Remark 20.2.466A24558044D5

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

Step 1. We prove that the following set is co-pluripolar:

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\colon 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 Krull's Hauptidealsatz and a fortiori a Cohen–Macaulay scheme. It follows from miracle flatness [Theorem 23.1, Mat89] that the natural projection $\pi _2\colon U\rightarrow X$ 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. Moreover, observe that

by proposition 5.16.22.

In the following, we will construct pluripolar sets $\Sigma _1\subseteq \Sigma _2 \subseteq \Sigma _3\subseteq \Sigma _4\subseteq \Lambda $ such that the behavior of $\pi _1$ is improved successively on the complement of $\Sigma _ i$.

Step 1.1. The usual Bertini theorem shows that there is a proper Zariski closed set $\Sigma _1\subseteq \Lambda $ such that $\pi _1$ has smooth fibers outside $\Sigma _1$. Enlarging $\Sigma _1$, we may guarantee that $\pi _1$ and $\mathcal{I}(\pi _2^*\varphi )$ are both flat outside $\Sigma _1$. See [Théorème 6.9.1, EGAIV2]. Then after further enlarging $\Sigma _1$ we may assume that $p^{-1}(H)$ avoids all associated points of $\mathcal{O}_ X/\mathcal{I}(\varphi )$, for all $H\in \Lambda \setminus \Sigma _1$. Let $\pi _{1,H}$ denote the fiber of $\pi _1$ at $H$ and write $i_ H\colon \pi _{1,H}\rightarrow U$ for the inclusion morphism. We arrive at

for all $H\in \Lambda \setminus \Sigma _1$.³

Step 1.2. By Grauert's coherence theorem,

is coherent 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. 94B9B26CA79D

   $\Sigma _2\supseteq \Sigma _1$.

2. 0FC04CC0A41E

   The $\mathcal{F}^ i$'s are locally free outside $\Sigma _2$.

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

Step 1.3. In order to proceed, we need to make use of the Hodge metric $h_{\mathcal{H}}$ on $\mathcal{F}$ defined in [HPS18]. We briefly recall its definition in our setting. By [Section 22, HPS18], we can find a proper Zariski closed set $\Sigma _3\subseteq \Lambda $ such that

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   $\Sigma _3\supseteq \Sigma _2$,

2. 77E34EDD06F8

   $\pi _1$ is smooth outside $\Sigma _3$,

3. DC4E272A416D

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

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   for each $i$,

   \[ R^ i\pi _{1*}\left(\omega _{U/\Lambda }\otimes \pi _2^*L \right) \]

   is locally free outside $\Sigma _3$.

Then for any $H\in \Lambda \setminus \Sigma _3$,

See [Lemma 22.1, HPS18].

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

Observe that $h_{\mathcal{H}}(\alpha ,\alpha )<\infty $ if and only if $\alpha \in \mathrm{H}^0(H\cap X,\omega _{H\cap X}\otimes L|_{H\cap X}\otimes \mathcal{I}(\varphi |_{H\cap X}))$. Moreover, $h_{\mathcal{H}}(\alpha ,\alpha )>0$ if $\alpha \neq 0$. It is shown in [HPS18] (cf. [Theorem 3.3.5, PT18]) that $h_{\mathcal{H}}$ is indeed a singular Hermitian metric, and it extends to a positive metric on $\mathcal{F}$.

Step 1.4. The determinant $\det h_{\mathcal{H}}$ is singular at all $H\in \Lambda \setminus \Sigma _3$ such that

As the map $\pi _2$ is smooth, we have $\pi _2^*\mathcal{I}(\varphi )= \mathcal{I}(\pi _2^*\varphi )$ by proposition 5.16.22. Under the identification $\pi _{1,H}\cong H\cap X$, we have

Thus, we have the following inclusions:

the right-hand side being $\mathcal{F}|_ H$.

Recall that the first inclusion follows from theorem 5.16.26. Hence, $\det h_{\mathcal{H}}$ is singular at all $H\in |\mathcal{O}_{\mathbb {P}^ N}(1)|\setminus \Sigma _3$ such that

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 ([Theorem 3.3.5, PT18] and [Theorem 21.1, HPS18]), its determinant $\det h_{\mathcal{H}}$ is also positive ([Proposition 1.3, Rau15] and [Proposition 25.1, HPS18]), it follows that $\Sigma _4$ is pluripolar. As a consequence, $\mathcal{G}_{L}$ is co-pluripolar.

Step 2.

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

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

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

and

Then

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