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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?

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    $\begingroup$ I don't understand the question - all complex projective algebraic varieties are Kahler, via the pullback of the Fubini-Study form. Could you please clarify? $\endgroup$ – dhy Jun 19 '18 at 3:20
  • $\begingroup$ I think that you can even take $\omega$ to have integral cohomology class $\endgroup$ – user74900 Jun 19 '18 at 11:15
  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – Francois Ziegler Jun 19 '18 at 19:08
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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. 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.

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  • $\begingroup$ Thanks!, your answer completely clarify my question. However, I am interested in spaces that admit a "kaehlerian cohomology class" (your third bullet item) but are not Kaehler manifolds. I thought that projective complex varieties do the job but they are Kaehler as your answer suggest. Do you happen to know an example of what I am looking for? Thanks again!! $\endgroup$ – C. Zhihao Jun 20 '18 at 0:41
  • $\begingroup$ @C.Zhihao See e.g. Yi Lin (2007, Thm 4.4), or Christoph Bock (2016, Thm 9.1)(pdf). $\endgroup$ – Francois Ziegler Jun 20 '18 at 16:39

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