Fujita conjecture
Encyclopedia
In mathematics
, Fujita's conjecture is a problem in the theories of algebraic geometry
and complex manifold
s, unsolved .
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
and is very ample when
where n is the complex dimension
of M.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Fujita's conjecture is a problem in the theories of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s, unsolved .
In complex manifold theory, the conjecture states that for a positive holomorphic line bundle L on a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
complex manifold M with canonical line bundle
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
K, then
- LmK
is spanned by sections when
- m ≥ n + 1
and is very ample when
- m ≥ n + 2,
where n is the complex dimension
Complex dimension
In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...
of M.