Let $k$ be an algebraically closed field of characteristic zero. I am not sure what the right definition of a hyperkahler variety over $k$ is, but I think the following might be close enough.

Definition. A smooth proper connected scheme $X$ over $k$ is a hyperkahler variety (over $k$) if $\mathrm{h}^{2,0}(X) = 1$, the etale fundamental group $\pi_1^{et}(X)$ of $X$ is trivial, $\omega_X$ is trivial, and (added later) the generator of $\mathrm{H}^{2,0}(X)$ is everywhere non-degenerate.

With this definition, it is not clear whether every hyperkahler variety over $k$ is projective. If $X$ is a two-dimensional hyperkahler variety over $k$, then $X$ is projective. But what about $\dim X > 2$?

Is every hyperkahler variety over $k$ projective?

PS. Please feel free to correct my definition of a hyperkahler variety over $k$.

  • 7
    $\begingroup$ That is not the correct definition: a product of a hyperkaehler variety and a Calabi-Yau threefold will satisfy that condition by Kuenneth. You should add the hypothesis that the generator of $H^{2,0}(X)=H^0(X,\bigwedge^2 \Omega_{X/k})$ is everywhere nondegenerate. $\endgroup$ Jun 12 '18 at 21:30
  • $\begingroup$ Maybe the toric hyperkähler varieties (a.k.a. hypertoric varieties) of Hausel-Sturmfels (2002, Def. 6.1)? $\endgroup$ Jun 12 '18 at 22:02
  • 1
    $\begingroup$ FZ: It seems that the OP is interested in compact hyperkahler manifolds $\endgroup$
    – wnx
    Jun 12 '18 at 22:21
  • $\begingroup$ (Hyper)Kaehler or (hyper)Kähler and étale are right! $\endgroup$ Jun 13 '18 at 11:11

With the correct definition of hyperkähler (which as Jason said requires $H^0(X,\Omega^2_X)$ to be generated by a holomorphic symplectic form), there are examples constructed by Yoshioka in section 4.4 here. The field $k=\mathbb{C}$.

These are non-Kähler compact complex simply connected holomorphic symplectic manifolds which are bimeromorphic to a projective hyperkähler manifold via a Mukai flop. They are therefore Moishezon, but certainly not projective. They are also discussed for example in the book by Gross-Huybrechts-Joyce, example 21.7.

  • 2
    $\begingroup$ This does not anser the question, which asks for hyperkähler schemes. $\endgroup$
    – abx
    Jun 13 '18 at 10:28
  • 1
    $\begingroup$ I see, I was just looking at the title which asked for "algebraic" and I thought the OP wanted an algebraic space. Well, I don't know the answer to the actual question then. $\endgroup$
    – YangMills
    Jun 13 '18 at 16:34

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