On the homeomophism type of smooth projective fourfolds

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 hyperkähler 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 hyperkähler 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.

Correction of an error (exotic structure for certain Hk 4-fold was wrongly stated in the previous version, now added as an open problem), submitted

References in corpus (1)

• On Contractions of smooth varieties