17.5.27 Positive mass and bigness1080E9C054BA

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 $. $\square$

Proof. We prove a pointwise inequality for smooth forms, apply it to regularizations of the potentials of $S$ on small balls in $X\setminus A$, and globalize with a partition of unity.

Step 1: a pointwise inequality. Let $\sigma $ be a smooth positive definite real $(1,1)$-form. At a given point, choose linear coordinates on the tangent space that diagonalize the Hermitian matrices of $\omega $ and $\sigma $ simultaneously, so that $\omega =\sum _ j\gamma _ j$ and $\sigma =\sum _ j\lambda _ j\gamma _ j$ with $\lambda _ j>0$, where $\gamma _ j=\mathrm i\, \mathrm dz_ j\wedge \mathrm d\bar z_ j$, and put $\mathrm dV=\gamma _1\wedge \cdots \wedge \gamma _ n$. Since $\tau $ is strictly positive (section 5.3), the form $\tau -c\, \omega ^{n-1}$ is positive at this point for some $c>0$. The product of the strongly positive form $\gamma _ j$ with this positive form is a nonnegative multiple of $\mathrm dV$ (proposition 5.3.2), and $\gamma _ j\wedge \omega ^{n-1}=(n-1)!\, \mathrm dV$, so $\gamma _ j\wedge \tau \geq c(n-1)!\, \mathrm dV$; thus $\gamma _ j\wedge \tau =t_ j\, \mathrm dV$ with $t_ j>0$. Expanding the products, we obtain

Since $\lambda _ j\sum _ k\lambda _ k^{-1}\geq 1$ for every $j$, the product of these two ratios is at least $1/n$. If $\sigma $ is only semipositive, we apply this to $\sigma +s\omega $ and let $s\downarrow 0$. Thus, for every smooth semipositive real $(1,1)$-form $\sigma $, as densities with respect to any fixed smooth volume form,

Step 2: regularization on a ball. Since the local potentials of $S$ are locally bounded on the open set $X\setminus A$, every point of $X\setminus A$ is the center of a coordinate ball $B\Subset X\setminus A$ on which $S=\mathrm{dd}^{\mathrm{c}}v$ with $v$ plurisubharmonic and bounded. Fix such a ball, put $\eta =\mathrm{dd}^{\mathrm{c}}|z|^2$ as in section 7.8, and write $\nu =g\eta ^ n$ with $g$ smooth and positive on $B$. Since $v$ is bounded on $B$, the non-pluripolar powers of $S$ on $B$ are the Bedford–Taylor powers of $v$ (see the remark following definition 13.2.4); in particular $(\mathrm{dd}^{\mathrm{c}}v)^ n=mg\eta ^ n$.

Let $v_\varepsilon =v*\rho _\varepsilon $ be the regularizations of $v$ by the radial mollifiers of section 2.5. On each ball $B'$ concentric with $B$ and of smaller radius, the functions $v_\varepsilon $ are defined for $\varepsilon $ smaller than the difference of the radii, and by proposition 5.9.2(2) they are smooth and plurisubharmonic on $B'$ and decrease to $v$ as $\varepsilon \downarrow 0$. As recalled at the beginning of section 7.8, $(\mathrm{dd}^{\mathrm{c}}v_\varepsilon )^ n=\det \mathcal H(v_\varepsilon )\, \eta ^ n$. The last assertion of proposition 7.8.3, applied on $B$ to $v$ and the continuous density $mg$, therefore gives

The right side converges to $mg$ uniformly on compact subsets of $B$, by lemma 2.5.1(1) applied to the continuous function $(mg)^{1/n}$. Hence $\sqrt{F_\varepsilon g}$ is bounded below by functions converging uniformly to $\sqrt mg$ on compact sets, and for every nonnegative $\chi \in C^\infty _ c(B)$,

Moreover, for every $\chi \in C^\infty _ c(B)$, as $\varepsilon \downarrow 0$ along any sequence,

For the first convergence, note that $\int \chi \, \mathrm{dd}^{\mathrm{c}}v_\varepsilon \wedge \tau =\int v_\varepsilon \, \mathrm{dd}^{\mathrm{c}}(\chi \tau )$, by the definition of $\mathrm{dd}^{\mathrm{c}}$ on currents (section 5.3); since $v_\varepsilon \to v$ in $L^1$ on $\operatorname{Supp}\chi $ (lemma 2.5.1(2)), this tends to $\int v\, \mathrm{dd}^{\mathrm{c}}(\chi \tau )=\int \chi \, S\wedge \tau $.

For the second convergence, choose a ball $B'$ as above containing $\operatorname{Supp}\chi $, and a sequence $\varepsilon _ j\downarrow 0$. By theorem 7.3.7, applied on $B'$ to the closed positive current $\omega $ of bidimension $(n-1,n-1)$ and to $n-1$ copies of the decreasing sequence $(v_{\varepsilon _ j})$, the currents $(\mathrm{dd}^{\mathrm{c}}v_{\varepsilon _ j})^{n-1}\wedge \omega $ converge weakly to the Bedford–Taylor product $(\mathrm{dd}^{\mathrm{c}}v)^{n-1}\wedge \omega $. By proposition 7.2.2(1) and (4), these products are the exterior products with $\omega $ of the smooth forms $(\mathrm{dd}^{\mathrm{c}}v_{\varepsilon _ j})^{n-1}$ and of the current $(\mathrm{dd}^{\mathrm{c}}v)^{n-1}$, which equals $\langle S^{n-1}\rangle $ on $B$. (For $n=1$ both sides of the second convergence equal $\int \chi \, \omega $.)

Step 3: globalization. Let $\chi $ be a smooth function with compact support in $X\setminus A$ and $0\leq \chi \leq 1$. Cover $\operatorname{Supp}\chi $ by finitely many balls $B_ l$ as in Step 2, and write $\chi =\sum _ l\chi _ l$ with nonnegative $\chi _ l\in C^\infty _ c(B_ l)$; for instance, multiply $\chi $ by a partition of unity with compact supports subordinate to the cover of $X$ by the balls $B_ l$ and $X\setminus \operatorname{Supp}\chi $ (proposition 5.2.6). In each $B_ l$ we use the notation of Step 2 with an index $l$ and put

Let $\alpha _{l,\varepsilon }$ and $\beta _{l,\varepsilon }$ be the densities of $\mathrm{dd}^{\mathrm{c}}v_{l,\varepsilon }\wedge \tau $ and $(\mathrm{dd}^{\mathrm{c}}v_{l,\varepsilon })^{n-1}\wedge \omega $ with respect to $\eta _ l^ n$. By (17.5.35) with $\sigma =\mathrm{dd}^{\mathrm{c}}v_{l,\varepsilon }$, $\alpha _{l,\varepsilon }\beta _{l,\varepsilon }\geq F_{l,\varepsilon }g_ l/n$, so the Cauchy–Schwarz inequality gives

Summing over $l$ and applying the Cauchy–Schwarz inequality to the finite sum $\sum _ l\sqrt{a_{l,\varepsilon }b_{l,\varepsilon }}$, we obtain

Now let $\varepsilon \downarrow 0$ along a sequence. By Step 2, the two sums on the left converge to $\int _ X\chi \, S\wedge \tau $ and $\int _ X\chi \langle S^{n-1}\rangle \wedge \omega $. These are at most the corresponding integrals over $X$ without $\chi $, because $S\wedge \tau $ and $\langle S^{n-1}\rangle \wedge \omega $ are positive measures and $0\leq \chi \leq 1$. By (17.5.36) and $g_ l\eta _ l^ n=\nu $, the lower limit of the right side is at least $(m/n)(\int _ X\chi \nu )^2$. Therefore

Finally, we remove the cutoff. Fix a metric $d$ inducing the topology of $X$ (lemma 5.2.7), and for $k\geq 1$ let $K_ k=\{ x\in X:d(x,A)\geq 1/k\} $, with $K_ k=X$ if $A=\varnothing $. These compact subsets of $X\setminus A$ increase to $X\setminus A$, because $A$ is closed. By proposition 5.2.6 there are smooth functions $\chi _ k\colon X\to [0,1]$ with compact support in $X\setminus A$ and $\chi _ k=1$ on $K_ k$. Since $\nu $ has a smooth density and $A$ has Lebesgue measure zero, $\nu (K_ k)\to \nu (X\setminus A)=\int _ X\nu $, so $\int _ X\chi _ k\nu \to \int _ X\nu $. Applying the last inequality to $\chi =\chi _ k$ and letting $k\to \infty $ proves (17.5.34). Both factors on its left are finite: the first is the pairing of a current with a smooth form on the compact manifold $X$, and the second is finite by theorem 13.5.4. $\square$

Proof. By proposition 13.9.2, big classes have positive volume, and every closed positive $(1,1)$-current $T$ satisfies $\int _ X\langle T^ n\rangle \leq \operatorname {vol}(\{ T\} )$. It therefore suffices to show that a pseudo-effective class $\alpha $ with $m=\operatorname {vol}(\alpha )>0$ is big, and we do so by verifying Lamari's criterion (lemma 17.5.28). Write $\alpha =\{ \theta \} $ with $\theta $ smooth and closed, choose $C>0$ such that $\theta +C\omega $ is positive definite, and put

the integrand is a positive multiple of the volume form at every point, by the positivity of mixed discriminants of positive definite matrices (lemma 5.3.3(2),(3)).

For $0<\varepsilon \leq 1$ let $\theta _\varepsilon =\theta +\varepsilon \omega $. Since $\alpha $ is pseudo-effective, there is $q\in \mathrm{PSH}(X,\theta )$ (corollary 10.2.19), and $\theta _\varepsilon +\mathrm{dd}^{\mathrm{c}}q\geq \varepsilon \omega $; thus $\{ \theta _\varepsilon \} =\alpha +\varepsilon \{ \omega \} $ is big. By proposition 16.21.2(4), $P_{\theta _\varepsilon }[V_{\theta _\varepsilon }]=V_{\theta _\varepsilon }$, and the mass of $V_{\theta _\varepsilon }$ is

the inequality holds because $\operatorname {vol}(\alpha +\varepsilon \{ \omega \} )$ decreases to $\operatorname {vol}(\alpha )$ as $\varepsilon \downarrow 0$, by definition 13.9.1, proposition 13.9.2. Hence $V_{\theta _\varepsilon }$ is a model potential of mass $m_\varepsilon $.

Fix a smooth strictly positive real $(n-1,n-1)$-form $\tau $ with $\mathrm{dd}^{\mathrm{c}}\tau =0$, normalized by $\int _ X\omega \wedge \tau =1$. The measure $m_\varepsilon \omega \wedge \tau $ has a smooth density with respect to $\omega ^ n$ and total mass $m_\varepsilon $. By corollary 17.5.26, applied in the big class $\{ \theta _\varepsilon \} $ with the reference $V_{\theta _\varepsilon }$, there are $u_\varepsilon \in \mathrm{PSH}(X,\theta _\varepsilon )$ and finite constants $B_\varepsilon $ such that

The constants $B_\varepsilon $ need not be bounded uniformly in $\varepsilon $; we only use that $u_\varepsilon $ is locally bounded outside a closed set of measure zero. By lemma 16.13.2, applied to the big class $\{ \theta _\varepsilon \} $, there is a closed pluripolar set $A_\varepsilon \subset X$ such that $V_{\theta _\varepsilon }$ is locally bounded on $X\setminus A_\varepsilon $; being pluripolar, $A_\varepsilon $ has Lebesgue measure zero in every chart (see the remarks after definition 5.47.2). Hence $u_\varepsilon $, which lies between $V_{\theta _\varepsilon }-B_\varepsilon $ and $V_{\theta _\varepsilon }$, is locally bounded on $X\setminus A_\varepsilon $, and so are the local potentials of $S_\varepsilon $, which differ from $u_\varepsilon $ by smooth local potentials of $\theta _\varepsilon $ (lemma 5.4.2).

We may thus apply lemma 17.5.33 to $S_\varepsilon $, with $A_\varepsilon $, $\nu =\omega \wedge \tau $ and $m_\varepsilon $ in place of $A$, $\nu $ and $m$:

We bound the two factors on the left. Since $\mathrm{dd}^{\mathrm{c}}\tau =0$, the first factor equals the cohomological pairing $\int _ X\theta _\varepsilon \wedge \tau =\int _ X\theta \wedge \tau +\varepsilon $. If $n\geq 2$, the form $\theta _\varepsilon +C\omega $ is positive definite, so theorem 13.5.4 gives

the last inequality holds because $\theta +(C+1)\omega =(\theta +(C+\varepsilon )\omega )+(1-\varepsilon )\omega $ and products of positive $(1,1)$-forms are positive. If $n=1$, the second factor equals $\int _ X\omega =C_0$. As the first factor is nonnegative, in both cases

Letting $\varepsilon \downarrow 0$ gives $\int _ X\theta \wedge \tau \geq m/(nC_0)$. Removing the normalization of $\tau $, we conclude that every smooth strictly positive $\mathrm{dd}^{\mathrm{c}}$-closed $(n-1,n-1)$-form $\tau $ satisfies (17.5.30) with $\delta =m/(nC_0)>0$. By lemma 17.5.28, the class $\alpha $ is big. $\square$

Proof. By proposition 16.11.4(1), $u\in \mathcal A_\theta (f)$; that is, $u\in \mathrm{PSH}(X,\theta )$ and $u\leq f$ quasi-everywhere. If $\{ \theta \} $ is not big, then $\mu _ u(X)=0$ by theorem 17.5.37, since otherwise $\theta +\mathrm{dd}^{\mathrm{c}}u$ would be a closed positive current of positive non-pluripolar mass in $\{ \theta \} $. Hence $\mu _ u=0$, and in particular the first assertion holds.

If $\{ \theta \} $ is big, the first assertion is theorem 16.13.4.

For the rooftop, the function $\min _ ju_ j$ is quasi-continuous, as a finite minimum of potentials (see the beginning of section 16.11), and it is bounded above; its envelope is $w$. Since $w$ is not identically $-\infty $, the family $\mathcal A_\theta (\min _ ju_ j)$ is nonempty, and it is uniformly bounded above because the obstacle is. By the first assertion, $\mu _ w$ is carried by $\{ w=\min _ ju_ j\} \subset \bigcup _ j\{ w=u_ j\} $. The obstacle is upper semicontinuous, so $w\leq u_ j$ everywhere by proposition 16.11.4(1),(2), and lemma 16.13.1 gives $\mathbf1_{\{ w=u_ j\} }\mu _ w\leq \mathbf1_{\{ w=u_ j\} }\mu _{u_ j}$ for every $j$. Summing over $j$ proves the rooftop inequality. $\square$

Proof. The potential $V_\theta +\min _ Xf$ lies below $f$, so $\mathcal A_\theta (f)$ is nonempty, and it is bounded above by $\max _ Xf$. Put $u=P_\theta (f)$. Since $f$ is continuous, $u\leq f$ everywhere by proposition 16.11.4(1),(2). The scalar case of theorem 16.20.2, with all $\theta _ j=\theta $, $u_ j=u$ and $f_ j=f$, gives

and corollary 17.5.38 shows that $\langle \theta _ u^ n\rangle $ is carried by $\{ u=f\} $. This proves the first identity. For $f=0$ the envelope $P_\theta (0)$ is $V_\theta $, by definition 10.3.7, which gives the second. If $\{ \theta \} $ is not big, the left sides vanish by corollary 17.5.38, and hence so do the right sides. $\square$