Diophantine Approximation on Varieties IV: Derivated algebraic distance and derivative metric Bezout Theorem

0.36 MB·English

by  Heinrich Massold

Diophantine Approximation on varieties IV:
Derivated algebraic distance and derivative metric
9
0
0 B´ezout Theorem
2

n
a
Heinrich Massold
J

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2 January 26, 2009

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Contents
.
h
t
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1 Introduction 2
m
[
2 Sharp decomposition of the algebraic distance 6

1
v
3 Affine Differentiation 12
9
8 3.1 Projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8
3.2 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3
.
1
4 The derivated algebraic distance 20
0
9 4.1 Hypersurfaces and points . . . . . . . . . . . . . . . . . . . . . . . . . 20
0
4.2 Effective cycles in projective space and projective subspaces . . . . . 21
:
v
4.2.1 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
i
X 4.2.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . 25
r
a
5 Reduction of the derivated algebraic distance to derivated algebraic
distances to points 26
6 Proof of the main Theorems 32
6.1 Combinatorics of the intersection points . . . . . . . . . . . . . . . . 32
6.2 Finish of proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A Proof of Lemmas 4.7, 4.8 and 4.9 40
1 Introduction
Let t = ( t+1) be the t-dimensional projective space over Spec , (W) t a
projPectivePsuZbspace of codimension q, and X Z ( t ) an effectZivePcycle.⊂WPeCsay
eff
that X is reguar with respect to (W) if the i∈rreducibPleCcomponents of codimension
P
< t q intersect (W) properly, and the irreducible components of codimension
− P
t q do not meet (W). For regular cycles the algebraic distance
≥ − P
D(Y, (W))
P ∈ R
is defined in [Ma1], section 4.1. Further for p+q t+1, and effective cycles X,Y
≤
of pure codimensions p,q that intersect properly, the algebraic distance
D(X,Y)
∈ R
is defined in [Ma1], Definition 4.1. For x,y points in t( ) denote by x,y their
P C | |
Fubini-Study distance, i. e. sin(x,y). The logarithm of the distance is a nonpositive
number.
There are the following Theorems for the algebraic distance.
1.1 Theorem For properly intersecting cyles , Z ( N) with base exten-
eff
sions X,Y to the equality X Y ∈ PZ
C
D(Y,Z) = h( . ) degZh( ) degYh( ) σ log2degY degZ
t
Y Z − Y − Z −
holds.
Proof [Ma1], Scholie 4.3.
1.2 Theorem With the previous Definition, let , be effective cycles intersecting
X Y
properly, and θ a point in N( ) (supp(X Y )).
P C \ C ∪ C
1. There are effectively computable constants c,c′ only depending on t and the
codimenion of X such that
deg(X)log θ,X( ) D(θ,X)+cdegX log θ,X( ) +c′degX,
| C | ≤ ≤ | C |
2. If = div(f) is an effective cycle of codimension one,
X
h( ) log f +Dσ , and
D L2 t
X ≤ | |
D(θ,X)+h( ) = log f θ +Dσ ,
N−1
X |h | i|
where the σ′s are certain constants, and f θ is taken to be the norm of
i |h | i|
the evaluation of f SymD(E) = Γ( N,O(D)) at a vector of length one
∈ P
representing θ.
2
3. For p + q N + 1, assume that , and have pure codimension p, and q
≤ X Y
respectively. There exists an effectively computable positive constant d, only
depending on t, and a map
f : I degX degY
X,Y
→ ×
from the unit interval I to the set of natural numbers less or equal degX
times the set of natural numers less or equal degY such that f (0) =
X,Y
(0,0),f (1) = (degX,degY), and the maps pr f : I degX,pr
X,Y 1 X,Y 2
◦ → ◦
f : I degY are monotonously increasing, and surjective, fullfilling: For
X,Y
→
every T I, and (ν,κ) = f (T), the inequality
X,Y
∈
νκlog θ,X +Y +D(θ,X.Y)+h( . )
| | X Y ≤
κD(θ,X)+νD(θ,Y)+degYh( )+degXh( )+ddegX degY
X Y
holds.
4. In the situation of 3, if further θ,X +Y = θ,X , then
| | | |
D(θ,X.Y)+h( . ) D(θ,Y)+degYh( )+degXh( )+d′degXdegY
X Y ≤ X Y
with d′ a constant only depending on N.
Proof [Ma1] Theorem 2.2.
Let now (W) t( ) be a subspace of dimension q, and ∂I a real differential
P ⊂ P C
operator on the Grassmannian G of t+1 q-dimensional subspaces of t+1
t+1−q,t+1
− C
with respect to some affine chart, where I = (i ,...,i is a multiindex of
1 2q(t+1−q)
order I = i + + i . (More details will be given later on). Defines the
1 2q(t+1−q)
| | ···
derivated algebraic distance of order S of X to (W) as
P
DS(Y, (W))) := sup log ∂I(expD(X, (W))) .
P | P |
|I|≤S
There are the following Theorems for this derivated algebraic distance.
1.3 Theorem Fors,D , and f Γ( t,O(D))let F be the polynomialof degree
∈ N ∈ P
at most D in t variables that corresponds to f with respect to affine coordinates of
t centered at θ. Then, with some positive constant c only depending on t,
P
∂s
DS(divf,θ) sup log f (0) +c(s+D)log(SD).
≤ s≤S|J|=s (cid:12)(cid:18)(∂z1)j1···(∂zt)jt (cid:19) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3
in the following Theorems the O-notation always signifies that the respective in-
equalities hold modulo a fixed contant only depending on t and codimnstions of
cylces times the term inside the O-bracket.
1.4 Theorem Let Z be an effective cycle of pure codimension p in t, and θ a
P
point not contained in the support of Z, and let , denote the Fubini-Study distance
|· ·|
in t.
P
There is a projective subspace (F) t of codimension t p intersecting Z
properly such that with z ,...,zP th⊂e pPoiCnts in the intersectio−n (F).Z counted
1 degZ
P
with multiplicity, z ,θ z ,θ , and for S < degZ/3 the equalities
1 degZ
| | ≤ ··· ≤ | |
degZ
DS(Z,θ) = log z ,θ +O(SlogdegZ),
i
| |
i=S+1
X
degZ
2 log z ,θ D3S(Z,θ)+O((degZ +S)log(SdegZ))
i
| | ≤
i=S+1
X
hold.
1.5 Corollary The derivated algebraic distance is a negative number modulo
O((degZ +S)log(SdegZ)).
Next, for n denote n the set of natural numbers less or equal n including 0,
∈ N
and for Z ,Z effective cyles, let f = (f ,f ) : degZ +degZ degZ degZ
0 1 0 1 0 1 0 1
→ ×
be a a path from (1,1) to (degZ ,degZ ) such that in each step exactly one of the
0 1
coordinates increases. If in the kth step the coordinate i increases, set i = i.
k
1.6 Theorem For any effective cycles Z ,Z Z ( t) that intersect properly,
0 1 eff
∈ P
and θ t a point not contained in the support of Z .Z , there is a path f such
0 1
that ∈ PC
n
2D(Z0,Z1)+2D(Z0.z1,θ) ≤ Dfik(k)(Zik,θ)+
k=1
X
O((degZ degZ +S)log(SdegZ degZ )).
0 1 0 1
1.7 Corollary With any l degZ +degZ , and (ν ,ν ) = f(l),
0 1 0 1
≤
2D(Z ,Z )+2D(Z .Z ,θ)
0 1 0 1
≤
ν D3ν1(Z ,θ)+ν D3ν0(Z ,θ)+O((degZ degZ +S)log(SdegZ degZ )).
0 1 1 0 0 1 0 1
4
This immediately implies
1.8 Corollary For every S degZ /3,
1
≤
2D(Z ,Z )+2D(Z .Z ,θ) max(SD(Z ,θ),D3S−1(Z ,θ))+O(degX degY).
0 1 0 1 0 1
≤
1.9 Theorem For any S degXdegY/9, there are natural numbers ν ,ν with
0 1
≤
ν ν S, and a path f such that and a function h : degZ +degZ with
0 1 S 0 1
≤ → N
h (k) = 0 for k k such that
S 0
≥
1.
n
2D(X,Y)+2DS(X.Y,θ) ≤ D3(fik(k)−hS(k))(Zik,θ).
k=1
X
2. For any k degZ +degZ greater or equal k , and (ν¯ ,ν¯ ) = f(k),
0 1 0 0 1
≤
2(ν¯ ν )(ν¯ ν )log Z +Z ,θ +2DS(Z .Z ,θ)+D(Z ,Z )
0 0 1 1 0 1 0 1 0 1
− − | | ≤
(ν¯ ν )D3κ0(Z ,θ)+(ν¯ ν )D3κ0(Z ,θ)+
0 0 1 1 1 0
− −
O((degZ degZ +S)log(SdegZ degZ )).
0 1 0 1
1.10 Corollary For any S degZ degZ /9, there are numbers ν ν with ν ν
0 1 0 1 0 1
≤ ≤
S such that for any l degZ +degZ greater or equal k with (ν¯ ,ν¯ ) = f(l),
0 1 0 0 1
≤
2D(Z ,Z )+2DS(Z .Z ,θ)
0 1 0 1
≤
(ν¯ ν )D3ν¯0(Z ,θ)+(ν¯ ν )D3ν¯1(Z ,θ)+O((degZ degZ +S)log(SdegZ degZ )).
1 1 0 0 1 0 1 0 1
− −
1.11 Corollary Let S degZ /3, S degZ /3 be natural numbers, and S =
0 0 1 1
≤ ≤
S S . Then,
0 1
2D(Z ,Z )+2DS(Z .Z ,θ) max(S D9S0(Z ,θ),S D9S1(Z ,θ))+
0 1 0 1 1 0 0 1
≤
O((degZ degZ +S)log(SdegZ degZ )).
0 1 0 1
Remarks: 1. I strongly conjecture that Theorem 1.4 as well as Theorems 1.6,
1.9, and their corollaries still hold if the factor 2 before D(θ,X.Y) and DS(θ,X.Y)
is dropped, and possibly also if the 3 in the exponent on the right hand side is
replaced by some smaller number greater or equal 1. In order to obtain this, one
would only have to improve Lemma 4.8 in this respect; however, I don’t know
right now how to do that. For the applications of the Theorems and Corollaries to
5
Diophantine Approximation and algebraic independence theory this improvement
would be insubstantial.
2. Throughout this paper, constants entailed by the notaion O( ) always depend
···
only on t, and the dimensions of cycles involved in the context. As, there are
always only finitely many cycles involved, the constants can also be assumed to be
depending only on t.
3. To my knowledge, in the literature, one special case of the above Theorems and
Corollarys is known, namely Corollary 1.8 in the case t = 1, codimZ = codimZ =
0 1
1. See [LR], preuve du corollaire 3.
Recall that in [Ma1], section 4 there were given 3 alternative definitions of the
algebraic distance. The algebraic distance D (θ, ) is not additive on the cycle
∞
·
group, and has some other deficencies; therefore it is probably not possible to prove
the derivative metric B´ezout for D . Proofs will be given for D and D .
∞ Ch 1
This paper heavily depends on part one ([Ma1]) of this series on diophantine ap-
proximation on varieties. and can possible not be read independentyl of it. It does
not however presuppose any knowledge of part 2 and part 3.
2 Sharp decomposition of the algebraic distance
Recall the following notations from [Ma1]. If G = G is the Grassmannian of q-
q,t
dimensional subspaces of t+1, then for a subspace (W) t of dimension r q
C P ⊂ P ≤
the sub Grassmannian of G consisting of the spaces that contain W is denoted G ,
W
and for (F) t a subspace of dimension p q the sub Grassmannian of spaces
P ⊂ P ≥
being contained in F is denoted GF.
Let (F) be a subspace of codimension r, and π the map
P ⊂ PC
π : t (F⊥) (F), [v,w] [v], v F,w F⊥.
P \P → P 7→ ∈ ∈
For any sub variety X (F) of codimension p, the closure X := π−1(X) is
F
⊂ P
a subvariety of codimension p in t with the same degree as X . This induces
F
P
a map π∗ : Zp( (F)) Zp( t),X X with left inverse X X. (F). For
F
P → P 7→ 7→ P
two effective cycles X Zp( (F)),Y Zq( (F)), denote by D•P(F)(XF,YF) their
∈ P ∈ P
algebraic distance as cycles in (F).
P
In [Ma1], Theorem 4.11, and Proposition 4.16, for Z an effective cycle in t , of
codimension p and (F) (W) subspaces of codimensions r t p,PqC> r
P ⊃ P ≥ −
respectively, regular with respect to X the relations
D( (W),Z) D( (W), (F).Z)+c degZ D( (W),Z) D( (F),Z)+c degZ
1 2
P ≤ P P ≤ P − P
with c ,c constants depending only on p,q,r, and t, were proved, and thereby
1 2
the algebraic distance of Z to (W) modulo O(degZ) is reduced to the algebraic
P
distance of (W) to Z. (F) and the algebraic distance of (F) ot Z.
P P P
6
If one wants to consider derivatives of algebraic distances, this decomposition is
not good enough, because the derivatives of two functions may have an arbitrarily
big difference even if there values don’t differ very much. One needs the following
sharper decomposition.
2.1 Proposition With the above notations, there is a positive constant c only
1
depending on p,q,r, and t such that
D( (W),Z)+c degZ = D( (W),Z. (F))+D(Z, (F)).
1
P P P P
The proof will use two Lemmas.
2.2 Lemma Let p + r t, X Zp ( ), and (F) ( ) a subspace of
≤ ∈ eff PC P ⊂ P C
codimension r that intersects X properly. Let further (W) (F) be a subspace
P ⊂ P
of codimension q r that is regular with respect to X. Then
≥
D(Z , (W)) D(Z, (W)) = D(Z , (F)) D(Z, (F).
F F
P − P P − P
Proof If q t+1 p, this is [Ma1], Proposition 5.1.
≤ −
If q > t+1 p, the Lemma will be proved for D , and D successively, firstly for
Ch G
−
D : Let (V) t be a subspace of codimension t+1 p that does not intersect
Ch
Z, and fulPlfills ⊂(WPC) (V) (F). By [Ma1], Propos−ition 5.1, and Proposition
P ⊂ P ⊂ P
4.14.2,
D(Z , (F)) D(Z, (F)) = D(Z , (V)) D(Z, (V)) =
F F
P − P P − P
D (Z , (V)) D (Z, (V)). (1)
Ch F Ch
P − P
Next, we repeat the construction of the cycle deformation in [Ma1], section 5. For
λ ,
∈ C
(F,π,ψ ,X ) (2)
λ λ
isdefined asfollows. Let X beasubvariety ofcodimension pin t, further F t+1
P ⊂ C
a sub vector space that is regular with respect to X, and F⊥ t+1 the orthogonal
⊂ C
complement of F with respect to the canonical inner product on t+1, and (F⊥)
the correspondingCprojective subspace of t. Consider the map froCm above P C
P
π : (E ) (F⊥) (F ), [v w] [v].
P C \P C → P C ⊕ 7→
For each λ ∗, there is the automorphism
∈ C
ψ : (E ) (E ), [v w] [λv +w].
λ
P C → P C ⊕ 7→
For any effective cycle X in t that intersects (F) properly, define, Φ as the
P P
subvariety of ( t )t+1−p given as the Zariski closure of the set
PC ×AC
(ψ (x),...,ψ (x),λ) ( t )t+1−p ∗ x X( ). . (3)
λ λ
{ ∈ PC ×C | ∈ C }
7
Then, Φ intersects (F) 1 properly. Further, for λ , and y the coordinate
P ×A ∈ C
of the affine line, the divisor Φ corresponding to the restriction of the function
λ
y λ to Φ is a proper intersection of Φ and the zero set of y λ and is of the form
− −
Zt+1−p λ , for some subvariety Z of t , and for λ = 0, we have Z = ψ (X).
Tλhe spe×cia{liz}ation Φ equals λ PC 6 λ λ
0
Φ = π∗(X. (F) = π∗(X . (F)),
0 λ
P P
for arbitrary λ ∗. (See [BGS], p.994.)
∈ C
Recall the correspondence from [Ma1], section 5,
C¯ = C 1
×A
(cid:26) Z
F G
(cid:26) Z
(cid:26)= Z~
( t)t+1−p 1 (ˇt)t+1−p 1
P ×A P ×A
and the map ∆ : t 1 ( t)t+1−p 1 from the proof of [Ma1], Proposition
P ×A → P ×A
4.16.3: C is the Chow correspondence from (17); and F,G,∆ are defined by taking
the identity on the second factor.
With (F,π,ψ ,Z ) for an effective cycle Z of codimension as above, and the corre-
λ λ
sponding cycle Φ t 1, the cycle Φ¯ := G F∗∆ Φ intersects ( (Wˇ ))t+1−p 1
∗ ∗
properly, and the⊂inPter×seAction of Φ¯ with (ˇt)t+1−p y is likewPise proper.×FAor
P × { }
any λ the cycle G F∗∆ (Z λ ) equals Ch(Z ) λ . Take g the
normal∈izeCd Green form o∗f (W∗ˇ )t+λ1×−p{in}(ˇt)t+1−p, andλdefi×ne{ } P(Wˇp)
P P
ϕ (λ) = δ g µ¯(q−1)(t+1−p),
W Ch(Zλ) (Wˇ)t+1−p
Z(ˇt)t+1−p P
P
ϕ (λ) = δ g µ¯(t−p)(t+1−p),
V Ch(Zλ) (Vˇ)t+1−p
Z(ˇt)t+1−p P
P
where µ¯ = c (O(1,...,1)) (see [Ma1], section 3.3 for details). By definition,
1
1 1
D ( (W),Z ) = ϕ (λ), D ( (V),Z ) = ϕ (λ),
Ch P λ (t+1−p)(t−p) W Ch P λ (t+1−p)(r−1) V
t−p,...,t−p r−1,...,r−1
(4)
(cid:0) (cid:1) (cid:0) (cid:1)
where (t+1−p)(r−1) is the multinomial coefficient ((t+1−p)(r−1))!. From the proof of
r−1,...,r−1 (r−1)!t+1−p
[Ma1], Proposition 4.16.3, it is clear that there are smooth real functions χ ,χ :
(cid:0) (cid:1) W F
such that ϕ (λ) = χ (log λ ), ϕ (λ) = χ (log λ ), and further that
W W V V
R → R | | | |
χ′ (0) = χ′ = 0, and χ′′ ,χ′′ are nonnegative; consequently ϕ (λ) ϕ (0),
W V W V W ≥ W
ϕ (λ) ϕ (0). Finally, with µ being the Fubini-Study form on the ith factor of
V V i
≥
( t)t+1−p,
P
ddc([ϕ ]) = pr (δ pr∗(µt+1−q µt+1−q µ¯q(t+1−p)−1)) =
W 2∗ Φ¯ 1 1 ··· t+1−p
8
(t+1 p)(t p)
− − pr δ pr∗(µt µt ) ,
t p,...,t p 2∗ Φ¯ 1 1··· t+1−p
(cid:18) − − (cid:19)
(cid:0) (cid:1)
and similarly
(t+1 p)(r 1)
ddc([ϕ ]) = − − pr δ pr∗(µt mt ) ,
V r 1,...,r 1 2∗ Φ¯ 1 1··· t+1−p
(cid:18) − − (cid:19)
(cid:0) (cid:1)
hence
1 1
ddc([ϕ ]) = ddc([ϕ ]) = pr δ pr∗(µt mt ) ,
(t+1−p)(t−p) W (t+1−p)(r−1) V 2∗ Φ¯ 1 1··· t+1−p
t−p,...,t−p r−1,...,r−1
(cid:0) (cid:1) (5)
(cid:0) (cid:1) (cid:0) (cid:1)
which does not depend on (W), and (V) but only on the numbers p and t, the
P P
subspace (F), and the cycle Z. We get
P
1 1
χ′′ = χ′′,
(t+1−p)(t−p) W (t+1−p)(r−1) V
t−p,...,t−p r−1,...,r−1
(cid:0) (cid:1) (cid:0) (cid:1)
consequently, since χ′ (0) = χ (0)′ = 0,
W V
1 1
(ϕ (0) ϕ (λ) = (ϕ (0) ϕ (λ)). (6)
(t+1−p)(t−p) W − W (t+1−p)(r−1) F − F
t−p,...,t−p r−1,...,r−1
(cid:0) (cid:1) (cid:0) (cid:1)
Together with (4), this implies
D( (W),Z ) D( (W,Z) = D( (W),Z ) D( (W),Z ) =
F 0 1
P − P P − P
D( (V),Z ) D( (V),Z),
F
P − P
which together with (1) entails the Lemma.
Proof of Proposition 2.1 for D : Again, if (V)isasubspace ofcodimension
G
P
t + 1 p interesecting Z properly such that (V) (F), then, by the proof of
− P ⊂ P
[Ma1], Proposistion 4.14,
D( (F),Z ) D( (F),Z) = D( (V),Z ) D( (V),Z) =
F F
P − P P − P
D ( (V,Z )) D ( (V,Z)).
G F G
P − P
With (F,π,ψ ,X ) as in (2), and the correspondence
λ λ
C¯ = C 1
×A
(cid:26) Z
F G
(cid:26) Z
(cid:26)= Z~
t 1 G 1
P ×A ×A
9
(7)
where again F, and G are defined by taking the identity on the second factor, and
the intersections of Φ¯ with G y are proper. Thus, we can proceed just in the
× { }
case D , and define
Ch
(t+1−p)(t+1−q)
ϕ (λ) = δ g µ , ϕ (λ) = δ g ,
W VZλ GW G V VZλ GV
ZG ZG
where µ = c (L ) with L the canonical line bundle on G. (see [Ma1], section
G 1 G G
3.2.) We have again
[ϕ ] = (pr ) (δ pr∗(g µ(t+1−p)(t+1−q))), [ϕ ] = pr (δ pr∗(g )),
W 2 ∗ Φ 1 GW G V 2∗ Φ 1 GV
and the calculations
ddc(δΦ¯pr1∗(gGW))+δΦ¯.pr1∗(GW) = δΦ¯pr1∗(ω(gGW)),
ddc(δΦ¯pr1∗(gGV)+δΦ¯.pr1∗(GV) = δΦ¯pr1∗(ω(gGV))
imply this time
ddc([ϕ ]) = pr (δ pr∗(ω(g )µ(t+1−p)(t+1−q))),
W 2∗ Φ¯ 1 W G
ddc([ϕ ]) = pr (δ pr∗(ω(g )).
V 2∗ Φ¯ 1 V
Now, G is the single point V in G, hence ω(g ) is the canonical generator of
V V
the one dimensional space of harmonic forms Hp(t+1−p),p(t+1−p)(G). Further, by the
(t+1−p)(t+1−q)
intersection theoryonGtheformω(g )µ likewise equals thisgenerator.
W G
We get
ddc([ϕ ]) = ddc([ϕ ]),
W V
and the rest of the proof is in complete analogy to the case D .
Ch
2.3 Lemma Let X Zp ( t ), and (W) t a subspace of codimension q >
t p that does not me∈et tehfef sPupCport of PX. Fin⊂alPlyC (F) a subspace of codimension
− P
r t p containing (W), and intersecting X properly. Then, for certain constants
≤ − P
c ,c depending only on p,q,r,t.
3 6
1.
D( (W),X ) = D (F)( (W),X . (F))+c degX.
F P F 3
P P P
2.
D (F)( (W),X . (F)) = D( (W),X . (F))+c degX.
P F F 6
P P P P
10