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Let $X$ be a smooth projective variety with negative Kodaira dimension over $\mathbb{C}$.

Is there an integer $n\geq 1$, a smooth projective hyperkahler variety $H$, and a finite morphism $H\to X^n$?

Is the answer positive for Fano varieties?

Motivation: Varieties $X$ with negative Kodaira dimension should admit an entire curve $\mathbb C\to X(\mathbb{C})$. If the answer to the above question is positive, we reduce this to the case of hyperkahler.

Guess: The answer is probably negative. But what's the easiest example?

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    $\begingroup$ I suppose you want your morphism $H\rightarrow X^n$ to be surjective. Now usually hyperkähler varieties are assumed to be simply connected by definition; then a trivial counter-example is $X=C\times \mathbb{P}^1$, with $C$ a curve of genus $\geq 1$. The question makes more sense if you assume that $X$ is Fano. $\endgroup$
    – abx
    Commented Sep 1, 2017 at 19:16
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    $\begingroup$ Since every hyperKaehler variety has even dimension, you should specify that $n$ is even. $\endgroup$
    – Jason Starr
    Commented Sep 3, 2017 at 19:31

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For smooth del-Pezzo surfaces the answer is yes. There is a smooth $K3$ surface which is a branched double cover of $\mathbb{P}^{2}$ blown up in $9$ points (example 1.2.ii) here http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf). Hence we get a finite morphism to any del-Pezzo.

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