A59F8B8F06A8
A natural source of quasi-plurisubharmonic functions is the metrics on line bundles.
Let $X$ be a connected Kähler manifold and $L$ be a holomorphic line bundle on $X$. Usually, we do not distinguish $L$ from the associated invertible sheaf $\mathcal{O}_ X(L)$.
Definition 5.20.1.C5880C6F61FA Let $V$ be a $1$-dimensional complex linear space. A Hermitian form $h$ on $V$ is a map $h\colon V\times V\rightarrow \mathbb {C}$ such that
CF86155BCCFB
$h$ is $\mathbb {C}$-linear in the second variable and conjugate linear in the first, and
B8DA9F74F30F
\[ |v|_ h^2\coloneqq h(v,v)\in \mathbb {R}_{>0} \]
for each $v\in V\setminus \{ 0\} $.
We usually identify $h$ with the quadratic form $V\rightarrow \mathbb {R}$ sending $v$ to $|v|_ h^2$. We write $|v|_ h=\sqrt{|v|_ h^2}$ for any $v\in V$.
The singular Hermitian form on $V$ is the map $V\rightarrow \{ 0,\infty \} $ sending $0$ to $0$ and other elements to $\infty $.
Definition 5.20.2.90891EB78EA0 Let $V_1$ and $V_2$ be $1$-dimensional complex linear spaces. Given two maps $h_ i\colon V_ i\rightarrow [0,\infty ]$ ($i=1,2$), each of which is either a Hermitian form or a singular Hermitian form, we define the tensor product $h_1\otimes h_2\colon V_1\otimes V_2\rightarrow [0,\infty ]$ as follows:
F0878993B90B
If either $h_1$ or $h_2$ is singular, we define $h_1\otimes h_2$ as the singular Hermitian form;
6CC728E84DA0
otherwise, define $h_1\otimes h_2$ as the usual tensor product: For any $v_1\in V_1$, $v_2\in V_2$, set
\[ h_1\otimes h_2(v_1\otimes v_2)=h_1(v_1)h_2(v_2). \]
Definition 5.20.3.293669D813B8 A Hermitian metric $h$ on $L$ is a family of Hermitian forms $(h_ x)_{x\in X}$, such that
8F75F9950CF9
for each $x\in X$, $h_ x$ is a Hermitian form on $L_ x$, and
56AE61E47B2A
for each local holomorphic section $s$ of $\mathcal{O}_ X(L)$, the map $x\mapsto |s(x)|^2_{h_ x}$ is smooth.
The pair $(L,h)$ is called a Hermitian line bundle. We shall write $\mathrm{dd}^{\mathrm{c}}h=c_1(L,h)$ for the first Chern form of $h$, normalized so that
\[ [c_1(L,h)]=c_1(L). \]
The map $x\mapsto |s(x)|_{h_ x}$ will be denoted by $|s|_ h$.
To be more precise, if $U\subseteq X$ is an open subset on which $L$ admits a nowhere vanishing holomorphic section $s$, then we define
\[ (\mathrm{dd}^{\mathrm{c}}h)|_ U=\mathrm{dd}^{\mathrm{c}}\left(-\log |s|_ h^2\right). \]
Proposition 5.20.4 (Lelong–Poincaré).0C02CD3B5D59 Let $s\in \mathrm{H}^0(X,L)$ be non-zero and $h$ be a Hermitian metric on $L$. Then
1322FD7BDF4F
5.20.5
\begin{equation} c_1(L,h)+\mathrm{dd}^{\mathrm{c}}\log |s|_ h^2=[Z(s)], \end{equation}
where $Z(s)$ is the zero divisor defined by $s$ and $[\bullet ]$ denote the associated current of integration.
Proof. In a holomorphic frame $e$ of $L$, write $s=fe$ and $|e|_h^2=\exp(-\rho)$. Then $\log|s|_h^2=\log|f|^2-\rho$ and $c_1(L,h)=\mathrm{dd}^{\mathrm c}\rho$. The normalized Poincaré--Lelong formula gives $\mathrm{dd}^{\mathrm c}\log|f|^2=[Z(f)]$. Substitution proves
\[c_1(L,h)+\mathrm{dd}^{\mathrm c}\log|s|_h^2=[Z(s)].\]
Changing the frame multiplies $f$ by a nowhere-vanishing holomorphic function; its logarithmic squared modulus is pluriharmonic, so the local identities agree on overlaps. This also checks the factor $1/(2\pi)$ in the normalization.
See
[(3.11), Dem12]. Again, we want to check that our conventions are compatible by investigating the following simple example.
Example 5.20.6.0D9E7B0EE3DC Let $X=\mathbb {P}^1$ and $L=\mathcal{O}_{\mathbb {P}^1}(1)$. The homogeneous coordinates on $\mathbb {P}^1$ will be denoted by $[X_0:X_1]$. At a point $x=[X_0:X_1]\in \mathbb {P}^1$, the fiber $L_ x$ is identified with the dual of $[x]$, where $[x]\subseteq \mathbb {C}^2$ is the line represented by $x$.
In order to introduce the Hermitian metric $h$ on $L$, we fix the standard Hermitian norm $\| \bullet \| $ on $\mathbb {C}^2$. Then given $\lambda \in L_ x=[x]^{\vee }$, we introduce
\[ |\lambda |_{h_ x}=\frac{|\lambda (\tilde{x})|}{\| \tilde{x}\| }, \]
where $\tilde{x}$ is an arbitrary non-zero element in $[x]$. The reader can easily verify that $h$ is indeed a Hermitian metric on $L$. The Hermitian metric $h$ is known as the Fubini–Study metric.
A holomorphic section $s\in \mathrm{H}^0(X,L)$ can be formally identified with a linear form $a_0X_0+a_1X_1$: At $x\in X$, the corresponding linear form on $[x]$ is given by sending $(X_0,X_1)$ to $a_0X_0+a_1X_1$.
Next we compute $\mathrm{dd}^{\mathrm{c}}h=c_1(L,h)$. For this purpose, we cover $\mathbb {P}^1$ by $\mathbb {C}=\mathbb {P}^1\setminus \{ \infty \} $ and $\mathbb {P}^1\setminus \{ 0\} $. Both are holomorphic coordinate charts with coordinate function $z=X_0/X_1$ and $z^{-1}=X_1/X_0$ respectively.
We claim that on $\mathbb {C}$,
3C9B623060E7
5.20.7
\begin{equation} \mathrm{dd}^{\mathrm{c}}h=\mathrm{dd}^{\mathrm{c}}\log (1+|z|^2). \end{equation}
In fact, let $t$ be the nowhere vanishing section of $L$ on $\mathbb {C}$ corresponding to $X_1$. Then for $z\in \mathbb {C}$, we have an obvious lift $(z,1)\in [z]$, so
\[ |t|_ h^2(z)=\frac{1}{|z|^2+1}. \]
So (5.20.7) follows.
In order to obtain a non-trivial case of the Lelong–Poincaré formula, we need to consider a section which vanishes at some points in $\mathbb {C}$. Let $s$ be the holomorphic section of $L$ corresponding to $X_0$. Then
\[ \log |s|_ h^2(z)=\log \frac{|z|^2}{|z|^2+1} \]
for any $z\in \mathbb {C}$ using the same argument as above. Therefore, we find that restricted to $\mathbb {C}$, we have
\[ c_1(L,h)+\mathrm{dd}^{\mathrm{c}}\log |s|_ h^2=\mathrm{dd}^{\mathrm{c}}f=[0], \]
where $f(z)=\log |z|^2$. So the Lelong–Poincaré formula (5.20.5) is verified in this case.
The Kähler form $\mathrm{dd}^{\mathrm{c}}h$ on $\mathbb {P}^1$ is also known as the Fubini–Study metric.
Definition 5.20.8.A43467D63B4A A (singular) plurisubharmonic metric (or psh metric for short) $h$ on $L$ is a family $(h_ x)_{x\in X}$ such that
DCF7D606ADB1
for each $x\in X$, $h_ x$ is either a Hermitian form on $L_ x$ or the singular Hermitian form on $L_ x$, and
32709CF6AC92
there is a Hermitian metric $h_0$ on $L$ and $\varphi \in \mathrm{PSH}(X,c_1(L,h_0))$ such that for each $x\in X$ and each $v\in L_ x$, we have
6C8BC3AC251D
5.20.9
\begin{equation} |v|^2_{h_ x}= \left\{ \begin{aligned} 0,& \quad \textup{if }v=0;\\ |v|^2_{h_{0,x}}\mathrm{e}^{-\varphi (x)},& \quad \textup{if }v\neq 0. \end{aligned} \right. \end{equation}
The (first) Chern current of $h$ is by definition
\[ \mathrm{dd}^{\mathrm{c}}h=c_1(L,h)\coloneqq c_1(L,h_0)+\mathrm{dd}^{\mathrm{c}}\varphi . \]
The set of psh metrics on $L$ will be denoted by $\mathrm{PSH}(L)$.
We shall write the plurisubharmonic metric defined by (
5.20.9) as $h_0\exp (-\varphi )$. With these conventions, $c_1(L,h)$ does not depend on the choice of $h_0$.
Given a local holomorphic section $s$ of $L$, we shall use the following notations interchangeably to denote the map $x\mapsto h_ x(s)$:
\[ |s|^2_ h=h(s)=h(s,s). \]
Observe that once a Hermitian metric $h_0$ on $L$ is given, the construction in 2 gives a bijection between $\mathrm{PSH}(X,c_1(L,h_0))$ and the set of plurisubharmonic metrics on $L$.
Based on this correspondence, plurisubharmonic metrics have the same functoriality property as quasi-plurisubharmonic functions in proposition 5.13.17. In particular, given a connected closed submanifold $Y$ of $X$, $h|_ Y$ is a plurisubharmonic metric as long as it is not singular everywhere.
Definition 5.20.11.9834FE8F2B0D Given two holomorphic line bundles $L_1$, $L_2$ on $X$ and plurisubharmonic metrics $h_1$ on $L_1$ and $h_2$ on $L_2$, we define the tensor product plurisubharmonic metric $h_1\otimes h_2$ on $L_1\otimes L_2$ as follows: for each $x\in X$, define
\[ (h_1\otimes h_2)_ x=h_{1,x}\otimes h_{2,x} \]
in the sense of definition 5.20.2.
We can easily verify that $h_1\otimes h_2$ is indeed a plurisubharmonic metric on $L_1\otimes L_2$.
Example 5.20.12.CC1C409253F3 We continue with our example example 5.20.6. Let $X=\mathbb {P}^1$ and $L=\mathcal{O}_{\mathbb {P}^1}(1)$. Let $h^0$ denote the Fubini–Study metric on $L$ as defined in example 5.20.6. Note that we have changed the notation from $h$ to $h^0$. Let $\omega =\mathrm{dd}^{\mathrm{c}}h^0$.
We construct $\varphi \in \mathrm{PSH}(X,\omega )$ as follows: On $\mathbb {C}$, define
6B76079BBD9C
5.20.13
\begin{equation} \varphi (z)=\log \frac{|z|^2}{1+|z|^2}. \end{equation}
Then $\varphi \in \mathrm{PSH}(\mathbb {C},\omega |_{\mathbb {C}})$ by (5.20.7). Setting $\varphi (\infty )=0$, we can easily verify that $\varphi \in \mathrm{PSH}(\mathbb {P}^1,\omega )$.
We then get a plurisubharmonic metric $h=h^0\exp (-\varphi )$. To be more explicit, the fiber metric $h$ at $0$ is singular, $h_{\infty }=h^0_{\infty }$, while for $z\in \mathbb {C}\setminus \{ 0\} $ and $\lambda \in [z]^{\vee }$, we have
\[ |\lambda |_{h_ z}=\frac{|\lambda (z,1)|}{|z|}. \]
In the remainder of this section, we assume that $X$ is compact.
Definition 5.20.14.9FE903B6E5A8Assume that $L$ is a pseudo-effective line bundle on $X$. A Fubini–Study metric on $L$ is a psh metric $h$ on $L$ of the following form: There exists $m\in \mathbb {Z}_{>0}$, finitely many sections $s_1,\ldots ,s_ N\in \mathrm{H}^0(X,L^ m)$ and $\lambda _1,\ldots ,\lambda _ N\in \mathbb {Q}$ such that for any local nowhere vanishing holomorphic section $s$ of $L$, we have
\[ |s|_ h^2=\min _{i=1,\ldots ,N}\left| \frac{s^{\otimes m}}{\mathrm{e}^{\lambda _ i/2}s_ i}\right|^{2m^{-1}}. \]
We write $\mathrm{FS}(L)$ for the set of Fubini–Study metrics on $L$.
If we fix a reference smooth Hermitian metric $h_0$ on $L$ with $\theta =\mathrm{dd}^{\mathrm{c}}h_0$, we can write $h=h_0\exp (-\varphi )$ with
\[ \varphi =\frac{1}{m}\max _{i=1,\ldots ,N}\left(\log |s_ i|^2_{h_0^{m}}+\lambda _ i\right). \]
Similarly, we write $\mathrm{FS}(X,\theta )$ for the set of such functions.
Definition 5.20.15.8FCD7BEEDD56 Assume that $L$ is a pseudo-effective line bundle on $X$. The set $\widetilde{\mathrm{FS}}(L)$ of generalized Fubini–Study metrics is the smallest subset of $\mathrm{PSH}(L)$ containing $\mathrm{FS}(L)$ which is closed under the following two operations:
B253418A2EBA
$\mathbb {Q}$-convex combinations: if $h_1,h_2\in \widetilde{\mathrm{FS}}(L)$ and $t\in (0,1)$, then
\[ h_1^ t \otimes h_2^{1-t}\in \widetilde{\mathrm{FS}}(L); \]
CCB3DD349505
minima: if $h_1,h_2\in \widetilde{\mathrm{FS}}(L)$, then
\[ \min \{ h_1,h_2\} \in \widetilde{\mathrm{FS}}(L). \]
We shall need the following Ohsawa–Takegoshi type extension theorem.
Theorem 5.20.16.2985932B2B64 Assume that $L$ is big and $T$ is a holomorphic line bundle on $X$. Fix a Hermitian metric $h_ T$ on $T$. Fix a Hermitian metric $h$ on $L$ and let $\theta =c_1(L,h)$. Take a Kähler form $\omega $ on $X$. Let $Y\subseteq X$ be a connected closed submanifold of dimension $m$. Suppose that $\varphi \in \mathrm{PSH}(X,\theta - \delta \omega )$ for some $\delta >0$ and $\varphi |_ Y\not\equiv -\infty $. Then there exists $k_0(\delta ,h_ T)>0$ such that for all $k \geq k_0$ and $s \in \mathrm{H}^0(Y, T\otimes L|_ Y^ k \otimes \mathcal{I}(k\varphi |_ Y))$, there exists an extension $\tilde s \in \mathrm{H}^0(X, T\otimes L^ k \otimes \mathcal{I}(k\varphi ))$ such that
\[ \int _ X |\tilde{s}|^2_{h^ k\otimes h_ T}\mathrm{e}^{-k\varphi }\, \omega ^ n \leq C \int _ Y |s|^2_{(h^ k\otimes h_ T)|_ Y} \mathrm{e}^{-k\varphi |_ Y}\, \omega |_ Y^ m, \]
where $C>0$ is an absolute constant, independent of the data $(\varphi ,s,k)$.
This is a special case of
[Theorem 1.4, His12].
Let us also recall a simple result about the Bergman kernel approximations:
Proposition 5.20.17.2DDBBF280789 Let $(L,h)$ be a Hermitian line bundle on $X$ and set $\theta =c_1(L,h)$. Let $(T,h_ T)$ be a Hermitian line bundle on $X$. Assume that $\varphi \in \mathrm{PSH}(X,\theta )$ is a potential with analytic singularities such that $\theta _{\varphi }$ is a Kähler current. Fix a Kähler form $\omega $ on $X$. For each $k\geq 1$, we let
1EE7A6DB1803
5.20.18
\begin{equation} \varphi _ k \coloneqq \frac{1}{k} \log \sup _{\substack {s \in \mathrm{H}^0(X,L^ k \otimes T)\\ \int _ X |s|^2_{h^ k \otimes h_ T}\mathrm{e}^{-k\varphi } \omega ^ n \leq 1}} |s|^2_{h^ k \otimes h_ T}. \end{equation}
Then for all sufficiently large $k$,
\[ \varphi \preceq \varphi _ k\preceq \alpha _ k \varphi , \]
where $\alpha _ k\in (0,1)$ is an increasing sequence with limit $1$.
Proof.
First we prove $\varphi \preceq \varphi _ k$. Since $\theta _{\varphi }$ is a Kähler current, there is $\delta >0$ such that $\varphi \in \mathrm{PSH}(X,\theta -\delta \omega )$. Let $x\in X$ with $\varphi (x)>-\infty $. Applying theorem 5.20.16 to the point $Y=\{ x\} $, for all sufficiently large $k$ and every unit vector $e_ x\in (L^ k\otimes T)_ x$, there is a section $\tilde{s}\in \mathrm{H}^0(X,L^ k\otimes T\otimes \mathcal{I}(k\varphi ))$ with $\tilde{s}(x)=e_ x$ and
\[ \int _ X |\tilde{s}|^2_{h^ k\otimes h_ T}\mathrm{e}^{-k\varphi }\omega ^ n\leq C\mathrm{e}^{-k\varphi (x)}. \]
After rescaling $\tilde{s}$, the definition of $\varphi _ k$ gives
\[ \varphi _ k(x)\geq \varphi (x)-\frac{\log C}{k}. \]
This proves $\varphi \preceq \varphi _ k$.
The other part of the inequality follows from 5.18.15 and (5.18.17).
More precisely, if $s_1,\ldots ,s_ N$ is an orthonormal basis of $\mathrm{H}^0(X,L^ k\otimes T\otimes \mathcal{I}(k\varphi ))$, with respect to the $L^2$-norm:
\[ s\mapsto \left( \int _ X |s|^2_{h^ k\otimes h_ T}\mathrm{e}^{-k\varphi }\, \omega ^ n \right)^{1/2}, \]
then by (5.18.17),
\[ s_ i\in \mathrm{H}^0\left(X,L^ k\otimes T\otimes \mathcal{I}_{\infty }\left((k-c^{-1}n)_+\varphi \right)\right), \]
where $c>0$ depends only on $\varphi $. So there is $C_ k>0$ so that
\[ \sum _{i=1}^ N |s_ i|^2_{h^ k\otimes h_ T}\mathrm{e}^{-(k-c^{-1}n)_+\varphi }\leq C_ k\quad \textup{quasi-everywhere}. \]
In other words,
\[ \varphi _ k\preceq \frac{1}{k}\left( k-c^{-1}n \right)_+ \varphi . \]
It suffices to take $\alpha _ k=(1-k^{-1}c^{-1}n)_+$.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.