On the homeomophism type of smooth projective fourfolds

Keiji Oguiso; Thomas Peternell

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arXiv · 1707.05657

On the homeomophism type of smooth projective fourfolds

Keiji Oguiso·Thomas Peternell

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Abstract

In this paper we study smooth complex projective $4$-folds which are topologically equivalent. First we show that Fano fourfolds are never oriented homeomorphic to Ricci-flat projective fourfolds and that Calabi-Yau manifolds and hyperk\"ahler manifolds in dimension $\ge 4$ are never oriented homeomorphic. Finally, we give a coarse classification of smooth projective fourfolds which are oriented homeomorphic to a hyperk\"ahler fourfold which is deformation equivalent to the Hilbert scheme $S^{[2]}$ of two points of a projective K3 surface $S.$ We also present an explicit example of a smooth projective fourfold oriented homeomorphic to $S^{[2]},$ which has positive Kodaira dimension.

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Keiji Oguiso, Thomas Peternell. 2017-07-18. On the homeomophism type of smooth projective fourfolds. https://arxiv.org/abs/1707.05657

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