Language and Manifold Note 1 Algebraic Cycle

  

Language and Manifold

 

Note 1

Algebraic Cycle

 

 

TANAKA Akio

 

1 <Algebraic cycle>

Field     k

Polynomial    f ( x ) = k [ x ]

Algebraic closure k     Polynomial that has one root at least.    

n-dimensional affine space Aⁿ = kⁿ

n-dimensional complex affine space     Aⁿ_c = { ( z₁, …, z_n ) | z_i ∈ C }

Finite polynomials over coordinates z₁, …, z_n     f₁ ( z₁, …, z_n ) , …, f_n ( z₁, …, z_n )

Affine algebraic manifold     V ( f₁, …, f_m ) = { ( z₁, …, z_n ) ∈ Aⁿ | f₁ ( z₁, …, z_n ) = …= f_m ( z₁, …, z_n ) = 0 }

n-dimensional projective space Pⁿ     Continued ratio ( Z₁ : … : Z_n ) (( Z₁ : … : Z_n ) ≠( 0, …, 0 )

Homogeneous polynomial    F₁ ( Z₀ : … : Z_n ), …, F_m ( Z₀ : … : Z_m )

Projective manifold     V ( F₁, …, F_m ) = { ( Z₀, …, Z_n ) ∈ Pⁿ | F₀ ( Z₀, …, Z_n ) = …= F_mn( Z₀, …, Z_n ) = 0 }

Nonsingular connected projective manifold X     V ( F₀, …, F_m )

Homogeneous polynomial    G₁ ( Z₀,… : Z_n ), …, G_k ( Z₀, … : Z _n)

W = V ( F₁, …, F_m, G₁, …, G_k )

Algebraic cycle  ∑_ia_iW_i     (W_i is irreducible submanifold. a_i∈Z )

 

[References]

<Projective space>

Algebraic Linguistics / Linguistic Note / 7 Projective Space / Tokyo July 26, 2007

 

Tokyo November 19, 2007

Sekinan Research Field of Language

www.sekinan.org