Blowing-up of locally monomially foliated space

(1)

HAL Id: hal-01100918

https://hal.archives-ouvertes.fr/hal-01100918

Preprint submitted on 7 Jan 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Blowing-up of locally monomially foliated space

Aymen Braghtha

To cite this version:

Aymen Braghtha. Blowing-up of locally monomially foliated space. 2015. �hal-01100918�

(2)

Blowing-up of locally monomially foliated space

Aymen Braghtha January 7, 2015

Abstract

In this paper, we prove that the blowing-up preserve the local monomiality of foliated space.

1 Locally monomial foliations

Let M be an analytic manifold of dimension n and D ⊂ M be a divisor with normal crossings. We denote respectively by O M and Θ M [log D] the sheaf of holomorphic functions and the sheaf of vector fields on M which are tangent to D.

A singular foliation on (M, D) is coherent subsheaf F of Θ M [log D] which is reduced and integrable (see [1] and [2]). The dimension (or the rank) of F is given by

s = max

p∈M dim F(p)

where F(p) ⊂ T p M denote the vector subspace generated by the evaluation of F at p.

Let F be a field (we usually take F = Q, R or C ). We shall say that F is F -locally monomial if for each point p ∈ M there exists

1. a local system of coordinates x = (x 1 , . . . , x n ) at p 2. an s-dimensional vector subspace V ⊂ F ⁿ

such that D is locally given by

D p = {x i = 0 : i ∈ I}, for some I ⊂ {1, . . . , n}

and F p is the O M,p -module generated by the abelian Lie algebra

L(V ) = { X n

i=1

a i x i

∂

∂x i

: a ∈ V } M

i I:e ¯

i

∈M

F ∂

∂x i

where I ¯ = {1, . . . , n} \ I. We shall say that the triple (M, D, F) is locally monomially foliated space and that (x, I, V ) is a local presentation at p.

Lemma 1.1. Let (x, I, V ) be a local presentation for (M, D, F) at a point p. Then, for each vec- tor m ∈ V ^⊥ , the (possibly multivalued) function f (x) = x ^m is a first integral of F.

2 Blowing-up

Let Y ⊂ M be a smooth submanifold of codimension r. We shall say that Y has normal crossings with (M, D, F ) if for each point p ∈ Y there exists a local presentation (x, I, V ) at p such that Y is given by

Y = {x 1 = x 2 = . . . = x r = 0}.

1

(3)

Proposition 1.2. Let Φ : M f → M be the blowing-up with a center Y which has normal crossings with (M, D, F). Let

D e = Φ ⁻¹ (D) and F ⊂ e Θ _M _f [log D] e

denote the total transform of D and the strict transform of F respectively. Then, the triple ( M , f D, e F e ) is a locally monomially foliated space.

The proof is based on the following result on linear algebra.

3 Some linear algebra

Let F be a field and let V ⊂ F ⁿ be a vector subspace of dimension s. Let us fix a disjoint partition of indices {1, . . . , n} = I 1 ⊔ I 2 write F ⁿ = F ^I

¹

⊕ F ^I

²

and let π I : F ⁿ → F ^I denote the projection in the corresponding subspace F ^I generated by {e i : i ∈ I}.

Lemma 3.1. There exists a basis for V such that for each vector v in this basis, either v = π I

₂

(v) or v = π I

₂

(v) + e i

for some i ∈ I 1

Proof. Up to a permutation of coordinates, we can suppose that I 1 = {1, . . . , n 1 } and I 1 = {n 1 + 1, . . . , n} (with the convention that n 1 = 0 if I 1 = ∅)

Let M = [m 1 , . . . , m s ] be a s ×n matrix whose rows are an arbitrary basis of V . By a finite number of elementary row operations and permutations of columns (which leave invariant the subsets I 1 and I 2 ), we can suppose that the matrix M has the form

M =

Id k

₁

×k

₁

A k

₁

×l

₁

0 k

₁

×k

₂

B k

₁

×l

₂

0 k

₂

×k

₁

0 k

₂

×l

₁

Id k

₂

×k

₂

C k

₂

×l

₂

where k i + l i = |I i | for i = 1, 2, k 1 + k 2 = s and Id k,l and 0 k,l denote the k × l identity and zero matrix respectively.

Now, it suffices to define v i = e i + P

n

₁

+k

₂

+1≤j m i,j e j , for i = 1, . . . , k 1 and v i = e i+n

₁

+ P

n

₁

+k

₂

+1≤j m i,j e j for i = k 1 + 1, . . . , s.

References

[1] Paul Baum and Raoul Bott, Singularities of holomorphic foliations. J. Differential Geometry, 7:

279-342, 1972.

[2] Yoshiki Mitera and Junya Yoshizaki. The local analytical triviality of a complex analytic singular foliation. Hokkaido Math. J., 33(2):275-297, 2004

Blowing-up of locally monomially foliated space

HAL Id: hal-01100918

https://hal.archives-ouvertes.fr/hal-01100918

Preprint submitted on 7 Jan 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Blowing-up of locally monomially foliated space

Aymen Braghtha

To cite this version:

Aymen Braghtha. Blowing-up of locally monomially foliated space. 2015. �hal-01100918�

Blowing-up of locally monomially foliated space

Aymen Braghtha January 7, 2015

Abstract

In this paper, we prove that the blowing-up preserve the local monomiality of foliated space.

1 Locally monomial foliations

Let M be an analytic manifold of dimension n and D ⊂ M be a divisor with normal crossings. We denote respectively by O M and Θ M [log D] the sheaf of holomorphic functions and the sheaf of vector fields on M which are tangent to D.

A singular foliation on (M, D) is coherent subsheaf F of Θ M [log D] which is reduced and integrable (see [1] and [2]). The dimension (or the rank) of F is given by

s = max

p∈M dim F(p)

where F(p) ⊂ T p M denote the vector subspace generated by the evaluation of F at p.

Let F be a field (we usually take F = Q, R or C ). We shall say that F is F -locally monomial if for each point p ∈ M there exists

1. a local system of coordinates x = (x 1 , . . . , x n ) at p 2. an s-dimensional vector subspace V ⊂ F n

such that D is locally given by

D p = {x i = 0 : i ∈ I}, for some I ⊂ {1, . . . , n}

and F p is the O M,p -module generated by the abelian Lie algebra

L(V ) = { X n

i=1

a i x i

∂

∂x i

: a ∈ V } M

i I:e ¯

∈M

F ∂

∂x i

where I ¯ = {1, . . . , n} \ I. We shall say that the triple (M, D, F) is locally monomially foliated space and that (x, I, V ) is a local presentation at p.

Lemma 1.1. Let (x, I, V ) be a local presentation for (M, D, F) at a point p. Then, for each vec- tor m ∈ V ⊥ , the (possibly multivalued) function f (x) = x m is a first integral of F.

2 Blowing-up

Let Y ⊂ M be a smooth submanifold of codimension r. We shall say that Y has normal crossings with (M, D, F ) if for each point p ∈ Y there exists a local presentation (x, I, V ) at p such that Y is given by

Y = {x 1 = x 2 = . . . = x r = 0}.

1

Proposition 1.2. Let Φ : M f → M be the blowing-up with a center Y which has normal crossings with (M, D, F). Let

D e = Φ −1 (D) and F ⊂ e Θ M f [log D] e

denote the total transform of D and the strict transform of F respectively. Then, the triple ( M , f D, e F e ) is a locally monomially foliated space.

The proof is based on the following result on linear algebra.

3 Some linear algebra

Let F be a field and let V ⊂ F n be a vector subspace of dimension s. Let us fix a disjoint partition of indices {1, . . . , n} = I 1 ⊔ I 2 write F n = F I

⊕ F I

and let π I : F n → F I denote the projection in the corresponding subspace F I generated by {e i : i ∈ I}.

Lemma 3.1. There exists a basis for V such that for each vector v in this basis, either v = π I

(v) or v = π I

(v) + e i

for some i ∈ I 1

Proof. Up to a permutation of coordinates, we can suppose that I 1 = {1, . . . , n 1 } and I 1 = {n 1 + 1, . . . , n} (with the convention that n 1 = 0 if I 1 = ∅)

Let M = [m 1 , . . . , m s ] be a s ×n matrix whose rows are an arbitrary basis of V . By a finite number of elementary row operations and permutations of columns (which leave invariant the subsets I 1 and I 2 ), we can suppose that the matrix M has the form

M =

Id k

×k

A k

×l

0 k

×k

B k

×l

0 k

×k

0 k

×l

Id k

×k

C k

×l

where k i + l i = |I i | for i = 1, 2, k 1 + k 2 = s and Id k,l and 0 k,l denote the k × l identity and zero matrix respectively.

Now, it suffices to define v i = e i + P

n

+k

+1≤j m i,j e j , for i = 1, . . . , k 1 and v i = e i+n

+ P

n

+k

1. a local system of coordinates x = (x 1 , . . . , x n ) at p 2. an s-dimensional vector subspace V ⊂ F ⁿ

Lemma 1.1. Let (x, I, V ) be a local presentation for (M, D, F) at a point p. Then, for each vec- tor m ∈ V ^⊥ , the (possibly multivalued) function f (x) = x ^m is a first integral of F.

D e = Φ ⁻¹ (D) and F ⊂ e Θ _M _f [log D] e

Let F be a field and let V ⊂ F ⁿ be a vector subspace of dimension s. Let us fix a disjoint partition of indices {1, . . . , n} = I 1 ⊔ I 2 write F ⁿ = F ^I

⊕ F ^I

and let π I : F ⁿ → F ^I denote the projection in the corresponding subspace F ^I generated by {e i : i ∈ I}.