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Trudy Mat. Inst. Steklova, 2004, Volume 246, Pages 328–351 (Mi tm165)  

This article is cited in 25 scientific papers (total in 25 papers)

Letters of a Bi-rationalist V: Mld's and Termination of Log Flips

V. V. Shokurov

Johns Hopkins University

Abstract: Termination of log flips and, more generally, of log quasiflips under the descending chain condition (dcc) of boundary multiplicities follows from two expected properties of the minimal log discrepancy (mld) function on algebraic log varieties: (1) the semicontinuity of mld's on any fixed log variety and (2) the ascending chain condition (acc) of mld's on the log varieties of given dimension with boundary multiplicities under the dcc. This reduces the global statement on termination to two local ones. All known cases of termination follow from this reduction. In particular, this gives the log termination in dimension 3, as well as the special and canonical termination up to dimension 4. To prove the log termination in dimension 4, one only needs the acc in dimension 4 for the mld values in the interval $[0,1]$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 246, 315–336

Bibliographic databases:
UDC: 512.7
Received in February 2004

Citation: V. V. Shokurov, “Letters of a Bi-rationalist V: Mld's and Termination of Log Flips”, Algebraic geometry: Methods, relations, and applications, Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences, Trudy Mat. Inst. Steklova, 246, Nauka, MAIK Nauka/Inteperiodika, M., 2004, 328–351; Proc. Steklov Inst. Math., 246 (2004), 315–336

Citation in format AMSBIB
\Bibitem{Sho04}
\by V.~V.~Shokurov
\paper Letters of a~Bi-rationalist V: Mld's and Termination of Log Flips
\inbook Algebraic geometry: Methods, relations, and applications
\bookinfo Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences
\serial Trudy Mat. Inst. Steklova
\yr 2004
\vol 246
\pages 328--351
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm165}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2101303}
\zmath{https://zbmath.org/?q=an:1107.14012}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2004
\vol 246
\pages 315--336


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    3. Fujino O., “On termination of 4-fold semi-stable log flips”, Publ. Res. Inst. Math. Sci., 41:2 (2005), 281–294  crossref  mathscinet  zmath  isi  scopus
    4. Alexeev V., Hacon Ch., Kawamata Yu., “Termination of (many) 4-dimensional log flips”, Invent. Math., 168:2 (2007), 433–448  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Birkar C., “Ascending chain condition for log canonical thresholds and termination of log flips”, Duke Math. J., 136:1 (2007), 173–180  crossref  mathscinet  zmath  isi  elib  scopus
    6. Kawakita M., “On a comparison of minimal log discrepancies in terms of motivic integration”, J. Reine Angew. Math., 620 (2008), 55–65  crossref  mathscinet  zmath  isi  scopus
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    10. Kovacs S.J., “Young person's guide to moduli of higher dimensional varieties”, Proceedings of Symposia in Pure Mathematics: Algebraic Geometry Seattle 2005, Proceedings of Symposia in Pure Mathematics, 80, no. 1- 2, 2009, 711–743  crossref  mathscinet  zmath  isi
    11. Birkar C., Cascini P., Hacon Ch.D., McKernan J., “Existence of minimal models for varieties of log general type”, J. Amer. Math. Soc., 23:2 (2010), 405–468  crossref  zmath  isi  scopus
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    18. Kawakita M., “Discreteness of Log Discrepancies Over Log Canonical Triples on a Fixed Pair”, J. Algebr. Geom., 23:4 (2014), 765–774  crossref  mathscinet  zmath  isi  scopus
    19. Kawakita M., “a Connectedness Theorem Over the Spectrum of a Formal Power Series Ring”, Int. J. Math., 26:11 (2015), 1550088  crossref  mathscinet  zmath  isi  scopus
    20. Nakamura Yu., “on Semi-Continuity Problems For Minimal Log Discrepancies”, J. Reine Angew. Math., 711 (2016), 167–187  crossref  mathscinet  zmath  isi  scopus
    21. Nakamura Yu., “On minimal log discrepancies on varieties with fixed Gorenstein index”, Mich. Math. J., 65:1 (2016), 165–187  crossref  mathscinet  zmath  isi  elib  scopus
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    23. Kawakita M., “Divisors Computing the Minimal Log Discrepancy on a Smooth Surface”, Math. Proc. Camb. Philos. Soc., 163:1 (2017), 187–192  crossref  mathscinet  zmath  isi  scopus
    24. Hacon Ch. Moraga J., “On Weak Zariski Decompositions and Termination of Flips”, Math. Res. Lett., 27:5 (2020), 1393–1421  crossref  mathscinet  isi
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