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Termination of (many) 4-dimensional log flips

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Abstract

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(K X +D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and termination of flips in a relative birational setting. We also prove termination of directed flips with big K X +D. As a consequence, we prove existence of minimal models of 4-dimensional dlt pairs of general type, existence of 5-dimensional log flips, and rationality of Kodaira energy in dimension 4.

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Valery Alexeev

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112-0090, USA

    Christopher Hacon

  3. Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan

    Yujiro Kawamata

Authors
  1. Valery Alexeev
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  2. Christopher Hacon
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  3. Yujiro Kawamata
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Correspondence to Yujiro Kawamata.

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Alexeev, V., Hacon, C. & Kawamata, Y. Termination of (many) 4-dimensional log flips. Invent. math. 168, 433–448 (2007). https://doi.org/10.1007/s00222-007-0038-1

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  • DOI: https://doi.org/10.1007/s00222-007-0038-1

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