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On the Homeomorphism Type of Smooth Projective Fourfolds

  • Keiji Oguiso
  • Thomas PeternellEmail author
Article
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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 hyperkähler manifolds in dimension ≥ 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.

Keywords

Homeomorphism type Smooth projective fourfolds Fano fourfolds Calabi-Yau manifold 

Mathematics Subject Classification (2010)

14F45 14J35 14J32 14J45 14J40 

Notes

Acknowledgments

Most thanks of the first named author go to Fabrizio Catanese for his invitation to Universität Bayreuth with full financial support from the ERC 2013 Advanced Research Grant-340258-TADMICAMT. The main part of this work started there. Very special thanks go to Frédéric Campana whose comments and remarks greatly helped to improve the paper. In particular, this concerns Theorems 3.1 and 1.5. We would like to express our thanks to Fabrizio Catanese, Yujiro Kawamata, Yongnam Lee, Bong Lian, and Kiwamu Watanabe for fruitful discussions. Last but not least, we thank the referee for his very valuable comments.

Funding Information

The first named author is supported by the ERC 2013 Advanced Research Grant-340258-TADMICAMT, JSPS Grant-in-Aid (S) No. 25220701, JSPS Grant-in-Aid (S) 15H05738, JSPS Grant-in-Aid (B) 15H03611, and KIAS Scholar Program.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Mathematical Sciencesthe University of TokyoTokyoJapan
  2. 2.Korea Institute for Advanced StudySeoulSouth Korea
  3. 3.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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