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.
In complex geometry, the conjecture states that for a positive holomorphic line bundle on a compact complex manifold , the line bundle (where is a canonical line bundle of ) is
where is the complex dimension of .
Note that for large the line bundle is very ample by the standard Serre's vanishing theorem (and its complex analytic variant). The Fujita conjecture provides an explicit bound on , which is optimal for projective spaces.
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 implies global generation.