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Equality in the Bogomolov–Miyaoka–Yau inequality in the non-general type case

  • Feng Hao and Stefan Schreieder EMAIL logo


We classify all minimal models X of dimension n, Kodaira dimension n-1 and with vanishing Chern number c1n-2c2(X)=0. This solves a problem of Kollár. Completing previous work of Kollár and Grassi, we also show that there is a universal constant ϵ>0 such that any minimal threefold satisfies either c1c2=0 or -c1c2>ϵ. This settles completely a conjecture of Kollár.

Funding statement: Both authors are supported by the DFG grant “Topologische Eigenschaften von Algebraischen Varietäten” (project no. 416054549). The first author is also supported by grant G097819N of Nero Budur from the Research Foundation Flanders.


The second author thanks I. Cheltsov, S. Filipazzi, Th. Peternell and C. Shramov for useful discussions. We thank the referees for their careful reading and for a question that lead us to discover the uniqueness statement in item d of Corollary 1.2.


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Received: 2020-03-31
Revised: 2020-12-24
Published Online: 2021-03-12
Published in Print: 2021-06-01

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