PomeDefinition 5.22.1.9CBA378B705D A quasi-plurisubharmonic function $u$ on $X$ is $\theta $-plurisubharmonic if
\[ \theta +\mathrm{dd}^{\mathrm{c}}u\geq 0 \]
as a current. We denote the set of these functions by $\mathrm{PSH}(X,\theta )$.
5.22 Plurisubharmonic potentials17A37A0D3690
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Throughout the chapter, $X$ is a connected compact complex manifold of dimension $n\geq 1$, $\theta $ is a smooth real $(1,1)$-form and $\omega $ is a Hermitian form on $X$, unless a statement specifies another setting. There are two distinct extensions of the Kähler setting: one can study a closed background on a non-Kähler manifold, or allow the background itself to be nonclosed. In both cases positivity is expressed through quasi-plurisubharmonic functions.
On an open set $U$ in a complex manifold, a plurisubharmonic function is a locally integrable upper semicontinuous function satisfying the submean inequality on every complex line in local coordinates. We write $\mathrm{PSH}(U)$ for the set of these functions. A function is quasi-plurisubharmonic if it is locally the sum of a plurisubharmonic function and a smooth real function; the set of such functions is denoted by $\mathrm{QPSH}(U)$. The following relative positivity condition makes sense on any such open set as well.
Since $\omega >0$, every constant function belongs to $\mathrm{PSH}(X,\omega )$.
Proposition 5.22.2.887F604607E9 Every real distribution $u$ on $X$ with $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$ has a unique $\theta $-plurisubharmonic representative. On each relatively compact coordinate ball with coordinates $z=(z_1,\ldots ,z_ n)$, there is a constant $A>0$ such that $u+A|z|^2$ is plurisubharmonic. Conversely, every quasi-plurisubharmonic function $u$ on $X$ belongs to $\mathrm{PSH}(X,C\omega )$ for some constant $C>0$.
Proof. On a coordinate ball with compact closure in a larger chart, smoothness of $\theta $ gives
\[ \theta \leq A\mathrm{dd}^{\mathrm{c}}|z|^2 \]
for a sufficiently large $A$. Consequently,
\[ \mathrm{dd}^{\mathrm{c}}(u+A|z|^2)\geq A\mathrm{dd}^{\mathrm{c}}|z|^2-\theta \geq 0. \]
The local distributional characterization of plurisubharmonic functions, [Chapter I, Theorem 5.8, DemCADG], gives a unique plurisubharmonic representative of $u+A|z|^2$. Subtracting the smooth correction gives a quasi-plurisubharmonic representative of $u$ on the ball. These representatives agree on overlaps: after a common local smooth correction, two of them become plurisubharmonic functions that are equal almost everywhere, hence everywhere. They therefore glue to a unique $\theta $-plurisubharmonic representative.
For the converse, cover $X$ by finitely many open sets $V_ j\Subset U_ j$, where each $U_ j$ is a chart on which $u=p_ j+f_ j$, with $p_ j$ plurisubharmonic and $f_ j$ smooth. On the compact set $\overline V_ j$, the smooth form $\mathrm{dd}^{\mathrm{c}}f_ j$ is bounded below by $-C_ j\omega $ for some constant $C_ j$, so $\mathrm{dd}^{\mathrm{c}}u\geq \mathrm{dd}^{\mathrm{c}}f_ j\geq -C_ j\omega $ on $V_ j$. Any constant $C>0$ larger than all the $C_ j$ gives $\mathrm{dd}^{\mathrm{c}}u\geq -C\omega $ on $X$. $\square$
For $u\in \mathrm{PSH}(X,\theta )$, choose coordinates $z$ and a constant $A>0$ as in proposition 5.22.2, and put $p=u+A|z|^2$. Then
\[ \theta +\mathrm{dd}^{\mathrm{c}}u=\mathrm{dd}^{\mathrm{c}}p+(\theta -A\mathrm{dd}^{\mathrm{c}}|z|^2). \]
Thus local plurisubharmonic theory applies to $p$, while the smooth remainder $\theta -A\mathrm{dd}^{\mathrm{c}}|z|^2$ records the original background; this remainder must be kept when forming products or integrating by parts.
Proposition 5.22.3.F62C0F30E87C If $\theta =\mathrm{dd}^{\mathrm{c}}\rho $ on an open set for a smooth function $\rho $, then $\mathrm{d}\theta =0$ on that set. In particular, at a point where $\mathrm{d}\theta \neq 0$, no neighborhood admits such a representation.
Proof. Apply $\mathrm{d}$ and use $\mathrm{d}\mathrm{dd}^{\mathrm{c}}=0$. $\square$
Conversely, a closed real $(1,1)$-form has smooth local $\mathrm{dd}^{\mathrm{c}}$-potentials on sufficiently small coordinate balls, by the local $\partial \bar\partial $-lemma; this local fact does not require $X$ to admit a Kähler metric.
The exterior derivative also constrains the regularity of a potential whose current is pointwise proportional to a Hermitian form. In dimension at least two, differentiating such an identity expresses the derivative of the proportionality factor through smooth coefficients.
Lemma 5.22.4.0DED015E67EF Let $X$ be a complex manifold of dimension $n\geq 2$, let $\theta $ be a smooth real $(1,1)$-form and let $\gamma $ be a Hermitian form on $X$. Suppose that real-valued $u,a\in L^1_{\mathrm{loc}}(X)$ satisfy
\[ \theta +\mathrm{dd}^{\mathrm{c}}u=a\gamma \]
as currents. Then $a$ and $u$ are smooth.
Proof. Taking the exterior derivative and multiplying by $\gamma ^{n-2}$ gives
\[ \mathrm{d}a\wedge \gamma ^{n-1} =\mathrm{d}\theta \wedge \gamma ^{n-2} -a\, \mathrm{d}\gamma \wedge \gamma ^{n-2}. \]
At each point, $\xi \mapsto \xi \wedge \gamma ^{n-1}$ is an isomorphism from real one-forms to real $(2n-1)$-forms, as one sees in a unitary frame. Its inverse depends smoothly on the point. Let $B$ and $C$ be the images of $\mathrm{d}\theta \wedge \gamma ^{n-2}$ and $\mathrm{d}\gamma \wedge \gamma ^{n-2}$ under this inverse. They are smooth real one-forms, and
\[ \mathrm{d}a=B-aC \]
in the sense of distributions. Since $a\in L^1_{\mathrm{loc}}$, the right-hand side is locally integrable, so $a\in W^{1,1}_{\mathrm{loc}}$. Differentiating the equation and using the smoothness of its coefficients, we obtain inductively $a\in W^{k,1}_{\mathrm{loc}}$ for every integer $k$, and the local Sobolev embedding shows that $a$ is smooth. Consequently $\mathrm{dd}^{\mathrm{c}}u=a\gamma -\theta $ is smooth; taking its trace in local coordinates and applying elliptic regularity shows that $u$ is smooth as well. $\square$
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