8]. We briefly recall its definition in our setting. By [8, Section 22], we can find a proper Zariski closed set \(\Sigma _3\subseteq \Lambda \) such that

  1. (1)

    \(\Sigma _3\supseteq \Sigma _2\).

  2. (2)

    \(\pi _1\) is submersive outside \(\Sigma _3\).

  3. (3)

    Both \({\mathcal {F}}\) and \(\pi _{1*}\left( \omega _{U/\Lambda }\otimes \pi _2^*{\mathcal {L}} \right) /{\mathcal {F}}\) are locally free outside \(\Sigma _3\).

  4. (4)

    For each i,

    $$\begin{aligned} R^i\pi _{1*}\left( \omega _{U/\Lambda }\otimes \pi _2^*{\mathcal {L}} \right) \end{aligned}$$

    is locally free outside \(\Sigma _3\).

Then for any \(H\in \Lambda {\setminus } \Sigma _3\),

$$\begin{aligned} H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi |_{H\cap X}))\subseteq {\mathcal {F}}|_H\subseteq H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}). \end{aligned}$$

See [8, Lemma 22.1].

Now we can give the definition of the Hodge metric on \(\Lambda {\setminus } \Sigma _3\). Given any \(H\in \Lambda {\setminus } \Sigma _3\), any \(\alpha \in {\mathcal {F}}|_H\), the Hodge metric is defined as

$$\begin{aligned} h_{{\mathcal {H}}}(\alpha ,\alpha ):=\int _{X\cap H} |\alpha |^2_{he^{-\varphi }|_{X\cap H}}\in [0,\infty ]. \end{aligned}$$

Observe that \(h_{{\mathcal {H}}}(\alpha ,\alpha )<\infty \) if and only if \(\alpha \in H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi |_{H\cap X}))\). Moreover, \(h_{{\mathcal {H}}}(\alpha ,\alpha )>0\) if \(\alpha \ne 0\). It is shown in [8] (c.f. [12, Theorem 3.3.5]) that \(h_{{\mathcal {H}}}\) is indeed a singular Hermitian metric and it extends to a positive metric on \({\mathcal {F}}\).

Step 2.6.4. The determinant \(\det h_{{\mathcal {H}}}\) is singular at all \(H\in \Lambda {\setminus } \Sigma _3\) such that

$$\begin{aligned} H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi |_{H\cap X}))\ne {\mathcal {F}}|_H. \end{aligned}$$

As the map \(\pi _2\) is smooth, we have \(\pi _2^*{\mathcal {I}}(\varphi )= {\mathcal {I}}(\pi _2^*\varphi )\) by Lemma 2.2. Under the identification \(\pi _{1,H}\cong H\cap X\), we have

$$\begin{aligned} \pi _2^*{\mathcal {I}}(\varphi )|_{\pi _{1,H}}\cong {\mathcal {I}}(\varphi )|_{H\cap X}. \end{aligned}$$

Thus we have the following inclusions:

$$\begin{aligned} H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi |_{H\cap X}))\subseteq H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi )|_{H\cap X})\!=\! {\mathcal {F}}|_H. \end{aligned}$$

Recall that the first inclusion follows from the Ohsawa–Takegoshi \(L^2\)-extension theorem. Hence \(\det h_{{\mathcal {H}}}\) is singular at all \(H\in |{\mathcal {O}}_{{\mathbb {P}}^N}(1)|{\setminus } \Sigma _3\) such that

$$\begin{aligned} H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi |_{H\cap X}))\ne H^0(H\cap X,\omega _{H\cap X}\otimes {\mathcal {L}}|_{H\cap X}\otimes {\mathcal {I}}(\varphi )|_{X\cap H}). \end{aligned}$$

Let \(\Sigma _4\) be the union of \(\Sigma _3\) and the set of all such H. Since the Hodge metric \(h_{{\mathcal {H}}}\) is positive ( [12, Theorem 3.3.5] and [8, Theorem 21.1]), its determinant \(\det h_{{\mathcal {H}}}\) is also positive ([13, Proposition 1.3] and [8, Proposition 25.1]), it follows that \(\Sigma _4\) is pluripolar. As a consequence, \({\mathcal {G}}_{{\mathcal {L}}}\) is co-pluripolar.

Step 2.7

Fix an ample invertible sheaf \({\mathcal {S}}\) on X. The same result holds with \({\mathcal {L}}\otimes {\mathcal {S}}^{\otimes a}\) in place of \({\mathcal {L}}\). Thus the set

$$\begin{aligned} A:=\bigcap _{a=0}^{\infty }{\mathcal {G}}_{{\mathcal {L}}\otimes {\mathcal {S}}^{\otimes a}} \end{aligned}$$

is co-pluripolar. For each \(H\in W\) such that \(X\cap H\) is smooth and \({\mathcal {I}}(\varphi |_{X\cap H})\ne {\mathcal {I}}(\varphi )|_{X\cap H}\), let \({\mathcal {K}}\) be the following cokernel:

$$\begin{aligned} 0\rightarrow {\mathcal {I}}(\varphi |_{X\cap H})\rightarrow {\mathcal {I}}(\varphi )|_{X\cap H}\rightarrow {\mathcal {K}}\rightarrow 0. \end{aligned}$$

By Serre vanishing theorem, taking a large enough, we may guarantee that

$$\begin{aligned} H^1(X\cap H,\omega _{X\cap H}\otimes ({\mathcal {L}}\otimes {\mathcal {S}}^{\otimes a})|_{X\cap H}\otimes {\mathcal {I}}(\varphi |_{X\cap H}))=0 \end{aligned}$$

and

$$\begin{aligned} H^0(X\cap H,\omega _{X\cap H}\otimes ({\mathcal {L}}\otimes {\mathcal {S}}^{\otimes a})|_{X\cap H}\otimes {\mathcal {K}})\ne 0. \end{aligned}$$

Then

$$\begin{aligned} \begin{array}{c} H^0(X\cap H,\omega _{X\cap H}\otimes ({\mathcal {L}}\otimes {\mathcal {S}}^{\otimes a})|_{X\cap H}\otimes {\mathcal {I}}(\varphi |_{X\cap H}))\ne H^0(X\cap H,\omega _{X\cap H}\otimes ({\mathcal {L}}\otimes {\mathcal {S}}^{\otimes a})|_{X\cap H}\\ \otimes {\mathcal {I}}(\varphi )|_{X\cap H}). \end{array} \end{aligned}$$

Thus \(H\not \in A\). We conclude that \({\mathcal {G}}\) is co-pluripolar. \(\square \)

Remark 2.8

As pointed out by the referee, in [4], Fujino and Matsumura also treated the case when X is not projective. It is of interest to understand if Theorem 2.4 can be extended to non-projective complex manifolds as well.

Note that the argument for \(\pi {:}\,U\rightarrow \Lambda \) in the proof of Theorem 2.4 works for more general fibrations. With essentially the same proof, we can similarly prove an analytic Bertini type theorem for fibrations.

Corollary 2.9

Let \(\pi {:}\,U\rightarrow W\) be a surjective morphism of projective varieties. Let \((L,\phi )\) be a Hermitian pseudo-effective line bundle on U, namely L is a holomorphic line bundle on U and \(\phi \) is a plurisubharmonic metric on L. Then there is a pluripolar subset \(\Sigma \subseteq W\) such that for all \(w\in W{\setminus } \Sigma \), \(U_w:=\pi ^{-1}(w)\) is smooth and we have \({\mathcal {I}}(\phi |_{U_w})={\mathcal {I}}(\phi )|_{U_w}\).

Corollary 2.10

Let X be a projective manifold of pure dimension \(n\ge 1\). Let \(\Lambda \) be a base-point free linear system. Let \(\varphi \) be a quasi-psh function on X. Then there is a pluripolar subset \(\Sigma \subseteq \Lambda \) such that for any \(H\in \Lambda {\setminus } \Sigma \) and any real number \(k>0\),

$$\begin{aligned} {\mathcal {I}}(k\varphi |_H)={\mathcal {I}}(k\varphi )|_H \end{aligned}$$
(2.2)

and we have a short exact sequence for all \(k>0\),

$$\begin{aligned} 0\rightarrow {\mathcal {I}}(k\varphi )\otimes {\mathcal {O}}_X(-H)\rightarrow {\mathcal {I}}(k\varphi )\rightarrow {\mathcal {I}}(k\varphi |_H)\rightarrow 0. \end{aligned}$$
(2.3)

Proof

First observe that by the strong openness theorem [6] in order to verify (2.2) for all real \(k>0\), it suffices to verify it for k lying in a countable subset \(K\subseteq {\mathbb {R}}_{>0}\).

Applying Theorem 2.4 to each \(k\varphi \) with \(k\in K\) and each connected component of X, we find that there is a pluripolar set \(\Sigma _1\subseteq \Lambda \) such that for any \(H\in \Lambda {\setminus } \Sigma _1\) and any \(k\in K\), (2.2) holds. On the other hand, the union of the sets of associated primes of \({\mathcal {I}}(k\varphi )\) for \(k>0\) is a countable set, hence the set A of \(H\in \Lambda \) that avoids them is co-meager. It suffices to take \(\Sigma =\Sigma _1\cup (\Lambda {\setminus } A)\). \(\square \)

Following the terminology of [2], given quasi-psh functions \(\varphi \) and \(\psi \) on X, we say \(\varphi \sim _{{\mathcal {I}}}\psi \) if for all real \(k>0\), \({\mathcal {I}}(k\varphi )={\mathcal {I}}(k\psi )\).

Corollary 2.11

Let X be a projective manifold of pure dimension \(n\ge 1\). Let \(\varphi \), \(\psi \) be quasi-psh functions on X such that \(\varphi \sim _{{\mathcal {I}}}\psi \). Let \(\Lambda \) be a base-point free linear system. Then there is a pluripolar subset \(\Sigma \subseteq \Lambda \) such that for any \(H\in \Lambda {\setminus } \Sigma \), \(\varphi |_H\) and \(\psi |_H\) are both quasi-psh functions on H and we have \(\varphi |_H\sim _{{\mathcal {I}}} \psi |_H\).