Additional meaning and embedding

1.Derived category

Category theoretic mirror symmetry conjecture

Fukaya category is
Fuk(X, ω).

2.
General symplectic manifold is

.
3.

Derived category of A_∞

4.
Conjecture

When

and

have physical mirror relation,

there exists the next triangle category's equivalence

2.Embedding

Derived Category Language 2

Kawamata Conjecture

Conjecture

is birational map between smooth objective algebraic manifolds.

And

At This condition,

there exists next fully faithful embedding.

3.Stable and mobile

Notes for KARCEVSKIJ Sergej

Condition of Meaning
TANAKA Akio
September 11, 2011

[Preparation]

Graded differential algebra

Minimal model of graded differential algebra

Degree of homogenious element x of graded differential algabra |x|

Basis of linear space is given by homogenious and elements x₁, ....., x_n

Λ (V) = Λ(V)_k =Λ (x₁, ....., x_n )

Operation of minimal model

Spherical surface Sⁿ, n≥2

de Rham complex *(Sⁿ)

When n iseaven number,

Volume element of S_n

M_n = Λ (x), |x| = n, dx = 0,

M_2n-1gives minimal model S_n to de Rham complex .

When n is odd number,

M_ngives minimal model S_n to de Rham complex .

[Interpretation]

Word is given by spherical surface.

Meaning of word is given by elements x₁, ....., x_n.

Word has minimal model.

Word becomes formal.

Fundamental group of word contains free group of rank b₁(M).

Here KARCEVSKIJ's "stable part" is identified to fundamental group and " mobile part" is identified to free group.

Refer to the paper Notes for KARCEVSKIJ Sergej, "Dusualisme asymétrique du signe linguistique" .

This paper has been published by Sekinan Research Field of Language.
All rights reserved.
© 2011 by The Sekinan Research Field of Language

4.Additional meaning

Stable and Unstable of Language

For the Supposition of KARCEVSKIJ Sergej

Completion of Language

TANAKA Akio

September 23, 2011

[Preparation]

n dimensional complex space Cⁿ

Open set

Whole holomorphic function over U

Ring sheaf for

U →O^an(U)

Complex analytic manifold C_anⁿ

Algebraic manifold Aⁿ multinomial of C_anⁿ

Ideal of multinomial ring a

[x₁, x₂, ..., x_n]

V(a) = {(a₁, a₂, ..., a_n)

Cⁿ f (a₁, a₂, ..., a_n) = 0,

a }

Whole closed set of V(a)

Fundamental open set D(f) = {(a₁, a₂, ..., a_n)

Cⁿ | f (a₁, a₂, ..., a_n) ≠ 0}

Arbitrary family of open set {Ui}

Easy sheaf F

Zariski topological space

Ring sheaf O

Affine space Aⁿ = (

, O)

Ring R

Set of whole maximum ideal Spm R1

Spm R Spectrum of R

Spm R is Noether- like.

◊

R is integral domain.

Whole of open sets without null set U_x

Quotient field K

Mapping from U_xto whole partial set of K O

O(V(a)^c) =

R_f

^c expresses complementary set.

O is easy sheaf of ring over Spm R that is whole set K.

◊

R is finite generative integral domain over k.

Triple (i) (ii) (iii) is called affine algebraic variety.

(i) Set Spm R

(ii) Zariski topology

(iii) Ring's sheaf O

O is called structure sheaf of affine algebraic variety.

◊

Ring homomorphism between definite generative integral domains

Upper is expressed by

Ring holomorphism O_X(U) → O_Y((^t

)^-1U)

Morphism from affine algebraic variety Y to X ( O_X(U) → O_Y((^t )^-1U), X→Y )

When

is surjection, ^t

is isomorphism overclosed partial set defined by p= Ker

Upper is called to closed immersion.

Ring holomorphism

Morphism between affine algebraic varieties

Kernel of

Image of

It is called that when

is injection

is dominant.

◊

R is medium ring between S and its quotient field K.

When

that is given by natural injection

is isomorphism over open set,

is called open immersion.

◊

When X is algebraic variety, longitude of maximum chain is equal to transcendental dimension of function field k(X).

It is called dimension of algebraic variety X, expressed by dim X.

◊

Defined generative field over k K

Space ( X, O_x )added ring that is whole sets of K that has open covers {U_i}

satisfies next conditions is called algebraic variety.

(i) Each U_iis affine algebraic variety that has quotient K .

(ii) For each i, j

I, intersection

is open partial set of

◊

Tensor product between ring and itself becomes ring by each elements products.

Elements

that defines surjective homomorphism is expressed by

Image

of closed embedding defined by

is called diagonal.

◊

Field K

Ringed space that have common whole set K (A, O_A) (B, O_B)

Topological space C

Open embedding

A and B have common partial set C.

Topological space glued A and B by C

Easy sheaf over W O_W

ahere, arbitrary open set Ø ≠

Ringed space

is called glue of A and B by C.

◊

Intedgral domains that have common quotient field K R, S

Element R a_m ≠ 0

Element S b_n ≠ 0

Spm T

Spm R, Spm T

Spm S

Glue defined by the upper is called simple.

◊

Affine algebraic varieties U¹, U²

Common open set of U¹, U²U^C

Diagonal embedding

When the upper is closed set, glue is called separated.

◊

For simple glue

, next is equivalent.

(*) It is separated.

(**) Ring

is generated by R and S.

◊

R and S are integral domains that have common quotient field K.

For partial ring T=RS generated by R and S, when <Definition> simple is satisfied, it is called "Spm R ad Spm S are simple glue."

◊

Projective space Pⁿ is simple glue.

◊

Algebraic Variety's morphism is glue of affine algebraic variety's ring homomorphism image.

Algebraic direct product is direct product of affine algebraic variety.

◊

Affine algebraic variety X

Ring over k R

is called R value point of X.

Whole

is called set of R value point of X, expressed by X(R).

Ring homomorphism over k

X(f) := X(R)

X(S)

Ring homomorphism

is function from ring category over k to category of set.

◊

Functors from ring category to set category F, G

Ring R

Family of

over ring R {

}

{

}

has functional morphism.

Functors F,G have isomorphism ( or natural transformation).

Functor from ring's category to set's category that is isomorphic to algebraic variety, is called representable or represent by X, or fine moduli.

◊

Functor from ring's category to set's category F

When

satisfies the next conditions, X is called coarse moduli.

(i) There is natural transformation

(ii) Natural transformation

Morphism that satisfies

is existent uniquely.

(iii) For algebraic close field k'

k, (k') is always bijection.

◊

Algebraic variety G that

is functor to group's category is called algebraic group.

◊

Finite generative ring over k A

When G = Spm A satisfies 3 conditions on the next triad is called affine algebraic group.

Triad

Conditions

(i)

are commutative for

(ii)There is identity map for A.

(iii) There is coincident with

for A.

◊

Projective space over C Pⁿ

(2n+1) dimensional spherical surface

{

}

Pⁿ has continuous surjection from

Pⁿis compact.

Map

is called closed map when

is closed set image

becomes closed set.

◊

Algebraic variety X is called complete when projection

is closed map for arbitrary manifold Y.

◊

Morphism from complete algebraic manifold X to separated algebraic manifold Y,

is closed map.

◊

Projective space Pⁿ is complete.

◊

Algebraic manifold that has closed embedding at Pⁿ is complete.

This algebraic manifold is called projective algebraic manifold.

[Interpretation]

Here language is expressed by P^n.

Word is expressed by projective algebraic manifold.

Meaning of word is expressed by closed embedding.

5.
Note

Provisional philosophic conjecture on additional meaning is the following.

From symplectic geometry to Fukaya category.
From Fukaya category to derived category.
Kawamata conjecture at derived category.
Smooth objective algebraic manifolds satisfies birational map.
Fully faithful embedding exists between two manifolds.

[References]
Extract of Zoho papers at Sekinan Library.
More details refer to Sekinan Zoho.

[2012]

Birational Language

[2011]

Notes for KARCEVSKIJ Sergej
Stable and Unstable of Language

[2009]

Symplectic Language Theory
Floer Homology Language

Tokyo

23 May 2016

SRFL Theory

Wednesday 24 July 2019