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arXiv · 1707.08659

Positivity of the diagonal

Brian Lehmann·John Christian Ottem
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Abstract

We study how the geometry of a projective variety $X$ is reflected in the positivity properties of the diagonal $\Delta_X$ considered as a cycle on $X \times X$. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of $X$. For example, when the diagonal is big, we prove that the Hodge groups $H^{k,0}(X)$ vanish for $k>0$. We also classify varieties of low dimension where the diagonal is nef and big.

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Brian Lehmann, John Christian Ottem. 2017-07-26. Positivity of the diagonal. https://arxiv.org/abs/1707.08659

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