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 geometry, the conjecture states that for a positive holomorphic line bundle L on a compact complex manifold M, the line bundle KMLm (where KM is a canonical line bundle of M) is

where n is the complex dimension of M.

Note that for large m the line bundle KMLm 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]


  • 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.

External links[edit]