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Dec 14, 2016 at 17:33 comment added Neil Strickland An outline of Voisin's example: take a non-algebraic torus $T$ with suitable endomorphism $\phi$ (all eigenvalues simple and not real), then consider the following four copies of $T$ in $T\times T$: the two axes, the diagonal and the graph of $\phi$. Blow up the (finitely many, transverse) intersection points, then blow up the proper transforms of the four $T$'s.
Dec 13, 2016 at 19:19 history edited Joe Silverman CC BY-SA 3.0
Added more material.
Dec 13, 2016 at 19:07 history answered Igor Rivin CC BY-SA 3.0