Let $M\subset \mathbb{P}^N$ be a compect Kahler manifold of dimension $n$, and let $\omega$ be the associated closed (1,1)-form. $A^{p,q}(M)$ is the set of $C^\infty$ complex differential forms of type (p,q). Then $L: A^{p,q}(M)\to A^{p+1,q+1}(M),\eta\mapsto \eta\wedge \omega $ induces an isomorphism $L^k:H^{n-k}(M)\to H^{n+k}(M)$.
My quesion: Does $L$ induces a map $L: H^{p,q}(M)\to H^{p+1,q+1}(M)$?
Yes.
Let $x,y\in A^{p,q}(M)$ represent the same cohomology class. That is, $$x=d\varphi+y$$
We have to check that $Lx$ and $Ly$ also represent the same cohomology class, which can then be denoted $L[x]\in H^{p+1,q+1}(M)$. The key observation is that since $\omega$ is closed, $d(\varphi\wedge\omega)=(d\varphi)\wedge\omega$.
This allows you to write $$x\wedge \omega= (d\varphi)\wedge\omega+y\wedge \omega$$ as $$Lx=d(\varphi\wedge\omega)+Ly$$