Suppose $M$ is a complex $n$-dimensioanl compact Kähler manifold and $\omega$ a Kähler class. Suppose $\alpha\in H^{1,1}(M,\mathbb{R})$ is a nef class belonging to the boundary of the Kähler cone of $M$. If for some $1\leq k\leq n-1$ we have 


$$\int\_M\alpha^k\omega^{n-k}=0,$$


can we conclude that $\alpha^k=0\in H^{k,k}(M,\mathbb{R})?$


If the answer is generally no, any counterexample?


Thanks in advances!



A question about nef classes on compact Kähler manifolds
@Kevin I think the answer is 'yes'. Here is a proof: if $\alpha$ is nef, then, for every $\varepsilon >0$ the class $\alpha+\varepsilon \omega$ is Kahler, and in particular the class $(\alpha+\varepsilon\omega)^k$ contains a positive $(k,k)$-current. We let $\varepsilon$ go to $0$, and obtain in the class $\alpha^k$ a positive $(k,k)$-current $T$. Since $\int\_X\alpha^k\wedge\omega^{n-k}=\int\_XT\wedge \omega^{n-k}=0$, it follows that $T\wedge\omega^{n-k}=0$, hence the trace measure of $T$ is $0$, hence $T$ has to be zero.


