This is problem 2 of Chapter 5 in Voisin's book Hodge Theory and Complex Algebraic Geometry I.
Let $X$ be a connected compact complex manifold of dimension $n$ and let $L$ be a holomorphic line bundle on $X$. We assume that there exists an integer $N>0$, such that $$H^0(X,L^{\otimes N})\neq 0.$$Show that if $H^n(X,L\otimes K_X)\neq0$, the line bundle $L$ is trivial.
The question is named after An application of Serre duality but I did not figure out how.