Let $X$ be a complex projective manifold of complex dimension $n$ and $A\to X$ an ample line bundle. Let $L\to X$ be a line bundle such that $$ c_1(L)^k\cdot c_1(A)^{n-k}>0,\quad k=1,\dots,n. $$ Is it true that then $L$ is big?

The answer is yes if $n\le 2$: for $n=1$ there is nothing to prove, and for $n=2$ the positivity of the top self intersection $c_1(L)^2>0$ says that $L$ or its dual is big. But then $c_1(L)\cdot c_1(A)>0$ implies that in fact $L$ is big.

The answer is again yes in all dimensions if $X$ is an abelian variety: in this case $L$ is moreover ample. This is because one can represent $c_1(L)$ and $c_1(A)$ by "constant" hermitian forms, the second being positive definite, and thus the intersection conditions simply tell that the elementary symmetric polynomials in the eigenvalues of the hermitian form representing $c_1(L)$ are all positive. Thus, $L$ is positively curved and hence ample.

I strongly suspect anyway that the result is false in general.

Could you give for instance a counterexample in dimension three?

Thanks in advance.