Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

When does the subadditivity theorem for multiplier ideals hold?


Authors: Shunsuke Takagi and Kei-ichi Watanabe
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3951-3961
MSC (2000): Primary 13B22; Secondary 14J17
DOI: https://doi.org/10.1090/S0002-9947-04-03436-1
Published electronically: February 4, 2004
MathSciNet review: 2058513
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.


References [Enhancements On Off] (What's this?)

  • [Ar] M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136. MR 33:7340
  • [DEL] J.-P. Demailly, L. Ein and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan. Math. J. 48 (2000), 137-156. MR 2002a:14016
  • [DV] P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction, Proc. Cambridge Phil. Soc. 30 (1934), 453-459.
  • [ELS] L. Ein, R. Lazarsfeld, and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), 241-252. MR 2002b:13001
  • [Fu] T. Fujita, Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17 (1994), 1-3. MR 95c:14053
  • [Gi] J. Giraud, Improvement of Grauert-Riemenschneider's Theorem for a normal surface, Ann. Inst. Fourier, Grenoble 32 (1982), 13-23. MR 84f:14025
  • [HY] N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3143-3174.
  • [Hi] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 (1964), 205-326. MR 33:7333
  • [How] J. A. Howald, Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. 353 (2001), 2665-2671 MR 2002b:14061
  • [La] R. Lazarsfeld, Positivity in Algebraic Geometry, in preparation.
  • [Li1] J. Lipman, Rational singularities with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195-279. MR 43:1986
  • [Li2] -, Desingularization of $2$-dimensional schemes, Ann. Math. 107 (1978), 151-207. MR 58:10924
  • [Li3] -, Adjoints of ideals in regular local rings, Math. Res. Lett. 1 (1994), no. 6, 739-755. MR 95k:13028
  • [Li4] -, Proximity inequalities for complete ideals in two-dimensional regular local rings, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), 293-306, Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994. MR 95j:13018

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13B22, 14J17

Retrieve articles in all journals with MSC (2000): 13B22, 14J17


Additional Information

Shunsuke Takagi
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo 153-8914, Japan
Email: stakagi@ms.u-tokyo.ac.jp

Kei-ichi Watanabe
Affiliation: Department of Mathematics, College of Humanities and Sciences, Nihon University, Setagaya-ku, Tokyo 156–0045, Japan
Email: watanabe@math.chs.nihon-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-04-03436-1
Received by editor(s): January 2, 2003
Received by editor(s) in revised form: June 3, 2003
Published electronically: February 4, 2004
Additional Notes: The authors thank MSRI for the support and hospitality during their stay in the fall of 2002. The second author was partially supported by Grants-in-Aid in Scientific Researches, 13440015, 13874006; and his stay at MSRI was supported by the Bunri Fund, Nihon University.
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society