Analytic Bertini theorem

Abstract

We prove an analytic Bertini theorem, generalizing a previous result of Fujino and Matsumura.

1 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.

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

• (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 (L, h), 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) $\Sigma _2\supseteq \Sigma _1$.

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

• (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) $\Sigma _3\supseteq \Sigma _2$.

• (2) $\pi _1$ is submersive outside $\Sigma _3$.

• (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) 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 $\varphi |_H\sim _{{\mathcal {I}}} \psi |_H$.