PomeProposition 10.11.1.7F12058085DC For each fixed $\varepsilon >0$, the currents in $\alpha [-\varepsilon \omega ]$ have uniformly bounded distributional mass.
10.11 Closed classes with Hermitian lower bounds827D49511DDA
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Testing a closed current against a Gauduchon form eliminates its potential. The resulting trace depends only on the class, and it bounds the mass of every almost positive representative.
Recall from subsection 20.4.9 that $H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$ is the space of smooth closed real $(1,1)$-forms modulo $\mathrm{dd}^{\mathrm{c}}$ of smooth real functions, and that a class is pseudoeffective if it contains a closed positive current. It is nef if, for every $\varepsilon >0$, it has a smooth representative bounded below by $-\varepsilon \omega $ (subsection 20.4.16). Fix a class $\alpha \in H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$ and a smooth closed representative $\theta $. By lemma 20.4.10, the closed real $(1,1)$-currents in $\alpha $ are exactly the currents $T=\theta +\mathrm{dd}^{\mathrm{c}}u$ with $u$ a real distribution, and the potential of a positive one is quasi-plurisubharmonic. A closed current $T$ with $T\geq \delta \omega $ for some $\delta >0$ is called a Kähler current; we also say that such a current is strict. The class $\alpha $ is big if it contains a Kähler current, that is, if $\theta $ is big in the sense of section 5.40. None of these properties depends on the choice of $\theta $, because replacing $\theta $ by $\theta +\mathrm{dd}^{\mathrm{c}}f$ with $f$ smooth only changes the potentials by $-f$.
For a pseudoeffective class $\alpha $ and $\varepsilon >0$, let
\[ \alpha [-\varepsilon \omega ] =\{ T\in \alpha :T\geq -\varepsilon \omega \} \]
be the set of closed currents in $\alpha $ whose negative part is bounded by $\varepsilon \omega $. Proof. Choose a Gauduchon metric $\gamma $. For $T=\theta +\mathrm{dd}^{\mathrm{c}}u$ in $\alpha [-\varepsilon \omega ]$, the current $T+\varepsilon \omega $ is positive, and since $\mathrm{dd}^{\mathrm{c}}\gamma ^{n-1}=0$, Stokes' theorem for currents gives
\[ \int _ X(T+\varepsilon \omega )\wedge \gamma ^{n-1} =\int _ X(\theta +\varepsilon \omega )\wedge \gamma ^{n-1}. \]
The right-hand side does not depend on $T$. Since $\gamma $ is comparable to any fixed reference metric, this trace bounds all coefficient measures of the positive current $T+\varepsilon \omega $. Subtracting the fixed smooth form $\varepsilon \omega $ gives the asserted bound for $T$. In dimension one the same argument applies with $\gamma ^{n-1}=1$ and the ordinary Stokes theorem. $\square$
Proposition 10.11.2.A8D846A01D78 The pseudoeffective cone in $H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$ is closed and convex and contains no nonzero line. For a fixed Gauduchon metric $\gamma $, the trace functional $\ell _\gamma $ of subsection 20.4.9 is strictly positive on its nonzero elements, and the slice of pseudoeffective classes with $\ell _\gamma =1$ is compact. This slice may be empty. Pullback and pushforward by a compact modification are continuous linear maps on Bott–Chern cohomology and preserve pseudoeffectivity.
Proof. The finite-dimensional space of smooth representatives in lemma 20.4.20 identifies Bott–Chern cohomology with a finite-dimensional vector space and gives it its usual topology; the trace functional is continuous for this topology. Its positivity and the absence of lines in the cone were proved in subsection 20.4.9, and convexity follows by adding positive currents.
Let pseudoeffective classes $\alpha _ j$ converge to $\alpha $, and choose positive closed representatives $T_ j$. Their masses $\ell _\gamma (\alpha _ j)$ are bounded, so by weak compactness a subsequence converges to a positive closed current $T$. Let $B_\gamma $ be the continuous smooth representative map of lemma 20.4.20. Along the subsequence, $B_\gamma (T_ j)$ converges both to $B_\gamma (T)$ and to the representative of $\alpha $, so $T$ represents $\alpha $. Thus the cone is closed. On its mass-one slice, (20.4.21) bounds $B_\gamma (T)$ in the finite-dimensional representative space. The slice is therefore closed and bounded there, hence compact.
The class maps of a modification are linear, and they are well-defined because classes can be represented equally by smooth forms and by currents. As linear maps between finite-dimensional spaces, they are continuous. Pullback preserves positive closed currents, since it can be computed on local plurisubharmonic potentials as in proposition 20.4.22, and proper pushforward preserves positivity and closedness. Applying these maps to positive representatives proves the assertions about the cones. $\square$
This compactness is linear. The functional $\ell _\gamma $ pairs a closed current with a fixed smooth form; it is not a top-degree volume of the class, and it gives no bound for the Monge–Ampère masses of potentials.
The same compactness shows that a nef class $\alpha $ is pseudoeffective. Indeed, choose smooth representatives $\theta _ j\in \alpha $ with $\theta _ j\geq -\varepsilon _ j\omega $, where $\varepsilon _ j\downarrow 0$. They have the same Gauduchon trace and hence uniformly bounded mass. A weak cluster value is a closed current, positive because the negative parts tend to zero, and it belongs to $\alpha $ because the smooth representative map of lemma 20.4.20 is continuous.
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