Control the convex combination of two classes on the boundary of the kahler cone Let $(X,w)$ be a compact kahler manifold, and $[\eta]$ be a class is on the boundary of the kahler cone. The claim is that one can find another class $[\beta]$ also on the boundary of the kahler cone such that $(1-t)[\beta] + t[\eta] > 0$ when $t \in (0, 1)$. How can one prove this?
Some thoughts:
according to  this question, one reasonable way to prove the claim is to find some $k < 0$ such that $[w] + k[\eta]$ not kahler. Then there is some chance for this choice to be on the boundary of the cone. Now consider $(1-t)([w] + k[\eta]) + t[\eta] = (1-t)[w] + (k + (1-k)t)[\eta]$. If $(k + (1-k)t)$ is positive then the resulting sum will be kahler. However, how can one gaurantee that 1).$[w] + k[\eta]$ is on the boundary of the cone  2). $(k + (1-k)t) > 0$?
 A: Assume $[\eta]$ is on the boundary of the Kähler cone and that $[\eta] \neq 0.$ 
Define $k := -\sup\{t > 0 : [\omega]-t[\eta] \hskip4pt {\rm Kähler}\}$.  We claim that $k > -\infty$.  
If $k = -\infty,$ then $\frac{1}{t}[\omega]-[\eta]$ is Kähler for all $t > 0,$ so that $-[\eta]$ is nef.  The claim will be proved once we show this implies $[\eta]=0.$ 
Approach 1, suggested by @YangMills in the comments below:
By weak compactness of currents, nef classes are pseudoeffective, i.e. they contain closed positive currents. So there are distributions $,$ such that $\eta+^{c} \geq 0$ and $−{\eta}+^{c}v \geq 0$. So $^{}(+) \geq 0$ i.e. $+$ is plurisubharmonic, hence constant. Thus $\eta$ and $-{\eta}$ differ by $^{}$ of a distribution, which must be smooth by regularity of $^{}$. So $[\eta]$ and $-[\eta]$ are cohomologous, hence $[\eta]=0.$
Approach 2, admittedly overkill but of a more algebro-geometric flavor: 
Since both $[\eta]$ and $[-\eta]$ are nef, it follows from the characterization of nefness given as part (iii) of Theorem 4.3 in the Demailly-Paun paper 
https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n3-p05.pdf
that for every Kähler class ${\omega}'$ on $X$ we have $$\int_{X}\eta~ \wedge ({\omega}')^{\dim(X)-1} = 0$$
By nondegeneracy we must then have $[\eta]=0.$ 
We then have that $k > -\infty$ as claimed.              
It is clear that $[\omega]+k[\eta]$ lies on the boundary of the Kähler cone.  In order to show that $$(1−)([\omega]+[])+[] = (1−)[\omega]+(+(1−))[]$$ is Kähler for all $t \in (0,1)$ it is enough to verify that 
$$\frac{k+(1-k)t}{1-t} > k$$ for all $t \in (0,1),$ which is straightforward. 
