In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Kodaira embedding theorem characterises
non-singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular or sounder, a group of boar, see List of animal names
* Singular (band), a Thai jazz pop duo
*'' Singular ...
projective varieties
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the ...
, over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, amongst
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...
s. In effect it says precisely which
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
s are defined by
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables ...
s.
Kunihiko Kodaira
was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese ...
's result is that for a compact Kähler manifold ''M'', with a Hodge metric, meaning that the
cohomology class
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
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
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 ...
of some high enough dimension ''N''.
The fact that ''M'' embeds as an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
follows from its compactness by
Chow's theorem.
A Kähler manifold with a Hodge metric is occasionally called a Hodge manifold (named after
W. V. D. Hodge
Sir William Vallance Douglas Hodge (; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer.
His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area no ...
), 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.
Kodaira also proved (Kodaira 1963), by recourse to
the classification of compact complex surfaces, that every compact Kähler surface is a ''deformation'' of a projective Kähler surface. This was later simplified by Buchdahl to remove reliance on the classification (Buchdahl 2008).
Kodaira embedding theorem
Let ''X'' be a compact Kähler manifold, and ''L'' a holomorphic line bundle on ''X''. Then ''L'' is a
positive line bundle if and only if there is a holomorphic embedding
of ''X'' into some projective space such that
for some ''m'' > 0.
See also
*
Fujita conjecture
*
Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures ...
*
Moishezon manifold
References
*
*
*
*
* A proof of the embedding theorem without the vanishing theorem (due to
Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth function, smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähl ...
) appears in the lecture note
here
*{{cite book , doi=10.1017/CBO9780511629327.004, chapter=Coherent Sheaves , title=Several Complex Variables and Complex Manifolds II , year=1982 , pages=127–198 , isbn=9780521288880
Theorems in complex geometry
Theorems in algebraic geometry