Fujita conjecture

In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds. It is named after Takao Fujita, who formulated it in 1985.

Statement

In complex geometry, the conjecture states that for a positive holomorphic line bundle L on a compact complex manifold M, the line bundle K_M ⊗ L^⊗m (where K_M 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 K_M ⊗ 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

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

• Ohsawa–Takegoshi L² extension theorem

References

• 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.