European Journal of Mathematics

, Volume 4, Issue 1, pp 335–355 | Cite as

Rationally connected non-Fano type varieties

  • Igor Krylov
Research Article


Varieties of Fano type are very well behaved with respect to the MMP, and they are known to be rationally connected. We study a relation between the classes of rationally connected varieties and varieties of Fano type. It is known that these classes are birationally equivalent in dimension 2. We give examples of rationally connected varieties of dimension \(\geqslant 3\) which are not birational to varieties of Fano type, thereby answering Question 5.2 of Cascini and Gongyo (Saitama Math J 30:27–38, 2013).


Algebraic geometry Birational geometry Minimal model program Birational rigidity 

Mathematics Subject Classification

14E30 14E07 14J45 14M22 



The author would like to thank John Ottem and Ivan Cheltsov, for bringing this question to his attention and for many helpful conversations, and Antony Manioca, Milena Hering, Johan Martens, and Michael Wemyss for useful suggestions. The author also expresses his gratitude to the referee for pointing out many ways to improve this paper. The author was supported by an Edinburgh PCDS scholarship.


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Authors and Affiliations

  1. 1.Mathematisches Institut, Lehrstuhl Mathematik VIIIUniversity of BayreuthBayreuthGermany

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