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!
@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.
Just for fun, here is an answer in the purely algebraic setting.
So, suppose that $X$ is irreducible projective algebraic of dimension $n$, $\alpha=c_1(\mathcal O_X(D))$ is the class of a nef divisor $D$, and $\omega=c_1(\mathcal O_X(A))$ is the class of an ample divisor.
What we want to show is that if there exists an integer $1\le k \le n$ such that $D^k\cdot A^{n-k}=0$, then for every irreducible subvariety $Z\subseteq X$ of dimension $k$, one has $D^k\cdot Z=0$.
By induction on $n$, the case $n=1$ being clear.
If $k=n$, then there is nothing to prove. Otherwise, fix a $k$-dimensional irreducible subvariety $Z\subseteq X$ and, by taking a large multiple $mA$ of $A$, choose a divisor $H\in|mA|$ containing $Z$, say $$ H=m_1 H_1+\cdots+m_N H_N, $$ where the $H_j$'s are the reduced irreducible components of $H$, $m_j>0$, and $Z\subseteq H_1$.
Now, $$ \begin{aligned} 0 & =D^k\cdot A^{n-k}=\frac 1m\, D^k\cdot H\cdot A^{(n-1)-k} \\ & =\frac 1m\sum m_j\, D^k\cdot H_j\cdot A^{(n-1)-k} \\ & =\frac 1m\sum m_j\, D_{H_j}^k\cdot A_{H_j}^{(n-1)-k} \end{aligned} $$ where the $D_{H_j}$'s and $A_{H_j}$'s are respectively the restrictions of $D$ and $A$ to $H_j$.
Since $D_{H_j}$ is nef and $A_{H_j}$ is ample, we obtain that each $D_{H_j}^k\cdot A_{H_j}^{(n-1)-k}$ must be zero. In particular, $D_{H_1}^k\cdot A_{H_1}^{(n-1)-k}=0$.
Since $\dim H_1=n-1$, by induction hypothesis we have that $D_{H_1}^k$ intersects in zero each subvariety of $H_1$ of dimension $k$. But then, $0=D_{H_1}^k\cdot Z=D^k\cdot Z$.
