On log canonical divisors that are log quasi-numerically positive

Shigetaka Fukuda

Open Mathematics (2004)

  • Volume: 2, Issue: 3, page 377-381
  • ISSN: 2391-5455

Abstract

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Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.

How to cite

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Shigetaka Fukuda. "On log canonical divisors that are log quasi-numerically positive." Open Mathematics 2.3 (2004): 377-381. <http://eudml.org/doc/268866>.

@article{ShigetakaFukuda2004,
abstract = {Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.},
author = {Shigetaka Fukuda},
journal = {Open Mathematics},
keywords = {the log canonical divisor; divisorial log terminal; numerically positive; semi-ample MSC (2000); 14E30; semi-ample},
language = {eng},
number = {3},
pages = {377-381},
title = {On log canonical divisors that are log quasi-numerically positive},
url = {http://eudml.org/doc/268866},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Shigetaka Fukuda
TI - On log canonical divisors that are log quasi-numerically positive
JO - Open Mathematics
PY - 2004
VL - 2
IS - 3
SP - 377
EP - 381
AB - Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.
LA - eng
KW - the log canonical divisor; divisorial log terminal; numerically positive; semi-ample MSC (2000); 14E30; semi-ample
UR - http://eudml.org/doc/268866
ER -

References

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  1. [1] F. Ambro:The moduli b-divisor of an lc-trivial fibration, math. AG/0308143, August 2003. Zbl1094.14025
  2. [2] O. Fujino: “Base point free theorem of Reid-Fukuda type”,J. Math. Sci. Univ. Tokyo, Vol. 7, (2000), pp. 1–5. Zbl0971.14009
  3. [3] T. Fujita: “Fractionally logarithmic canonical rings of algebraic surfaces”,J. Fac. Sci. Univ. Tokyo Sect. IA Math., Vol. 30, (1984), pp. 685–696. Zbl0543.14004
  4. [4] S. Fukuda:A note on the ampleness of numerically positive log canonical and anti-log canonical divisors, math. AG/0305357, May 2003. 
  5. [5] Y. Kawamata, K. Matsuda and K. Matsuki: “Introduction to the minimal model problem”, In:Algebraic geometry, Sendai (Japan), 1985, North-Holland, Amsterdam, 1987, pp. 283–360. 
  6. [6] S. Keel, K. Matsuki and J. McKerman: “Log abundance theorem for threefolds”,Duke Math. J., Vol. 75, (1994), pp. 99–119. http://dx.doi.org/10.1215/S0012-7094-94-07504-2 Zbl0818.14007
  7. [7] J. Kollár (Ed.).Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. 
  8. [8] J. Kollár and S. Mori:Birational geometry of algebraic varieties, Cambridge University Press, Cambridge, 1998. Zbl0926.14003
  9. [9] K. Matsuki:A correction to the paper “Log abundance theorem for threefolds”, math. AG/0302360, February 2003. 
  10. [10] V. Shokurov: “3-fold log flips”,Russian Acad. Sci. Izv. Math., Vol. 40, (1993), pp. 95–202. http://dx.doi.org/10.1070/IM1993v040n01ABEH001862 

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