Kodaira embedding theorem
Encyclopedia
In mathematics
, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex number
s, amongst compact
Kähler manifold
s. In effect it says precisely which complex manifold
s are defined by homogeneous polynomial
s.
Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an integral cohomology class, there is a complex-analytic embedding of M into complex projective space
of some high enough dimension N. The fact that M embeds as an algebraic variety
follows from its compactness by Chow's theorem. A Kahler manifold with a Hodge metric is occasionally called a Hodge manifold (named after W. V. D. Hodge
), so Kodaira's results states that Hodge manifolds are projective. The converse that projective manifolds are Hodge manifolds is more elementary and was already known.
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...
, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, amongst 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...
Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
s. In effect it says precisely which 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 are defined by homogeneous polynomial
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
s.
Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an integral cohomology class, there is a complex-analytic embedding of M into complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
of some high enough dimension N. The fact that M embeds as an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
follows from its compactness by Chow's theorem. A Kahler manifold with a Hodge metric is occasionally called a Hodge manifold (named after W. V. D. Hodge
W. V. D. Hodge
William Vallance Douglas Hodge FRS was a Scottish mathematician, specifically a geometer.His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major...
), so Kodaira's results states that Hodge manifolds are projective. The converse that projective manifolds are Hodge manifolds is more elementary and was already known.