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The minimal model program for b-log canonical divisors and applications

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Abstract

We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work in the setting of the b-log MMP. If we assume that the log MMP terminates, then so does the b-log MMP. Furthermore, the b-log MMP includes both the log MMP and the equivariant MMP as special cases. There are various interesting b-log varieties arising from different objects, including the Brauer pairs, or “non-commutative algebraic varieties which are finite over their centres.” The case of toric Brauer pairs is discussed in further detail.

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Acknowledgements

This paper is an outcome of the workshop ‘Mori program for Brauer pairs in dimension three’ held in 2014 at the American Institute of Mathematics. The authors are indebted to all the support from the institute and their hospitality. We also thank the anonymous referee for carefully reading the manuscript and providing us with many constructive comments. DC was partially supported by an ARC discovery project grant DP130100513. CI was partially supported by an NSERC Discovery grant. SJK was supported in part by NSF Grant DMS-1565352 and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics at the University of Washington. RK was partially supported by the National Science Foundation award DMS-1305377. SO was partially supported by Grant-in-Aid for Scientific Research (No. 60646909, 16H05994, 16K13746, 16H02141, 16K13743, 16K13755, 16H06337) from JSPS and the Inamori Foundation.

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Authors and Affiliations

  1. Department of Pure Mathematics, School of Mathematics and Statistics UNSW, Kensington, Australia

    Daniel Chan & Boris Lerner

  2. Department of Mathematics, University of Washington, Seattle, WA, USA

    Kenneth Chan & Sándor J. Kovács

  3. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    Louis de Thanhoffer de Völcsey

  4. School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada

    Colin Ingalls

  5. Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI, USA

    Kelly Jabbusch

  6. Department of Mathematics, Michigan State University, East Lansing, MI, USA

    Rajesh Kulkarni

  7. Brock University, St. Catharines, ON, Canada

    Basil Nanayakkara

  8. Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Japan

    Shinnosuke Okawa

  9. Universiteit Hasselt, Diepenbeek, Belgium

    Michel Van den Bergh

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  1. Daniel Chan
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Correspondence to Shinnosuke Okawa.

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Chan, D., Chan, K., de Thanhoffer de Völcsey, L. et al. The minimal model program for b-log canonical divisors and applications. Math. Z. 303, 87 (2023). https://doi.org/10.1007/s00209-023-03205-w

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