Hodge classes on self-products of a variety with an automorphism

Chad Schoen

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We study (rational) sweeping out of general hypersurfaces by varieties having small moduli spaces. As a consequence, we show that general $K$ -trivial hypersurfaces are not rationally swept out by abelian varieties of dimension at least two. As a corollary, we show that Clemens’ conjecture on the finiteness of rational curves of given degree in a general quintic threefold, and Lang’s conjecture saying that such varieties should be rationally swept-out by abelian varieties, contradict. ...