Fujita conjecture

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In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved as of 2017. It is named after Takao Fujita, who formulated it in 1985.

In complex manifold theory, the conjecture states that for a positive holomorphic line bundle L on a compact complex manifold M with canonical line bundle K, then


is spanned by sections when

mn + 1

and is very ample when

mn + 2,

where n is the complex dimension of M.


  • Fujita, Takao (1987), "On polarized manifolds whose adjoint bundles are not semipositive", Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, pp. 167–178, MR 946238 .
  • Helmke, Stefan (1997), "On Fujita's conjecture", Duke Mathematical Journal, 88 (2): 201–216, MR 1455517, doi:10.1215/S0012-7094-97-08807-4 .
  • 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 .
  • Smith, Karen E. (2000), "A tight closure proof of Fujita's freeness conjecture for very ample line bundles", Mathematische Annalen, 317 (2): 285–293, MR 1764238, doi:10.1007/s002080000094 .

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