Fujita conjecture

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

L^mK

is spanned by sections when

m ≥ n + 1

and is very ample when

m ≥ n + 2,

where n is the complex dimension of M.

References

• 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 .

External links

• http://mathoverflow.net/questions/138812/supporting-facts-to-fujita-conjecture