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5.13.13 The manifold case7BAC2CC7335C

Let $X$ be a complex manifold. Here, complex manifolds are assumed to be paracompact, namely, all connected components have countable bases.

Definition 5.13.14.D072C98719AA A plurisubharmonic function on $X$ is a function $\varphi \colon X\rightarrow [-\infty ,\infty )$ such that for any $x\in X$, there exists an open neighborhood $U$ of $x$ in $X$, an integer $n\in \mathbb {N}$, a domain $\Omega \subseteq \mathbb {C}^ n$ and a biholomorphic map $F\colon \Omega \rightarrow U$ such that $F^*(\varphi |_ U)\in \mathrm{PSH}(\Omega )$.

The set of plurisubharmonic functions on $X$ is denoted by $\mathrm{PSH}(X)$.

Example 5.13.15.67A5152A77BC When $X$ is a domain in $\mathbb {C}^ n$, the notions of plurisubharmonic functions in definition 5.13.14 and in definition 5.8.1 coincide.

Example 5.13.16.399FC71A61E3 Write $\{ X_ i\} _{i\in I}$ for the set of connected components of $X$. Then we have a natural bijection

\[ \mathrm{PSH}(X) \cong \prod _{i\in I} \mathrm{PSH}(X_ i). \]

Here the product is in the category of sets. In particular, if $X=\varnothing $, then $\mathrm{PSH}(X)$ is a singleton.

This example allows us to reduce to the case of connected manifolds when studying general plurisubharmonic functions.

Proposition 5.13.17.726DDFDDA8F2 Let $Y$ be a complex manifold and $f\colon Y\rightarrow X$ be a holomorphic map. Then for any $\varphi \in \mathrm{PSH}(X)$, exactly one of the following cases occurs:

  1. CAAB826BFF14

    $f^*\varphi $ is identically $-\infty $ on some connected component of $Y$;

  2. 13566C9390E6

    $f^*\varphi \in \mathrm{PSH}(Y)$.

Proof. Work on one connected component $Y_0$ of $Y$. In coordinate neighborhoods of $Y$ and $X$, holomorphic pullback says that $\varphi\circ f$ is either plurisubharmonic or identically $-\infty$. A plurisubharmonic function cannot be identically $-\infty$ on a nonempty open subset of a connected domain, by its local integrability. Thus a neighborhood on which the pullback is identically $-\infty$ cannot overlap a neighborhood on which it is plurisubharmonic. The two types form disjoint open subsets of $Y_0$; connectedness leaves only one type. If the first type never occurs on any component, the pullback is plurisubharmonic on $Y$. Zero-dimensional charts are covered by the finite-value convention.

This proposition follows from proposition 5.13.9 applied separately on each connected component of $Y$. theorem 5.13.8 implies immediately the general form of the rigidity theorem:

Theorem 5.13.18.FEFFC91695E0 Let $\varphi \colon X\rightarrow [-\infty ,\infty )$ be a measurable function. Then the following are equivalent:

  1. 3BB9EAC495FC

    $\varphi $ is locally integrable and $\mathrm{dd}^{\mathrm{c}}\varphi \geq 0$;

  2. FB5BA344B4A3

    $\varphi $ coincides almost everywhere with a plurisubharmonic function $\psi $ on $X$.

Moreover, the plurisubharmonic function $\psi $ in 2 is unique.

Proof. Restrict a locally integrable $\varphi$ with $\mathrm{dd}^{\mathrm c}\varphi\geq0$ to each coordinate chart and apply the local representative theorem. The resulting plurisubharmonic representatives agree almost everywhere on overlaps, hence everywhere there by the uniqueness part of the same theorem. They therefore glue to a plurisubharmonic function on $X$ representing $\varphi$.

Conversely, local integrability and positivity hold in every chart for a plurisubharmonic function and are preserved by almost-everywhere equality. The same overlap argument proves uniqueness globally. On a zero-dimensional component all assertions reduce to finiteness at each point.

Definition 5.47.2.5458CA6608A5 A subset $E\subseteq X$ is pluripolar if for any $x\in X$, there is an open neighborhood $U$ of $x$ in $X$ and a function $\psi \in \mathrm{PSH}(U)$ such that

\[ \psi |_{E\cap U}\equiv -\infty . \]

A subset $E\subseteq X$ is non-pluripolar if $E$ is not pluripolar.

A subset $F\subseteq X$ is co-pluripolar if $X\setminus F$ is pluripolar.

When $X$ has dimension $1$, a pluripolar set is called a polar set. We say some property about objects on $X$ holds quasi-everywhere if it holds outside a pluripolar set.

Theorem 5.13.19 (Josefson's theorem).DB05CEAF06C1 Let $E\subseteq \mathbb {C}^ n$ be a pluripolar set. Then there is $\varphi \in \mathrm{PSH}(\mathbb {C}^ n)$ such that $\varphi |_ E\equiv -\infty $.

Proof. Apply Josefson's theorem, proved earlier in the collection to $E$. It constructs a function $V\in\mathrm{PSH}(\mathbb C^n)$ with $E\subset\{V=-\infty\}$, in fact with the additional bound $V(z)\leq\log^+|z|$. Taking $\varphi=V$ gives the assertion. The earlier construction is a proof of this stronger statement, so no additional external theorem is being invoked here. In dimension zero a pluripolar set is empty and a finite constant suffices.

See [Corollary 4.41, GZ17] for the proof of a more general result. There is also a global version of Josefson's theorem:

Theorem 5.13.20.0ED3DB1BD7C2 Assume that $X$ is a compact complex manifold and $E\subseteq X$ is a pluripolar set. Then there is a quasi-plurisubharmonic function $\varphi $ on $X$ with $\varphi |_ E\equiv -\infty $.

For a proof, see [Vu19]. The notion of quasi-plurisubharmonic functions will be recalled in definition 5.17.1 below.
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