20.2.44 Analytic Bertini theorems6B19CA7EC662

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

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   $\Sigma _2\supseteq \Sigma _1$.

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   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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   $\Sigma _3\supseteq \Sigma _2$,

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

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   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. $\square$

Proof. For quasi-every $H$, theorem 20.2.45 gives

After removing the hyperplanes meeting the associated points of $\mathcal O_ X/\mathcal I(\varphi )$ improperly, the right-hand side is non-zero. Since $\mathcal I(-\infty )=0$, this implies $\varphi |_ H\not\equiv -\infty $. $\square$

Proof. First observe that the set $\{ x\in X:\nu (\varphi ,x)>0\} $ is a countable union of proper analytic subsets by Siu's semicontinuity theorem theorem 5.16.7. It follows that a very general element in $\Lambda $ is not contained in this set.

Fix an ample line bundle $L$ so that there is a smooth Hermitian metric $h_ L$ such that $c_1(L,h_ L)+\mathrm{dd}^{\mathrm{c}}\varphi $ is a Kähler current. Thanks to theorem 20.2.45 and the connectedness assumption on $\Lambda $, we can find a co-pluripolar set $\Lambda '\subseteq \Lambda $ such that each $H\in \Lambda '$ satisfies the following:

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   $H$ is smooth and connected;

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   $\nu (\varphi ,H)=0$;

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   $\mathcal{I}(k\varphi |_ H)=\operatorname{Res}_{H}(\mathcal{I}(k\varphi ))$ for all $k\in \mathbb {Z}_{>0}$.

Put $\alpha _ H\coloneqq c_1(L,h_ L)|_ H$. It follows from theorem 20.2.39 and theorem 18.1.46 that

Since $\varphi |_ H\preceq _ P \operatorname{Tr}_ H(\varphi )$ by proposition 20.2.12, we also have $\varphi |_ H\preceq _{\mathcal I}\operatorname{Tr}_ H(\varphi )$. Hence

The trace is $\mathcal I$-good, so the right-hand side agrees with its $P$-envelope. The equality of the masses above and theorem 16.3.15 then imply that the two sides have the same $P$-singularity type, hence

Proof. Write $T=\theta _{\varphi }$ for some smooth closed real $(1,1)$-form $\theta $ on $X$ and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Thanks to 13.13.47, we can find $\psi \in \mathrm{PSH}(X,\theta )$ such that $\theta _{\psi }$ is a Kähler current and $\psi \leq \varphi $. By corollary 20.2.49 and the connectedness assumption on $\Lambda $, we can find a co-pluripolar set $\Lambda '\subseteq \Lambda $ such that each $H\in \Lambda '$ satisfies:

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   $H$ is smooth and connected;

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   the restriction $\psi |_ H$ is not identically $-\infty $.

Since $\theta _\psi $ is a Kähler current, its restriction to such an $H$ has positive non-pluripolar mass. As $\psi |_ H\leq \varphi |_ H$, theorem 13.13.11 gives

We conclude the proof. $\square$

Proof. Write $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ and set $\psi =P_{\theta }[\varphi ]_{\mathcal{I}}$. Then $\psi \sim _{\mathcal{I}}\varphi $ and

By 20.2.51, for quasi-every $H\in \Lambda $, the hypersurface $H$ is connected and smooth, $\psi |_ H$ is well-defined, and

For such $H$, corollary 20.2.50 gives $\operatorname{Tr}_ H(\psi )\sim _{\mathcal{I}}\psi |_ H$. Since $\varphi \sim _{\mathcal{I}}\psi $, proposition 20.2.21 also gives $\operatorname{Tr}_ H(\varphi )\sim _ P\operatorname{Tr}_ H(\psi )$. The assertion now follows from example 20.2.17. $\square$

Proof. Thanks to 16.8.13, we may replace $\varphi $ by $\varphi \lor \psi $ and assume that $\varphi \sim _ P \psi $. It suffices to show that $\varphi |_ H\sim _ P \psi |_ H$ for quasi-every $H\in \Lambda $.

Take a smooth closed real $(1,1)$-form $\theta $ on $X$ so that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. It suffices to compare $\varphi $ and $\psi $ with $P_{\theta }[\varphi ]$, so without loss of generality, we may assume that $\psi $ is a model potential in $\mathrm{PSH}(X,\theta )_{>0}$. Up to adding a constant to $\varphi $, we may then assume that $\varphi \leq \psi $. It follows from 13.13.30 that we can find a sequence $(\eta _ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ such that

for all $j\geq 2$. By corollary 20.2.49, 20.2.51, we can find a co-pluripolar set $\Lambda '\subseteq \Lambda $ such that any $H\in \Lambda '$ satisfies:

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   $H$ is smooth and connected;

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   $\eta _ j|_ H\in \mathrm{PSH}(H,\theta |_ H)_{>0}$ for all $j\geq 2$ and $\psi |_ H\in \mathrm{PSH}(H,\theta |_ H)_{>0}$.

Therefore, taking proposition 16.4.2 into account, we arrive at

for all $j\geq 2$. The normalized envelopes $P_{\theta |_ H}[\eta _ j|_ H]$ are uniformly bounded in $L^1$, hence

We conclude that

and hence $\psi |_ H\preceq _ P \varphi |_ H$. $\square$

Proof. By corollary 16.9.36, we may assume that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$.

Using 13.13.30, we can find a decreasing sequence $(\epsilon _ j)_ j$ in $(0,1)$ with limit $0$ and $\eta _ j\in \mathrm{PSH}(X,\theta )_{>0}$ such that $\eta _ j\leq \varphi _ j$ and

By corollary 20.2.49, 20.2.51, we can find a co-pluripolar set $\Lambda '\subseteq \Lambda $ such that any $H\in \Lambda '$ satisfies:

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   $H$ is smooth and connected;

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   $\eta _ j|_ H\in \mathrm{PSH}(H,\theta |_ H)_{>0}$ for all $j\geq 1$ and $\varphi |_ H\in \mathrm{PSH}(H,\theta |_ H)_{>0}$.

Therefore, taking proposition 16.4.2 into account, we arrive at

Since the normalized envelopes $P_{\theta |_ H}[\eta _ j|_ H]$ are relatively compact in $L^1$, we have $\epsilon _ jP_{\theta |_ H}[\eta _ j|_ H]\to 0$ in $L^1$. Letting $j\to \infty $, we get

The reverse inequality is trivial, so we get an equality here. By theorem 13.13.11 and proposition 16.4.4, we conclude that

Therefore, using corollary 16.9.14, we conclude that $\varphi _ j|_ H\xrightarrow {d_ S}\varphi |_ H$. $\square$

Proof. Choose a closed smooth real $(1,1)$-form $\theta _0$ representing a big class such that $\varphi \in \mathrm{PSH}(X,\theta _0)$ and $\theta _0+\mathrm{dd}^{\mathrm{c}}\varphi $ is a Kähler current. Let $(\varphi _ j)_ j$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta _0)_{>0}$. Since $\varphi $ is $\mathcal I$-good, corollary 18.1.14, theorem 18.1.2, proposition 16.9.6 gives

Applying 20.2.54, we find a co-pluripolar set of $H\in \Lambda $ such that $H$ is connected and smooth, the restrictions $\varphi _ j|_ H$ and $\varphi |_ H$ are not identically $-\infty $, and

For such an $H$, the potentials $\varphi _ j|_ H$ have analytic singularities, hence are $\mathcal I$-good by example 18.1.10. It follows from corollary 18.1.9 that $\varphi |_ H$ is $\mathcal I$-good.

On the other hand, corollary 20.2.50 gives, after intersecting with another co-pluripolar set of parameters $H$, that $\nu (\varphi ,H)=0$ and

The trace $\operatorname{Tr}_ H(\varphi )$ is $\mathcal I$-good by proposition 20.2.8. Since both $\operatorname{Tr}_ H(\varphi )$ and $\varphi |_ H$ are $\mathcal I$-good, their $\mathcal I$-equivalence identifies their $\mathcal I$-envelopes, which agree with their $P$-envelopes. Hence

Finally corollary 20.2.52 shows, after intersecting with one more co-pluripolar set of $H\in \Lambda $, that $\operatorname{Tr}_ H(\varphi )$ has a representative in $\mathrm{PSH}(H,\theta |_ H)_{>0}$. $\square$

Proof. We need to show that $\mathcal{I}(kh)=\mathcal{I}(kh')$ for every positive integer $k$. It suffices to prove the case $k=1$, since our assumptions are invariant after replacing $h$ and $h'$ by $h^ k$ and $h'^ k$. We will therefore prove $\mathcal{I}(h)=\mathcal{I}(h')$. First observe that it suffices to prove that

as subsheaves of $f_*(K_{X}\otimes L)$. In fact, suppose that (20.2.57) holds. Let $H$ be a $f$-ample invertible sheaf on $X$. Then (20.2.57) also holds with $L\otimes H^ m$ in place of $L$. It follows from Grauert–Remmert's version of Serre vanishing theorem [Theorem 2.1(A), BS76] that $\mathcal{I}(h)=\mathcal{I}(h')$.

It remains to prove (20.2.57). Observe that both sides of (20.2.57) are locally free by [Corollary 1.5, Mat16]. We claim that it suffices to show that

for quasi-every $z\in \Delta ^*$. In fact, this implies that the same holds outside a countable subset of $\Delta ^*$. But the set where (20.2.58) fails has to be $S^1$-invariant, it has to be empty.

By cohomology and base change together with the analytic Bertini theorem 20.2.48, for quasi-every $z\in \Delta ^*$, we have

But we assumed that for quasi-every $z$, $h|_{X_ z}\sim _{\mathcal{I}}h'|_{X_ z}$, it follows that for quasi-every $z\in \Delta ^*$, (20.2.58) holds. The proof is complete. $\square$