I am not very familiar with the notion of projective algebraic varieties, I work mostly from an algebraic topology/differential geometry point of view, but I am trying to find a prove for the following fact.


A non-singular complex projective algebraic variety $X$ admits a rational 2-form $\omega$ such that the multiplication by $\omega^i$ induces an isomorphism $H^{n-i}(X; \mathbb{Q}) \cong H^{n+i}(X; \mathbb{Q})$ for all $i = 0, \ldots, n$.


Does this implies that $X$ is a Kähler manifold?



Kähler form on complex projective algebraic variety
To answer the original question, a non-singular complex projective variety has a canonical Kaehler manifold structure (given by the pullback of the Fubini--Study form and the standard complex structure on $\mathbb{C}P^n$). In the converse direction, a compact Kaehler manifold can be embedded into $\mathbb{C}P^n$ for some $n>0$ if the cohomology class of the Kaehler form is integral (Kodaira embedding theorem). 


An interesting question is: what compact complex manifolds admit Kaehler structure? A review of some obstructions to Kaehlerness can be found [here](https://arxiv.org/abs/1601.04337). Let's list some notable obstructions for complex manifold $M$ of real dimension $2n$ to be Kaehler


* the underlying smooth manifold of $M$ must admit symplectic structure
* odd Betti numbers $b\_{2i+1}(M)$ should be even.
* there must exist a cohomology class $[\omega]\in H^2(X, \mathbb{R})$ such that wedge product $[\omega]^{\wedge j}:H^{n-j}(M, \mathbb{R})\rightarrow H^{n+j}(M, \mathbb{R})$ induces isomorphism
* the DG algebra of real $C^{\infty}$-differential forms on $M$ must be formal.


There is also a number of restrictions on the fundamental group $\pi\_1(M)$ (see the book 'Fundamental Groups of Compact Kähler Manifolds' for nice exposition). One should note though that the requirements listed here are very far from being sufficient; apparently, we are still very far from having a complete characterization of compact complex manifolds admitting compatible Kaehler structure. 


