Fujita conjecture
In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved as of 2017[update]. It is named after Takao Fujita, who formulated it in 1985.
Statement[edit]
In complex geometry, the conjecture states that for a positive holomorphic line bundle L on a compact complex manifold M, the line bundle KM ⊗ L⊗m (where KM is a canonical line bundle of M) is
- spanned by sections when m ≥ n + 1 ;
- very ample when m ≥ n + 2,
where n is the complex dimension of M.
Note that for large m the line bundle KM ⊗ L⊗m is very ample by the standard Serre's vanishing theorem (and its complex analytic variant). Fujita conjecture provides an explicit bound on m, which is optimal for projective spaces.
Known cases[edit]
For surfaces the Fujita conjecture follows from Reider's theorem. For three-dimensional algebraic varieties, Ein and Lazarsfeld in 1993 proved the first part of the Fujita conjecture, i.e. that m≥4 implies global generation.
See also[edit]
References[edit]
- Ein, Lawrence; Lazarsfeld, Robert (1993), "Global generation of pluricanonical and adjoint linear series on smooth projective threefolds.", J. Amer. Math. Soc., 6 (4): 875–903, doi:10.1090/S0894-0347-1993-1207013-5, MR 1207013.
- Fujita, Takao (1987), "On polarized manifolds whose adjoint bundles are not semipositive", Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, pp. 167–178, doi:10.2969/aspm/01010167, ISBN 978-4-86497-068-6, MR 0946238.
- Helmke, Stefan (1997), "On Fujita's conjecture", Duke Mathematical Journal, 88 (2): 201–216, doi:10.1215/S0012-7094-97-08807-4, MR 1455517.
- Siu, Yum-Tong (1996), "The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi", Geometric complex analysis (Hayama, 1995), World Sci. Publ., River Edge, NJ, pp. 577–592, MR 1453639, Zbl 0941.32021.
- Smith, Karen E. (2000), "A tight closure proof of Fujita's freeness conjecture for very ample line bundles" (PDF), Mathematische Annalen, 317 (2): 285–293, doi:10.1007/s002080000094, hdl:2027.42/41935, MR 1764238, S2CID 55051810.