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 ${\displaystyle L}$ on a compact complex manifold ${\displaystyle M}$, the line bundle ${\displaystyle K_{M}\otimes L^{\otimes m}}$ (where ${\displaystyle K_{M}}$ is a canonical line bundle of ${\displaystyle M}$) is

• spanned by sections when ${\displaystyle m\geq n+1}$;
• very ample when ${\displaystyle m\geq n+2}$,

where ${\displaystyle n}$ is the complex dimension of ${\displaystyle M}$.

Note that for large ${\displaystyle m}$ the line bundle ${\displaystyle K_{M}\otimes L^{\otimes m}}$ is very ample by the standard Serre's vanishing theorem (and its complex analytic variant). The Fujita conjecture provides an explicit bound on ${\displaystyle m}$, 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 ${\displaystyle m\geq 4}$ 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.