PomeProposition 5.18.2.AB08C5138082 Let $\varphi ,\psi \in \mathrm{QPSH}(X)$ be potentials with analytic singularities. Then so are $\lambda \varphi $ ($\lambda \in \mathbb {Q}_{>0}$), $\varphi +\psi $ and $\varphi \lor \psi $.
5.18 Analytic singularities26AEEBDF865B
26AEEBDF865B
The simplest type of plurisubharmonic singularities is given by the so-called analytic singularities. The notion is fairly subtle and there are several mutually incompatible definitions in the literature, as we shall explain below.
Let $X$ be a complex manifold.
If $X$ is compact, each potential with analytic singularities has a type. The type is not uniquely determined. We refer to [Bou02] and [Bou02b] for the details. Some authors allow $c\in \mathbb {R}_{>0}$ in (5.18.1). We keep $c\in \mathbb {Q}_{>0}$ because the following closure property, which is essential in constructing Demailly approximations, would otherwise fail.Proof. The assertion for $\lambda \varphi $ follows by multiplying the rational coefficient in (5.18.1). The assertion for $\lor $ is proved in [Proposition 4.1.8, Dem15]. For the addition assertion, take local presentations as in (5.18.1) and multiply the corresponding finite systems of holomorphic generators. $\square$
Definition 5.18.3.670832106E8E Let $D$ be an effective $\mathbb {Q}$-divisor1 on $X$. We say $\varphi \in \mathrm{QPSH}(X)$ has log singularities (along $D$) on $X$ if for each $x\in X$, there is an open neighborhood $U$ of $x$ such that
05DF50833BB0$D|_ U$ has finitely many irreducible components and can be written as
\[ D|_ U=\sum _{i=1}^ N a_ i D_ i \]with $D_ i$ prime divisors on $U$, $a_ i\in \mathbb {Q}_{>0}$ and $s_ i$ holomorphic functions on $U$ defining $D_ i$, and
B8CC79988714we have
C6AC1C84A3B35.18.4\begin{equation} \varphi |_ U=\sum _{i=1}^ N a_ i\log |s_ i|^2+R, \end{equation}where $R$ is a bounded function on $U$.
Lemma 5.18.5.3B54149D4222 Suppose that $X$ is a compact Kähler manifold, $\theta $ is a closed smooth real $(1,1)$-form on $X$, and $\varphi \in \mathrm{PSH}(X,\theta )$. Suppose that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $D$ on $X$. Then the cohomology class $\{ \theta \} -\{ D\} $ is nef.
Moreover, if in addition $\theta _{\varphi }$ is a Kähler current2, then the cohomology class $\{ \theta \} -\{ D\} $ is Kähler.
Proof. The current $\theta _{\varphi }-[D]$ has bounded local potentials. Therefore, the class $\{ \theta \} -\{ D\} $ is nef.
If $\theta _{\varphi }$ is a Kähler current, say $\theta _{\varphi }\geq \omega $ for a Kähler form $\omega $, then $\theta _{\varphi }-[D]-\omega $ is positive and has bounded local potentials, and therefore represents a nef class. We conclude that $\{ \theta \} -\{ D\} $ is Kähler. $\square$
Proposition 5.18.6.4A902295BB8C Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a complex manifold $Y$. Suppose that $\varphi \in \mathrm{QPSH}(X)$ has analytic singularities (resp. has log singularities along an effective $\mathbb {Q}$-divisor $D$). Then $\pi ^*\varphi $ has analytic singularities (resp. has log singularities along $\pi ^*D$).
Definition 5.18.7.239591DA9F90 Let $\varphi \in \mathrm{QPSH}(X)$ be a potential with analytic singularities. A log resolution of $\varphi $ is a modification $\pi \colon Y\rightarrow X$ such that $\pi ^*\varphi $ has log singularities along a simple normal crossing effective $\mathbb {Q}$-divisor.
Theorem 5.18.8.D6BDA051B7C1 Assume that $X$ is compact. Suppose that $\varphi \in \mathrm{QPSH}(X)$ has analytic singularities. Then there is a log resolution of $\varphi $.
A general quasi-plurisubharmonic function can be nicely approximated by those with analytic singularities. We need a few preliminary definitions.
Definition 5.18.10.FFAA7E68EEC1 Let $\mathcal{I}\subseteq \mathcal O_ X$ be a coherent ideal sheaf and $c\in \mathbb {Q}_{>0}$. A function $\varphi \in \mathrm{QPSH}(X)$ is said to have gentle analytic singularities (of type $(c,\mathcal{I})$) if
802BF78106AD$\varphi $ has analytic singularities of type $(c,\mathcal{I})$;
99804F1A8886$\mathrm{e}^{{\varphi }/{c}}\colon X\to \mathbb {R}_{\geq 0}$ is a smooth function;
78E694C022ABthere is a proper bimeromorphic morphism $\pi \colon \widetilde{X}\to X$ from a Kähler manifold $\widetilde{X}$ and an effective $\mathbb {Z}$-divisor $D$ on $\widetilde{X}$ such that one can write $\pi ^*\varphi $ locally as
\[ \pi ^*\varphi =c\log |g|^2+h, \]where $g$ is a local equation of the divisor $D$ and $h$ is smooth.
Theorem 5.18.11.9348BF57481B Let $X$ be a compact Kähler manifold and $\theta $ be a closed real smooth $(1,1)$-form on $X$. Then any $\varphi \in \mathrm{PSH}(X,\theta )$ admits a quasi-equisingular approximation $(\varphi _ j)_{j\in \mathbb {Z}_{>0}}$.
Moreover, we can guarantee that for all $j>0$, $\varphi _ j$ has gentle analytic singularities of type $\left(2^{-j},\mathcal{I}(2^ j\varphi )\right)$.
Proof. Apply the gentle quasi-equisingular approximation theorem to $(X,\theta,\varphi)$. It constructs approximants of type $(2^{-j},\mathcal I(2^j\varphi))$, with a curvature error tending to zero, convergence in $L^1$, and the required multiplier-ideal inclusions. These are the defining conditions of a quasi-equisingular approximation in this section. Its approximants have gentle analytic singularities, so they also supply the additional regularity asserted here. The powers $2^j$ fix a subsequence of Bergman approximations; they impose no extra assumption on $\varphi$.
Proposition 5.18.12.36EC36A38FAB Let $X$ be a compact Kähler manifold and $\theta $ be a closed real smooth $(1,1)$-form on $X$. Consider $\varphi \in \mathrm{PSH}(X,\theta )$. Let $(\varphi _ j)_ j$ and $(\psi _ j)_ j$ be two quasi-equisingular approximations of $\varphi $. Then for any $\epsilon \in (0,1)$ and any $j>0$, we can find $k_0>0$ such that for any $k\geq k_0$, we have
\[ \psi _ k\preceq (1-\epsilon )\varphi _ j. \]
Proof. The final assertion of the comparison theorem for analytic approximants applies to the two given quasi-equisingular approximations. Its hypotheses are precisely compact Kähler $X$, a smooth closed background $\theta$, a $\theta$-plurisubharmonic limit, and the multiplier-ideal inclusions in the definition of quasi-equisingularity. For every fixed $j$ and $0<\varepsilon<1$ it yields $\psi_k\preceq(1-\varepsilon)\varphi_j$ for every sufficiently large $k$. The index is allowed to depend on $j$ and $\varepsilon$. In particular, no global pointwise ordering of two unrelated approximation sequences is being assumed.
Definition 5.18.13.99446B95B55B Let $\varphi \in \mathrm{QPSH}(X)$ be a potential with analytic singularities. Then we define $\mathcal{I}_{\infty }(\varphi )$ as the ideal sheaf consisting of germs $f$ of holomorphic functions such that $\log |f|^2\leq \varphi +C$ for some $C\in \mathbb {R}$.
Lemma 5.18.14.D9F25876A735 Assume that $X$ is compact. Let $\varphi \in \mathrm{QPSH}(X)$ be a potential with analytic singularities. The sheaf $\mathcal{I}_{\infty }(\varphi )$ is coherent.
Proof. By theorem 5.18.8, we may find a modification $\pi \colon Y\rightarrow X$ such that $\pi ^*\varphi $ has log singularities. Observe that
\[ \mathcal{I}_{\infty }(\varphi )=\pi _*\mathcal{I}_{\infty }(\pi ^*\varphi ), \]
so we may replace $X$ and $\varphi $ by $Y$ and $\pi ^*\varphi $ and assume that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $D$. We decompose $D$ into its irreducible components:
\[ D=\sum _{i=1}^ N a_ iD_ i. \]
In this case, observe that
\[ \mathcal{I}_{\infty }(\varphi )=\mathcal{O}_ X\left(-\sum _{i=1}^ N\left(\lceil a_ i \rceil D_ i\right)\right) \]
is clearly coherent. $\square$
Lemma 5.18.15.2E0432C779A0 Assume that $X$ is compact. Let $\varphi \in \mathrm{QPSH}(X)$ be a potential with analytic singularities. Then for any $\epsilon >0$, we can find $k_0>0$ such that for each $k\geq k_0$, we have
35E0DBB664F7
5.18.16
\begin{equation} \mathcal{I}\left(k(1+\epsilon )\varphi \right)\subseteq \mathcal{I}_{\infty }(k\varphi ). \end{equation}
Proof. We shall prove a more precise local result. Take $x\in X$, take an open neighborhood $U\subseteq X$ of $x$, on which
\[ \varphi =c\log \left( |g_1|^2 +\cdots +|g_ N|^2\right)+\mathcal{O}(1), \]
where $c\in \mathbb {Q}_{>0}$ and $g_1,\ldots ,g_ N$ are holomorphic functions on $U$. Then we claim that for any $\lambda >0$, we have
D1A302A4CDD6
5.18.17
\begin{equation} \mathcal{I}(\lambda \varphi )_ x\subseteq \mathcal{I}_{\infty }\left( \left( \lambda -c^{-1}n \right)_+\varphi \right)_ x, \end{equation}
where for any $a\in \mathbb {R}$, $a_+$ means $a\lor 0$. Note that (5.18.16) is a straightforward consequence of (5.18.17). The global result follows easily from the local result by the compactness of $X$.
Fix a smooth volume form $\mathrm{d}V$ on $X$. Take $f\in \mathcal{I}(\lambda \varphi )$. By strong openness theorem 5.16.21, we can find $\epsilon >0$ so that $f\in \mathcal{I}((\lambda +\epsilon /c)\varphi )$. Therefore, we can find an open neighborhood $W\subseteq U$ of $x$ so that
\[ \int _ W |f|^2\left( |g_1|^2 +\cdots +|g_ N|^2\right)^{-c\lambda -\epsilon }\, \mathrm{d}V<\infty . \]
But it follows from the Briançon–Skoda division theorem ([Sko72], [SB74], [Theorem 11.17, Dem12]) that
\[ f_ x\in \left(g_{1,x},\ldots ,g_{N,x} \right)^\alpha , \]
where $\alpha =\left(\lfloor c \lambda \rfloor -n+1\right)_+$. Here $\left(g_{1,x},\ldots ,g_{N,x} \right)$ denotes the ideal in $\mathcal{O}_{X,x}$ generated by the germs of $g_1,\ldots ,g_ N$ at $x$. Note that $\alpha \geq (c\lambda -n)_+$.
It follows that on a neighborhood of $x$, we have
\[ \log |f|^2\leq \alpha \log \left( |g_1|^2 +\cdots +|g_ N|^2\right)+\mathcal{O}(1)\leq c^{-1}\alpha \varphi +\mathcal{O}(1). \]
Hence (5.18.17) follows. $\square$
Theorem 5.18.18.2D610C4DD6ADLet $X$ be a connected compact Kähler manifold and $Y\subseteq X$ be a connected closed submanifold. Take a Kähler form $\omega $ on $X$ and $\varphi \in \mathrm{PSH}(Y,\omega |_ Y)$ such that $\omega |_ Y+\mathrm{dd}^{\mathrm{c}}\varphi $ is a Kähler current and $\mathrm{e}^{\varphi }$ is a Hölder continuous function on $Y$. Then there exists $\tilde{\varphi }\in \mathrm{PSH}(X,\omega )$ satisfying
1E31A11A83C4$\tilde{\varphi }|_ Y=\varphi $;
2229D01A1A95$\omega _{\tilde{\varphi }}$ is a Kähler current.
In addition, if $\varphi $ has analytic singularities, then so does $\tilde{\varphi }$.
EF5655B5F536
5.18.9
\begin{equation} \mathcal{I}(\lambda ' \varphi _ j)\subseteq \mathcal{I}(\lambda \varphi ). \end{equation}
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