1 Introduction

Let \(X\) be a compact complex manifold with \(\dim _{\mathbb {C}}X=n\). Following various authors (e.g. [2]), Xiao makes in [6] the following assumption:

\((H)\) :

there exists a Hermitian metric \(\omega \) on \(X\) such that

$$\begin{aligned} \partial \bar{\partial }\omega ^k=0\quad \text {for all k}=1, 2, \dots , n-1. \end{aligned}$$

It is clear that \((H)\) holds if \(X\) is a Kähler manifold. It is also standard and easy to check that condition \((H)\) is equivalent to either of the following two equivalent conditions:

$$\begin{aligned}\partial \bar{\partial }\omega =0 \quad \text{ and } \quad \partial \bar{\partial }\omega ^2=0 \iff \partial \bar{\partial }\omega =0 \quad \text{ and } \quad \partial \omega \wedge \bar{\partial }\omega =0.\end{aligned}$$

Following Xiao’s method in [6], itself inspired by earlier authors, especially Chiose [2], we prove the following statement in which a real Bott–Chern class of bidegree \((1,\,1)\) being nef means, as usual, that it contains \(C^{\infty }\) representatives with arbitrarily small negative parts (see inequalities (1)).

Theorem 1.1

Let \(X\) be a compact complex manifold with \(\dim _{\mathbb {C}}X=n\) satisfying the assumption \((H)\). Then, for any nef Bott-Chern cohomology classes \(\{\alpha \},\,\{\beta \}\in H^{1,\,1}_{BC}(X,\,\mathbb {R})\), the following implication holds:

$$\begin{aligned} \{\alpha \}^n - n\,\{\alpha \}^{n-1}.\,\{\beta \} >0 \Longrightarrow \text{ the } \text{ class }\,\,\{\alpha - \beta \}\,\, \hbox {contains a K}\ddot{ a}\hbox {hler current}.\end{aligned}$$

This answers affirmatively the qualitative part of a special version (i.e. the one for a difference of two nef classes) of Demailly’s transcendental Morse inequalities conjecture (see [1], Conjecture 10.1, \((ii)\)]) and will be crucial to the eventual extension of the duality theorem proved in [1], Theorem 2.2] to transcendental classes in the fairly general context of compact Kähler (not necessarily projective) manifolds. Although the method we propose here also produces a lower bound for the volume of the difference class \(\{\alpha - \beta \}\), this bound (that we will not present here) is weaker than the lower bound \(\{\alpha \}^n - n\,\{\alpha \}^{n-1}.\,\{\beta \}\) predicted in the quantitative part of Conjecture 10.1, \((ii)\) in [1].

Xiao proves in [6] the existence of a Kähler current in the class \(\{\alpha - \beta \}\) under the stronger assumption \(\{\alpha \}^n - 4n\,\{\alpha \}^{n-1}.\,\{\beta \} >0\) and the same assumption \((H)\) on \(X\). The two ingredients he uses are as follows.

Lemma 1.2

(Lamari’s duality lemma, [3], Lemme 3.3]) Let \(X\) be a compact complex manifold with \(\dim _{\mathbb {C}}X=n\) and let \(\alpha \) be any \(C^{\infty }\) real \((1,\,1)\)-form on \(X\). Then the following two statements are equivalent.

(i):

There exists a distribution \(\psi \) on \(X\) such that \(\alpha + i\partial \bar{\partial }\psi \ge 0\) in the sense of \((1,\,1)\)-currents on \(X.\)

(ii):

\(\int _X\alpha \wedge \gamma ^{n-1}\ge 0\) for any Gauduchon metric \(\gamma \) on \(X.\)

As an aside, we notice that this statement, when applied to \(d\)-closed real \((1,\,1)\)-forms \(\alpha \), translates to the pseudo-effective cone \(\mathcal{E}_X\subset H^{1,\,1}_{BC}(X,\,\mathbb {R})\) of \(X\) and the closure of the Gauduchon cone \(\overline{\mathcal{G}_X}\subset H^{n-1,\,n-1}_A(X,\,\mathbb {R})\) of \(X\) being dual under the duality between the Bott-Chern cohomology of bidegree \((1,\,1)\) and the Aeppli cohomology of bidegree \((n-1,\,n-1)\). (See [4] for the definition of the Gauduchon cone.)

Theorem 1.3

(The Tosatti–Weinkove resolution of Hermitian Monge–Ampère equations, [5]) Let \(X\) be a compact complex manifold with \(\dim _{\mathbb {C}}X=n\) and let \(\omega \) be a Hermitian metric on \(X\).

Then, for any \(C^{\infty }\) function \(F\,:\,X\rightarrow \mathbb {R}\), there exist a unique constant \(C>0\) and a unique \(C^{\infty }\) function \(\varphi \,:\,X\rightarrow \mathbb {R}\) such that

$$\begin{aligned}(\omega + i\partial \bar{\partial }\varphi )^n = C e^F\omega ^n, \omega + i\partial \bar{\partial }\varphi >0 \text{ and } \sup \limits _X\varphi =0.\end{aligned}$$

As a matter of fact, Yau’s classical theorem that solved the Calabi Conjecture, of which Theorem 1.3 is a generalisation to the possibly non-Kähler context, suffices for the proof of Theorem 1.1 whose assumptions imply that \(X\) must be Kähler (as already pointed out by Xiao in his situation based on [2], Theorem 0.2]) although this is not used either here or in Xiao’s work.

2 Xiao’s approach

In this section, we simply reproduce Xiao’s arguments (themselves inspired by earlier authors) up to the point where we will branch off in a different direction in the next section to handle certain estimates.

Let us fix a Hermitian metric \(\omega \) on \(X\) such that \(\partial \bar{\partial }\omega ^k=0\) for all \(k\). We also fix nef Bott–Chern \((1,\,1)\)-classes \(\{\alpha \}, \{\beta \}\). By the nef assumption, for every \(\varepsilon >0\), there exist \(C^{\infty }\) functions \(\varphi _{\varepsilon }, \psi _{\varepsilon }\,:\,X\rightarrow \mathbb {R}\) such that

$$\begin{aligned} \alpha _{\varepsilon }:=\alpha + \varepsilon \,\omega + i\partial \bar{\partial }\varphi _{\varepsilon }>0 \text{ and } \beta _{\varepsilon }:=\beta + \varepsilon \,\omega + i\partial \bar{\partial }\psi _{\varepsilon }>0 \text{ on } X.\end{aligned}$$
(1)

Note that \(\alpha _{\varepsilon }\) and \(\beta _{\varepsilon }\) need not be \(d\)-closed, but the property \(\partial \bar{\partial }\omega ^k=0\) yields:

$$\begin{aligned} \partial \bar{\partial }\alpha _{\varepsilon }^k = \partial \bar{\partial }\beta _{\varepsilon }^k=0 \text{ and } \partial \bar{\partial }(\alpha + \varepsilon \,\omega )^k = \partial \bar{\partial }(\beta + \varepsilon \,\omega )^k=0\end{aligned}$$
(2)

for all \(k=1, 2, \dots , n-1\). We normalise \(\sup \nolimits _X\varphi _{\varepsilon } = \sup \nolimits _X\psi _{\varepsilon }=0\) for every \(\varepsilon >0\).

Let us fix \(\varepsilon >0\). The existence of a Kähler current in the class \(\{\alpha - \beta \} = \{\alpha _{\varepsilon } - \beta _{\varepsilon }\}\) is equivalent to

$$\begin{aligned}\exists \,\delta >0,\,\exists \,\,\text{ a } \text{ distribution }\,\theta _{\delta }\,\,\text{ on }\,X \text{ such } \text{ that } \alpha _{\varepsilon } - \beta _{\varepsilon } + i\partial \bar{\partial }\theta _{\delta }\ge \delta \,\alpha _{\varepsilon },\end{aligned}$$

which, in view of Lamari’s duality Lemma 1.2, is equivalent to

$$\begin{aligned}\exists \delta >0 \text{ such } \text{ that } \int \limits _X(\alpha _{\varepsilon } - \beta _{\varepsilon })\wedge \gamma ^{n-1}\ge \delta \,\int \limits _X\alpha _{\varepsilon }\wedge \gamma ^{n-1}\end{aligned}$$

for every Gauduchon metric \(\gamma \) on \(X\). This is, of course, equivalent to

$$\begin{aligned}\exists \delta >0 \text{ such } \text{ that } (1-\delta )\,\int \limits _X\alpha _{\varepsilon }\wedge \gamma ^{n-1}\ge \int \limits _X\beta _{\varepsilon }\wedge \gamma ^{n-1}\end{aligned}$$

for every Gauduchon metric \(\gamma \) on \(X\).

Xiao’s approach is to prove the existence of a Kähler current in the class \(\{\alpha - \beta \} = \{\alpha _{\varepsilon } - \beta _{\varepsilon }\}\) by contradiction. Suppose that no such current exists. Then, for every \(\varepsilon >0\) and every sequence of positive reals \(\delta _m\downarrow 0\), there exist Gauduchon metrics \(\gamma _{m,\,\varepsilon }\) on \(X\) such that

$$\begin{aligned} (1-\delta _m)\,\int \limits _X\alpha _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1} < \int \limits _X\beta _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1} =1 \text{ for } \text{ all } m\in \mathbb {N}^{\star },\,\varepsilon >0.\end{aligned}$$
(3)

The last identity is a normalisation of the Gauduchon metrics \(\gamma _{m,\,\varepsilon }\) which is clearly always possible by rescaling \(\gamma _{m,\,\varepsilon }\) by a positive factor. This normalisation implies that for every \(\varepsilon >0\), the positive definite \((n-1,\,n-1)\)-forms \((\gamma _{m,\,\varepsilon }^{n-1})_m\) are uniformly bounded in mass, hence after possibly extracting a subsequence we can assume the convergence \(\gamma _{m,\,\varepsilon }^{n-1}\rightarrow \Gamma _{\infty ,\,\varepsilon }\) in the weak topology of currents as \(m\rightarrow +\infty \), where \(\Gamma _{\infty ,\,\varepsilon }\ge 0\) is an \((n-1,\,n-1)\)-current on \(X\). Taking limits as \(m\rightarrow +\infty \) in (3), we get

$$\begin{aligned} \int \limits _X\alpha _{\varepsilon }\wedge \Gamma _{\infty ,\,\varepsilon }\le 1 \text{ for } \text{ all } \varepsilon >0.\end{aligned}$$
(4)

Note that the l.h.s. of (3) does not change if \(\alpha _{\varepsilon }\) is replaced with any \(\alpha _{\varepsilon } + i\partial \bar{\partial }u\) (thanks to \(\gamma _{m,\,\varepsilon }\) being Gauduchon), while \(\alpha _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1}\) is (after division by \(\gamma _{m,\,\varepsilon }^n\)) the trace of \(\alpha _{\varepsilon }\) w.r.t. \(\gamma _{m,\,\varepsilon }\) divided by \(n\) (i.e. the arithmetic mean of the eigenvalues). To find a lower bound for the trace that would contradict (3), it is natural to prescribe the volume form (i.e. the product of the eigenvalues) of some \(\alpha _{\varepsilon } + i\partial \bar{\partial }u_{m,\,\varepsilon }\) by imposing that it be, up to a constant factor, the strictly positive \((n,\,n)\)-form featuring in the r.h.s. of (3). More precisely, the Tosatti-Weinkove Theorem 1.3 allows us to solve the Monge-Ampère equation

$$\begin{aligned}(\star )_{m,\,\varepsilon } \qquad \quad (\alpha _{\varepsilon } + i\partial \bar{\partial }u_{m,\,\varepsilon })^n = c_{\varepsilon }\,\beta _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1} \end{aligned}$$

for any \(\varepsilon >0\) and any \(m\in \mathbb {N}^{\star }\) by ensuring the existence of a unique constant \(c_{\varepsilon }>0\) and of a unique \(C^{\infty }\) function \(u_{m,\,\varepsilon }\,:\,X\rightarrow \mathbb {R}\) satisfying \((\star )_{m,\,\varepsilon }\) such that

$$\begin{aligned}\widetilde{\alpha }_{m,\,\varepsilon }:=\alpha _{\varepsilon } + i\partial \bar{\partial }u_{m,\,\varepsilon } >0, \sup \limits _X(\varphi _{\varepsilon } + u_{m,\,\varepsilon }) = 0.\end{aligned}$$

Note that \(c_{\varepsilon }\) is independent of \(m\) since we must have

$$\begin{aligned} c_{\varepsilon } = \int \limits _X\widetilde{\alpha }_{m,\,\varepsilon }^n = \int \limits _X(\alpha + \varepsilon \omega )^n\downarrow \int \limits _X\alpha ^n:=c_0>0,\end{aligned}$$
(5)

where the non-increasing convergence is relative to \(\varepsilon \downarrow 0\). Indeed, the second identity in (5) follows from \(\partial \bar{\partial }(\alpha + \varepsilon \omega )^k=0\) for all \(k=1, 2, \dots , n-1\) (cf. (2)). Thus, it is significant that \(c_{\varepsilon }\) does not change if we add any \(i\partial \bar{\partial }u\) to \(\alpha \), i.e. \(c_{\varepsilon }\) depends only on the Bott–Chern class \(\{\alpha \}\), on \(\omega \) and on \(\varepsilon \). Analogously, one defines

$$\begin{aligned} M_{\varepsilon }: = \int \limits _X\widetilde{\alpha }_{m,\,\varepsilon }^{n-1}\wedge \beta _{\varepsilon } = \int \limits _X(\alpha + \varepsilon \omega )^{n-1}\wedge (\beta + \varepsilon \omega )\downarrow \int \limits _X\alpha ^{n-1}\wedge \beta :=M_0\ge 0,\end{aligned}$$
(6)

where the non-increasing convergence is relative to \(\varepsilon \downarrow 0\). Clearly, \(M_{\varepsilon }\) is independent of \(m\) and depends only on the Bott-Chern classes \(\{\alpha \}\), \(\{\beta \}\), on \(\omega \) and on \(\varepsilon \). Note that the second integral in (6) equals \(\int _X(\alpha + \varepsilon \omega + i\partial \bar{\partial }\varphi _{\varepsilon })^{n-1}\wedge (\beta + \varepsilon \omega + i\partial \bar{\partial }\psi _{\varepsilon })\) which is positive since \(\alpha _{\varepsilon }, \beta _{\varepsilon }>0\) by (1). Since \(M_0\ge 0\), the hypothesis \(c_0 - nM_0>0\) made in Theorem 1.1 implies \(c_0>0\). This justifies the final claim in (5).

3 Estimates in the Monge–Ampère equation

We now propose an approach to the details of these estimates that differs from that of Xiao. We start with a very simple, elementary (and probably known) observation.

Lemma 3.1

For any Hermitian metrics \(\alpha , \beta , \gamma \) on a complex manifold, the following inequality holds at every point:

$$\begin{aligned} (\Lambda _{\alpha }\beta )\cdot (\Lambda _{\beta }\gamma ) \ge \Lambda _{\alpha }\gamma .\end{aligned}$$
(7)

Proof

Since (7) is a pointwise inequality, we fix an arbitrary point \(x\) and choose local coordinates about \(x\) such that

$$\begin{aligned} \beta (x) \!=\! \sum \limits _j i dz_j\!\wedge d\bar{z}_j, \quad \alpha (x) \!=\! \sum \limits _j \alpha _j\,i dz_j\!\wedge d\bar{z}_j \quad \text{ and } \quad \gamma (x) \!=\! \sum \limits _{j,\,k} \gamma _{j\bar{k}}\,i dz_j\!\wedge d\bar{z}_k. \end{aligned}$$

Then \(\alpha _j>0\) and \(\gamma _{j\bar{j}}>0\) for every \(j\). If we denote by the same symbol any \((1,\,1)\)-form and its coefficient matrix in the chosen coordinates, we have

$$\begin{aligned} \alpha ^{-1}\,\gamma = \left( \frac{1}{\alpha _j}\,\gamma _{j\bar{k}}\right) _{j,\,k},\hbox { hence }\text{ Tr }(\alpha ^{-1}\,\gamma ) = \sum \limits _j\frac{1}{\alpha _j}\,\gamma _{j\bar{j}}. \end{aligned}$$

Thus (7) translates to \((\sum _j\frac{1}{\alpha _j})\,\sum _k\gamma _{k\bar{k}} \ge \sum _j\frac{1}{\alpha _j}\,\gamma _{j\bar{j}}\) which clearly holds since \(\sum _{j\ne k}\frac{1}{\alpha _j}\,\gamma _{k\bar{k}} >0\) because all the \(\alpha _j\) and all the \(\gamma _{k\bar{k}}\) are positive. \(\square \)

Our main observation is the following statement.

Lemma 3.2

For every \(m\in \mathbb {N}^{\star }\) and every \(\varepsilon >0\), we have:

$$\begin{aligned} \left( \int \limits _X\widetilde{\alpha }_{m,\,\varepsilon } \wedge \gamma _{m,\,\varepsilon }^{n-1}\right) \cdot \left( \int \limits _X\widetilde{\alpha }_{m,\,\varepsilon }^{n-1}\wedge \beta _{\varepsilon }\right) \ge \frac{1}{n}\,\int \limits _X\widetilde{\alpha }_{m,\,\varepsilon }^n = \frac{c_{\varepsilon }}{n}.\end{aligned}$$
(8)

Proof

Let \(0<\lambda _1\le \lambda _2\le \dots \le \lambda _n\), resp. \(0<\mu _1\le \mu _2\le \dots \le \mu _n\), be the eigenvalues of \(\widetilde{\alpha }_{m,\,\varepsilon }\), resp. \(\beta _{\varepsilon }\), w.r.t. \(\gamma _{m,\,\varepsilon }\). We have:

$$\begin{aligned} \displaystyle \widetilde{\alpha }_{m,\,\varepsilon }^n \!=\! \lambda _1\dots \lambda _n\,\gamma _{m,\,\varepsilon }^n \quad \text{ and }\quad \widetilde{\alpha }_{m,\,\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1} \!=\! \frac{1}{n}\,(\Lambda _{\gamma _{m,\,\varepsilon }}\widetilde{\alpha }_{m,\,\varepsilon })\,\gamma _{m,\,\varepsilon }^n \!=\! \frac{\lambda _1 \!+\! \cdots \!+\! \lambda _n}{n}\, \gamma _{m,\,\varepsilon }^n. \end{aligned}$$

Similarly, \(\beta _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1} = \frac{1}{n}\,(\Lambda _{\gamma _{m,\,\varepsilon }}\beta _{\varepsilon })\,\gamma _{m,\,\varepsilon }^n = \frac{\mu _1 + \dots + \mu _n}{n}\, \gamma _{m,\,\varepsilon }^n.\)

Thus, the Monge–Ampère equation \((\star )_{m,\,\varepsilon }\) translates to

$$\begin{aligned} \lambda _1\dots \lambda _n = c_{\varepsilon }\,\frac{\mu _1 + \dots + \mu _n}{n}.\end{aligned}$$
(9)

In particular, the normalisation \(\int _X\beta _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1} = 1\) reads

$$\begin{aligned} \frac{1}{c_{\varepsilon }}\,\int \limits _X\lambda _1\dots \lambda _n\, \gamma _{m,\,\varepsilon }^n = \int \limits _X\frac{\mu _1 + \dots + \mu _n}{n}\, \gamma _{m,\,\varepsilon }^n =1.\end{aligned}$$
(10)

Note that we also have

$$\begin{aligned} \widetilde{\alpha }_{m,\,\varepsilon }^{n-1}\wedge \beta _{\varepsilon } = \frac{1}{n}\,(\Lambda _{\widetilde{\alpha }_{m,\,\varepsilon }}\beta _{\varepsilon })\,\widetilde{\alpha }_{m,\,\varepsilon }^n = \frac{1}{n}\,(\Lambda _{\widetilde{\alpha }_{m,\,\varepsilon }}\beta _{\varepsilon })\,\lambda _1\dots \lambda _n\,\gamma _{m,\,\varepsilon }^n.\end{aligned}$$
(11)

Putting all of the above together, we get:

$$\begin{aligned}&\left( \int \limits _X\widetilde{\alpha }_{m,\,\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1}\right) \cdot \left( \int \limits _X\widetilde{\alpha }_{m,\,\varepsilon }^{n-1}\wedge \beta _{\varepsilon }\right) \\&\quad = \left( \int \limits _X\frac{1}{n}\,(\Lambda _{\gamma _{m,\,\varepsilon }}\widetilde{\alpha }_{m,\,\varepsilon })\,\gamma _{m,\,\varepsilon }^n\right) \cdot \left( \int \limits _X \frac{1}{n}\,(\Lambda _{\widetilde{\alpha }_{m,\,\varepsilon }}\beta _{\varepsilon })\,\lambda _1\dots \lambda _n\,\gamma _{m,\,\varepsilon }^n\right) \\&\quad \mathop {\ge }\limits ^{(a)} \frac{1}{n^2}\,\left( \int \limits _X\left[ (\Lambda _{\gamma _{m,\,\varepsilon }}\widetilde{\alpha }_{m,\,\varepsilon })\,(\Lambda _{\widetilde{\alpha }_{m,\,\varepsilon }}\beta _{\varepsilon })\right] ^{\frac{1}{2}}\,(\lambda _1\dots \lambda _n)^{\frac{1}{2}}\,\gamma _{m,\,\varepsilon }^n\right) ^2\\&\quad \mathop {\ge }\limits ^{(b)} \frac{1}{n^2}\,\left( \int \limits _X(\Lambda _{\gamma _{m,\,\varepsilon }}\beta _{\varepsilon })^{\frac{1}{2}}\,(\lambda _1\dots \lambda _n)^{\frac{1}{2}}\,\gamma _{m,\,\varepsilon }^n\right) ^2 \mathop {=}\limits ^{(c)} \frac{1}{n^2}\,\left( \int \limits _X\frac{\sqrt{n}}{\sqrt{c_{\varepsilon }}}\,\lambda _1\dots \lambda _n\,\gamma _{m,\,\varepsilon }^n\right) ^2\\&\quad = \frac{1}{n\,c_{\varepsilon }}\,\left( \int \limits _X\widetilde{\alpha }_{m,\,\varepsilon }^n\right) ^2 \mathop {=}\limits ^{(d)} \frac{1}{n\,c_{\varepsilon }}\,\left( \int \limits _Xc_{\varepsilon }\,\beta _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1}\right) ^2 \mathop {=}\limits ^{(e)} \frac{c_{\varepsilon }}{n}.\end{aligned}$$

This proves (8). Inequality \((a)\) is an application of the Cauchy–Schwarz inequality, inequality \((b)\) has followed from (7), identity \((c)\) has followed from (9), identity \((d)\) has followed from \(\widetilde{\alpha }_{m,\,\varepsilon }^n = c_{\varepsilon }\,\beta _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1}\) (which is nothing but the Monge–Ampère equation \((\star )_{m,\,\varepsilon }\)), while identity \((e)\) has followed from the normalisation \(\int _X\beta _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1} = 1\) (cf. (3)). The proof of Lemma 3.2 is complete. \(\square \)

End of proof of Theorem 1.1

Now, \(\widetilde{\alpha }_{m,\,\varepsilon } = \alpha _{\varepsilon } + i\partial \bar{\partial }u_{m,\,\varepsilon }\) and \(\partial \bar{\partial }\gamma _{m,\,\varepsilon }^{n-1}=0\), so

$$\begin{aligned} \int \limits _X\widetilde{\alpha }_{m,\,\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1} = \int \limits _X\alpha _{\varepsilon }\wedge \gamma _{m,\,\varepsilon }^{n-1}\longrightarrow \int \limits _X\alpha _{\varepsilon }\wedge \Gamma _{\infty ,\,\varepsilon }\le 1 \text{ for } \text{ all } \varepsilon >0,\end{aligned}$$
(12)

where the above arrow stands for convergence as \(m\rightarrow +\infty \) and the last inequality is nothing but (4) (which, recall, is a consequence of the assumption that no Kähler current exists in \(\{\alpha - \beta \}\)—an assumption that we are going to contradict). On the other hand, the second factor on the l.h.s. of (8) is precisely \(M_{\varepsilon }\) defined in (6), so in particular it is independent of \(m\). Fixing any \(\varepsilon >0\), taking limits as \(m\rightarrow +\infty \) in (8) and using (12), we get

$$\begin{aligned} M_{\varepsilon }\ge \frac{c_{\varepsilon }}{n} \text{ for } \text{ every }\varepsilon >0.\end{aligned}$$
(13)

Taking now limits as \(\varepsilon \downarrow 0\) and using (6) and (5), we get

$$\begin{aligned}M_0\ge \frac{c_0}{n}, \text{ i.e. } \{\alpha \}^{n-1}.\,\{\beta \}\ge \frac{\{\alpha \}^n}{n}.\end{aligned}$$

The last identity means that \(\{\alpha \}^n - n\,\{\alpha \}^{n-1}.\,\{\beta \}\le 0\) which is impossible if we suppose that \(\{\alpha \}^n - n\,\{\alpha \}^{n-1}.\,\{\beta \}> 0\). This is the desired contradiction proving the existence of a Kähler current in the class \(\{\alpha - \beta \}\) under the assumption \(\{\alpha \}^n - n\,\{\alpha \}^{n-1}.\,\{\beta \}> 0\). \(\square \)