Lemma 17.5.28 (Lamari's criterion)D70C0784CA6E

Proof. Necessity. If $u\in \mathrm{PSH}(X,\theta )$, then $T=\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$. For $\tau $ as in the statement, the definition of $\mathrm{dd}^{\mathrm{c}}u$ as a current gives $\int _ X\mathrm{dd}^{\mathrm{c}}u\wedge \tau =\int _ Xu\, \mathrm{dd}^{\mathrm{c}}\tau =0$, hence $\int _ X\theta \wedge \tau =\int _ XT\wedge \tau $, which is nonnegative by the definition of a positive current, since $\tau $ is positive, hence strongly positive.

Sufficiency. Conversely, assume (17.5.29) for all forms $\tau $ as in the statement. We extend the functional $\tau \mapsto \int _ X\theta \wedge \tau $ from $\mathrm{dd}^{\mathrm{c}}$-closed forms to a positive linear functional on all smooth real $(n-1,n-1)$-forms, regard the extension as a positive $(1,1)$-current $T$, and show that $T-\theta $ is $\mathrm{dd}^{\mathrm{c}}$ of a distribution.

A positive extension. Let $F$ be the real vector space of smooth real $(n-1,n-1)$-forms on $X$, let $S=\{ \tau \in F:\mathrm{dd}^{\mathrm{c}}\tau =0\} $, and put $e=\omega ^{n-1}$, with $e=1$ if $n=1$. Since $\omega $ is closed, $e\in S$, and $e$ is strictly positive. Define $\ell _0(\tau )=\int _ X\theta \wedge \tau $ for $\tau \in S$. By hypothesis $\ell _0(\tau )\geq 0$ when $\tau \in S$ is strictly positive. If $\tau \in S$ is merely positive, then $\tau +te\in S$ is strictly positive for every $t>0$. Indeed, at each point $x$ the cone of positive forms contains a ball about $e(x)$, for some norm; since the cone is stable under sums and positive multiples, it also contains the ball of $t$ times that radius about $\tau (x)+te(x)$. Hence $\ell _0(\tau )+t\ell _0(e)\geq 0$, and letting $t\downarrow 0$ gives $\ell _0(\tau )\geq 0$.

We next compare the forms in $F$ with $e$. Fix finitely many holomorphic charts $U_1,\ldots ,U_ N$ and compact sets $K_ a\subset U_ a$ with $\bigcup _ aK_ a=X$. In $U_ a$ write $\eta \in F$ as $\eta =\sum _{I,J}\eta ^ a_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$, the sum running over increasing multi-indices of length $n-1$, and put

We claim that there is a constant $M>0$ such that

Let $Z$ be the finite-dimensional real vector space of real $(n-1,n-1)$-forms with constant coefficients on $\mathbb C^ n$, normed by the maximum modulus of the coefficients, and let $P\subset Z$ be the closed convex cone of positive forms (proposition 5.3.2). For $x\in U_ a$, the differential of the chart is a complex-linear isomorphism from the tangent space at $x$ onto $\mathbb C^ n$, under which $\mathrm dz_ j$ corresponds to the $j$th coordinate form. Since complex-linear isomorphisms preserve positivity, a real $(n-1,n-1)$-form $\eta $ is positive at $x$ if and only if its coefficient form $\eta ^ a(x)=\sum _{I,J}\eta ^ a_{IJ}(x)\, \mathrm dz_ I\wedge \mathrm d\bar z_ J\in Z$ lies in $P$.

Because $e$ is strictly positive, the continuous map $x\mapsto e^ a(x)$ sends the compact set $K_ a$ into the interior of $P$; the complement of this interior is closed and disjoint from the compact image, so it has positive distance $r_ a$ from it. Put $r=\frac12\min _ ar_ a$. If $\eta \in F$ satisfies $\| \eta \| \leq r$ and $x\in K_ a$, then $e^ a(x)\pm \eta ^ a(x)$ lies within distance $r<r_ a$ of $e^ a(x)$, hence in $P$; thus $e\pm \eta $ is positive at $x$. Since the sets $K_ a$ cover $X$, the forms $e\pm \eta $ are positive. Applying this to $r\eta /\| \eta \| $ for $\eta \neq 0$ proves (17.5.31) with $M=1/r$.

For $\eta \in F$, a form $\tau \in S$ with $\tau \geq \eta $ is called an $S$-majorant of $\eta $, and we put

Let $\eta \in F$ and $C=M\| \eta \| $, so that $-Ce\leq \eta \leq Ce$ by (17.5.31). The form $Ce$ is an $S$-majorant of $\eta $, so $p(\eta )\leq C\ell _0(e)$. If $\tau $ is an $S$-majorant of $\eta $, then $\tau +Ce\geq \eta +Ce\geq 0$, so $\tau +Ce\in S$ is positive and $\ell _0(\tau )\geq -C\ell _0(e)$. Thus $p$ is finite. It is sublinear: the sum of $S$-majorants of $\eta _1$ and $\eta _2$ is an $S$-majorant of $\eta _1+\eta _2$, and for $\lambda >0$ the $S$-majorants of $\lambda \eta $ are $\lambda $ times those of $\eta $. If $\eta \in S$, then $\eta $ is its own $S$-majorant, and every $S$-majorant $\tau $ of $\eta $ satisfies $\ell _0(\tau )-\ell _0(\eta )=\ell _0(\tau -\eta )\geq 0$ because $\tau -\eta \in S$ is positive. Hence $p=\ell _0$ on $S$.

Next we extend $\ell _0$ to a linear functional on $F$ dominated by $p$; this is the Hahn–Banach theorem, whose proof we recall. Let $\ell $ be a linear functional on a subspace $W\supset S$ which extends $\ell _0$ and satisfies $\ell \leq p$ on $W$, and let $\eta _0\in F\setminus W$. For $\eta _1,\eta _2\in W$,

so

Taking $\eta _1=\eta _2=0$ shows that the supremum is greater than $-\infty $ and the infimum is less than $+\infty $; hence both are finite, and we may choose a real number $c$ between them. Define $\ell (\eta +t\eta _0)=\ell (\eta )+tc$ for $\eta \in W$ and $t\in \mathbb R$. For $t>0$, the upper bound for $c$ with $\eta _2=\eta /t$, multiplied by $t$, gives $\ell (\eta )+tc\leq p(\eta +t\eta _0)$; for $t<0$, the lower bound for $c$ with $\eta _1=\eta /|t|$, multiplied by $|t|$, gives the same inequality. Thus the extension to $W+\mathbb R\eta _0$ is still dominated by $p$. The union of a chain of such extensions is again such an extension, so Zorn's lemma gives a maximal one, and by the step just proved its domain is $F$. We denote it again by $\ell $.

The functional $\ell $ is positive: if $\eta \in F$ is positive, then $0$ is an $S$-majorant of $-\eta $, so $\ell (-\eta )\leq p(-\eta )\leq 0$. It is also bounded: by (17.5.31), the form $M\| \eta \| e$ is an $S$-majorant of both $\eta $ and $-\eta $, so

We now turn $\ell $ into a current. Every smooth form on the compact manifold $X$ is a test form. For a complex form $\varphi $ of degree $2n-2$, let $\varphi '$ be its component of bidegree $(n-1,n-1)$, and put $T(\varphi )=\ell (\operatorname{Re}\varphi ')+\mathrm i\, \ell (\operatorname{Im}\varphi ')$, where $\operatorname{Re}\varphi '=(\varphi '+\overline{\varphi '})/2$ and $\operatorname{Im}\varphi '=(\varphi '-\overline{\varphi '})/(2\mathrm i)$ are real forms in $F$. This is a complex-linear functional: replacing $\varphi $ by $\mathrm i\varphi $ replaces $(\operatorname{Re}\varphi ',\operatorname{Im}\varphi ')$ by $(-\operatorname{Im}\varphi ',\operatorname{Re}\varphi ')$. Moreover, $T$ has the continuity property of a current of degree $2$. Indeed, let $Q$ be a compact subset of a chart $U$, and let $\varphi $ be supported in $Q$, with coefficients $\varphi _ J$ in real coordinates on $U$. On the compact set $K_ a\cap Q$, the coefficients of $\operatorname{Re}\varphi '$ and $\operatorname{Im}\varphi '$ in the chart $U_ a$ are linear combinations of the functions $\varphi _ J$ and $\overline{\varphi _ J}$ with coefficients that are smooth on $U\cap U_ a$, and they vanish on $K_ a\setminus Q$. Hence $\| \operatorname{Re}\varphi '\| +\| \operatorname{Im}\varphi '\| \leq C_ Q\sum _ J\sup _ Q|\varphi _ J|$ with a constant $C_ Q$ independent of $\varphi $, and (17.5.32) gives $|T(\varphi )|\leq MC_ Q\ell _0(e)\sum _ J\sup _ Q|\varphi _ J|$.

By construction $T$ vanishes on forms of bidegree other than $(n-1,n-1)$, so it is a current of bidegree $(1,1)$. It is real, because the $(n-1,n-1)$-component of $\overline\varphi $ is $\overline{\varphi '}=\operatorname{Re}\varphi '-\mathrm i\operatorname{Im}\varphi '$, so that $T(\overline\varphi )=\overline{T(\varphi )}$. It is positive, because a strongly positive test form $\varphi $ of bidegree $(n-1,n-1)$ is real and positive, so that $T(\varphi )=\ell (\varphi )\geq 0$. Since $\ell $ extends $\ell _0$, the real $(1,1)$-current $R=T-\theta $ annihilates $S$.

The current $R$ is $\mathrm{dd}^{\mathrm{c}}$-exact. We first show that $R$ is closed. If $\xi $ is a smooth form of bidegree $(n-2,n-1)$, then $\mathrm{dd}^{\mathrm{c}}\partial \xi =\frac{\mathrm i}{2\pi }\partial \bar\partial \partial \xi =-\frac{\mathrm i}{2\pi }\partial \partial \bar\partial \xi =0$, and $\mathrm{dd}^{\mathrm{c}}\bar\partial \xi '=\frac{\mathrm i}{2\pi }\partial \bar\partial \bar\partial \xi '=0$ for $\xi '$ of bidegree $(n-1,n-2)$. Since $\mathrm{dd}^{\mathrm{c}}$ is a real operator, the conjugate of a $\mathrm{dd}^{\mathrm{c}}$-closed complex $(n-1,n-1)$-form $\sigma $ is $\mathrm{dd}^{\mathrm{c}}$-closed, so the real forms $\operatorname{Re}\sigma $ and $\operatorname{Im}\sigma $ lie in $S$; as $R$ is complex-linear and annihilates $S$, it annihilates $\sigma $. In particular $R(\partial \xi )=R(\bar\partial \xi ')=0$. For a smooth $(2n-3)$-form $\gamma $, the $(n-1,n-1)$-component of $\mathrm d\gamma $ is $\partial \gamma ^{n-2,n-1}+\bar\partial \gamma ^{n-1,n-2}$; since $R$ has bidegree $(1,1)$, it follows that $R(\mathrm d\gamma )=0$. Therefore $\mathrm dR=0$.

Since $e\in S$, we also have $\int _ XR\wedge \omega ^{n-1}=0$. Let $\Lambda $ denote contraction with $\omega $, as in lemma 10.2.12; the identity $R\wedge \omega ^{n-1}=(\Lambda R)\omega ^ n/n$ shows that the trace $\Lambda R$ has mean zero. By lemma 10.2.5(2), applied to the Laplacian $L=\mathrm d^*\mathrm d$ of section 10.2, there is a real distribution $f$ with $Lf=-4\pi \Lambda R$. By lemma 10.2.12, $\Lambda \mathrm{dd}^{\mathrm{c}}f=\Lambda R$, so $R'=R-\mathrm{dd}^{\mathrm{c}}f$ is a closed real $(1,1)$-current with zero trace. By the last assertion of lemma 10.2.12, $R'$ is harmonic, and by lemma 10.2.5(1) it is smooth. If $n=1$, a $(1,1)$-form with zero trace vanishes, so $R'=0$. If $n\geq 2$, (10.2.13) gives $*R'=-R'\wedge \omega ^{n-2}/(n-2)!$, a smooth closed $(n-1,n-1)$-form; hence $*R'\in S$. Since $R$ annihilates $S$ and $\mathrm{dd}^{\mathrm{c}}{*R'}=0$,

Thus $R'=0$ and $R=\mathrm{dd}^{\mathrm{c}}f$ in every dimension.

Conclusion. The current $T=\theta +\mathrm{dd}^{\mathrm{c}}f$ is closed and positive, and it lies in the class $\{ \theta \} $, since it differs from $\theta $ by $\mathrm{dd}^{\mathrm{c}}$ of a distribution. By corollary 10.2.19, there is $u\in \mathrm{PSH}(X,\theta )$ with $\theta +\mathrm{dd}^{\mathrm{c}}u=T$. This proves sufficiency.

For the last assertion, $\{ \theta \} $ is big exactly when $\mathrm{PSH}(X,\theta -\delta \omega )\neq \varnothing $ for some $\delta >0$. Indeed, if $T\geq \delta \omega $ is a Kähler current in $\{ \theta \} $ (definition 10.4.1), then $T-\delta \omega $ is a closed positive current in the class $\{ \theta -\delta \omega \} $, so $T-\delta \omega =\theta -\delta \omega +\mathrm{dd}^{\mathrm{c}}u$ with $u\in \mathrm{PSH}(X,\theta -\delta \omega )$, by corollary 10.2.19. Conversely, if $u\in \mathrm{PSH}(X,\theta -\delta \omega )$, then $\theta +\mathrm{dd}^{\mathrm{c}}u\geq \delta \omega $ is a Kähler current in $\{ \theta \} $. Applying the first part to $\theta -\delta \omega $ shows that this holds if and only if (17.5.30) holds for that $\delta $. $\square$