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Suppose in $P^3$ we have $K3$ surface defined by $x^4+y^4+z^4+w^4=0$ can we find a complex subvariety that is a torus?

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    $\begingroup$ Yes, of course: take any line $\ell$ on the surface (e.g. $y=ix$, $w=iz$), and a general plane through $\ell$. Its residual intersection with the surface is a smooth plane cubic, hence an elliptic curve. $\endgroup$
    – abx
    Feb 16, 2015 at 16:37

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