On the homeomophism type of smooth projective fourfolds

TL;DR

This paper investigates the topological relationships among smooth complex projective fourfolds, establishing non-homeomorphism results for certain classes and classifying those homeomorphic to specific hyperk"ahler fourfolds, including an explicit example.

Contribution

It provides new topological distinctions among Fano, Calabi-Yau, and hyperk"ahler fourfolds and classifies fourfolds homeomorphic to a hyperk"ahler deformation equivalent to S^{[2]}, including an explicit example.

Findings

Fano fourfolds are not homeomorphic to Ricci-flat fourfolds.

Calabi-Yau and hyperk"ahler fourfolds are not homeomorphic.

Classification of fourfolds homeomorphic to a specific hyperk"ahler fourfold.

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.