On an invariant of b-operator for Reiffens ( p, 4) isolated - - PowerPoint PPT Presentation

On an invariant of b-operator for Reiffen’s (p, 4) isolated singularity.

Yayoi Nakamura (Kinki Univ.)

Algebra, Algorithms and Algebraic Analysis mini-workshop, 6th Sep. 2013 Rolduc, Netherland

1

b-function For a holomorphic function f defining an isolated singular- ity at the origin in Cn, there is a differential operator P(s) and a polynomial b(s) satisfying P(s)fs+1 = b(s)fs, the polynomial b(s) is called b-function and the operator P(s) is called b-operator.

X : an open neighbourhood of the origin O in Cn DX = D : the sheaf of holomorphic linear partial differen- tial operators of finite order on X OX = O : the sheaf of holomorphic functions on X D[s] = D ⊗

C

C[s]. x = (x1, . . . , xn) For f ∈ O, fi = ∂f

∂xi, A = n

• i=1

Ofi J (s) = {P(s) ∈ D[s] | P(s)fs = 0}.

invariant L(f) For a linear partial differential operator P(s) =

• sjPj(z, ∂) ∈ D[s],
• rdT(P(s)) = maxj(j+ordPj(z, ∂)) is called the total order
• f P(s).

There exists operators of the form P(s) =

ℓ

• j=0

sℓ−jPj(z, ∂) in J (s) s uch that ordTP(s) = ℓ, P0(z, ∂) = 1. L(f) denotes the minimum of ordTP(s) for P(s) ∈ J (s)

• f the form specified as above.

L(f) = 1 is a necessary and sufficient condition for the function to be quasihomogeneous.

• T. Yano developed a general theory of b-function and gave

various examples of b-function. He introduced the number L(f) and investigated a method to determine b-functions for f being isolated singularities with L(f) = 2 and L(f) = 3.

• T. Yano, On the Theory of b-functions, Publ. RIMS,

14 (1978), 111-202.

Reiffen’s singularity Let f = zq

1 + zp 2 + z1zp−1 2

with p, q ∈ N, q ≥ 4 and p ≥ q + 1. The hypersurface zq

1 + zp 2 + z1zp−1 2

= 0 in C2 defines a semi-quasihomogeneous singularity of weight (1

q, 1 p) with the

Milnor number (p − 1)(q − 1) and the Tjurina number (p − 1)(q − 1) − q + 3. This hypersurface is examined by H.-J.Reiffen as a singu- larity on which the holomorphic deRham complex is not exact( H.-J. Reiffen, Das Lemma von Poincar´ e f¨ ur holomorphe Differentialformen auf komplexen R¨ aumen,

• Math. Zeitchr., 101 (1967), 269–284. ).

Theorem Let f = z4

1 +zp 2 +z1zp−1 2

with p ∈ N and p ≥ 5. Then L(f) = 2.

• Proof. Let P(s) be the following differential operator;

((p − 1)3zp−4

2

− 4p3)s2 +

• −(p − 4)p3

p − 1 z2 ∂ ∂z1 + 3(p − 4)p2 p − 1 z2 ∂ ∂z2 +(p − 1)2(7p − 16) 4 zp−4

2

− p2(4p2 − 7p + 12) p − 1

• s

+

• (−(p − 1)3

42 zp−4

2

+ p3 4 )z2

1 + (p − 4)p3

4(p − 1) z1z2 + (p − 4)(p − 1)(p + 4) 42 zp−2

2

∂2 ∂z2

1

+

• −(p − 4)(p + 4)

4 z2

1 + (−3(p − 1)2

8 zp−3

2

+ (p + 4)pz2)z1 −(p − 4)(p − 1) 2 zp−2

2

+ (p − 4)p2 p − 1 z2

2

• ∂2

∂z1∂z2 + 3(p − 4) 2 z2

1 − (p − 4)(5p + 4)

4(p − 1) z1z2 − 9(p − 1) 42 zp−2

2

+ (p + 8)p p − 1 z2

2

∂2 ∂z2

2

+

• (−(p − 1)2(8p − 17)

16 zp−4

2

+ p2(5p2 − 8p + 12) 4(p − 1) )z1 −3(p − 1)(p − 2)(p − 4) 8 zp−3

2

+ p3(p − 4) 4(p − 1) z2 ∂ ∂z1 +

• −3p(p − 4)(p + 2)

4(p − 1) z1 − 3(p − 1)(7p − 13) 16 zp−3

2

+ p(15p2 − 16p + 64) 4(p − 1) z2 ∂ ∂z2 .

By the definition, the total order of the operator is 2. One can check that the operator P(s) annihialtes fs. Thus we have L(f) ≤ 2. Since f is not quasihomogeneous, L(f) ≥ 2. It completes the proof.

Yano’s method N = D[s]/J (s) = D[s]fs is a D[t, s]-module with actions

• f t and s given by

t : P(s) → P(s + 1)f, s : P(s) → P(s)s Put M = N/tN, ˜ M = (s + 1)M Then, M = D[s]/(J (s) + D[s]f) ˜ M = D[s]/(J (s) + D[s](A + Of)) b-function is the minimal polynomial of the action s to M s : M → M, P(s)fs → sP(s)fs.

If f(x) = 0, P(−1)f−1+1 = b(−1)f−1. Let ˜ b(s) ∈ C[s] satisfying b(s) = (s + 1)˜ b(s). Then, ˜ b(s) is the minimal polynomial of the action s to ˜ M. Thus, the determination of b-function is reduced to the study of ˜ M .

• T. Yano had given a method for computing b-function for

the case L(f) ≤ 3. Main idea of his method is as follows.

• Construct

˜ M and give presentation of ˜ M

• Apply the functor HomD(·, Bpt)
• Compute the representation matrix of s
• Compute the minimal polynomial of the matrix

Here, Bpt = Dδ, δ:delta function.

L(f) = 1 case. Assume that f is quasihomogeneous function. Then, ˜ M ∼ = D/DA. ˜ M has the presentation 0 ← − ˜ M ← − D

(fi)

← − Dn. Applying the functor HomD(·, Bpt), we have 0 − → F − → Bpt − → Bn

pt

where F = HomD( ˜ M, Bpt). Here, F = {η ∈ Bpt | gη = 0, g ∈ A}

In quasihomogeneous case, since s−X0 ∈ J (s) holds with the euler operator X0 satisfying X0f = f, P(s) = sjPj(x, D) ∈ D[s] and Pj(x, D)Xj

0 are congru-

ent modulo J (s). Thus the action of s in F is X0. This action compute the weighted degree of each classes in F.

Local cohomology classes η ∈ Bpt can be denoted η =

• λ

cλ[ 1 xλ] =

• (ℓ1,...,ℓn)

c(ℓ1,...,ℓn)[ 1 xℓ1

1 xℓ2 2 · · · xℓn n

] For the weight (w1, . . . , wn), we define the weighted degree

• f the cohomology class η by

wη = − max{

n

• i=1

wiℓi | cλ = 0}.

Let f = x4 + y5. A basis of the dual space of Milnor algebra is given by the algebraic local cohomology classes of the form [ 1 xiyj], 1 ≤ i ≤ 3, 1 ≤ j ≤ 4. We denote the class [ 1 xiyj] by i, j for simplicity. The operator X0 is given by X0 = 5 20x ∂ ∂x + 4 20y ∂ ∂y. Applying the operator X0 to each local cohomology class i, j, we have the weighted degree of each class i, j

i, j 1, 1 1, 2 2, 1 1, 3 2, 2 3, 1 −20X0i, j 9 13 14 17 18 19 1, 4 2, 3 3, 2 2, 4 3, 3 3, 4 21 22 23 26 27 31 The b-function of f = x4 + y5 b(s) = (s + 1)(20s + 9)(20s + 13)(20s + 17) (20s + 19)(20s + 21)(20s + 23)(20s + 27) (20s + 31)(10s + 7)(10s + 9)(10s + 11) (10s + 13)

L(f) = 2 case There are non-constant functions in ideal quotient A : f. Let aν (ν = 1, . . . , r) be the generators of A : f. Let aν,i(x) ∈ OX (i = 1, . . . , n) be functions satisfying aν(x)f +

n

• i=1

aν,i(x)fi = 0 for each aν(x). Set a′

ν = n i=1 aν,i(x) ∂

∂xi .

A representation of ˜ M is given by 0 ← ˜ M

• 1

s

• ←

− D2

X    

fi 0 f a′

ν aν    

← − Dn+r+1

X

, where    fi 0 f a′

ν aν

   = t

• f1 . . . f1 f a′

1 . . . a′ r

0 . . . 0 0 a1 . . . ar

• .

Applying functor HomDX(·, Bpt), we have 0 → F → B2

pt → Bn+r+1 pt

with F = HomDX( ˜ M, Bpt).

Let F1 = {u ∈ Bpt | (A + OXf)u = 0} and F2 = {v ∈ Bpt | (A : f)v = 0}. Set µ1 = dim F1 and µ2 = dim F2. Then µ1 = dim OX/(A+ OXf), µ2 = dim OX/(A : f) and thus µ1 + µ2 = µ := dim OX/A holds. Since F2 ⊂ F1, for a basis (u1, . . . , uµ2)

• f F2, we can take a basis of F1 as (u1, . . . , uµ2, uµ2+1, . . . , uµ1).

For each ui ∈ F1, there exists algebraic local cohomology class vi so that aν(x)v = −a′

ν(x, D)u

mod F2. Then

• ui
• , i = 1, . . . , µ2 and
• ui

vi

• , i = 1, . . . , µ1 form

the basis of F.

A first order differential operator A ∈ DX and a second

• rder differential operator B ∈ DX exist so that

s2 + As + B ∈ J (s). The action of s on F is represented by s :

• u

v

• →
• 1

−B −A u v

• .

Then, b-function b(s) = (s + 1)˜ b(s) is given as the minimal polynomial of representation matrix of s on the above basis

• f F.

Let f = x4 + y5 + xy4. f is a semiquasihomogeneous function with weight vector (1 4, 1 5) of weighted degree 1 and L(f) = 2. The following 12 cohomology classes constitute a basis of the dual sapce {η ∈ Bpt | fjη = 0, j = 1, . . . , n} of OX/A as a vector space: 1, 1 (− 9

20) η10 = 1, 2 (−13 20) η9 = 2, 1 (− 7 10)

η8 = 1, 3 (−17

20) η7 = 2, 2 (− 9 10) η6 = 3, 1 (−19 20)

η5 = 1, 4 (−21

20) η4 = 2, 3 (−11 10) η3 = 3, 2 (−23 20)

η2 = 2, 4 − 4

51, 5 + 1 54, 1

(−13

10)

η1 = 3, 3 (−27

20)

and 3, 4 − 4

52, 5 + 16 251, 6 + 1 55, 1 − 4 254, 2

(−31

20).

The number on the right hand side is the weighted degree

• f each cohomology class.

Then F1 = SpanC{1, 1, η1, . . . , η10} and F2 = SpanC{1, 1}. Using first order annihilators aν,1 ∂ ∂x+aν,2 ∂ ∂y+aν, ν = 1, 2 with (a1, a1,1, a1,2) = (−48x − 60y, 12x2 + 15xy, 9xy + 12y2) (a2, a2,1, a2,2) = (144y2 − 60x − 1200y, −36xy2 − 9y3 + 15x2 + 300xy, −27y3 + 9x2 + 240y2),

• ne can verify that the following 12 classes constitute a basis
• f space F = HomDX( ˜

M, Bpt):

• 1, 1
• ,
• 1, 1
• ,
• η10

−13

20η10

• ,
• η9

− 7

10η9 + 1 100η10

• ,
• η8

−17

20η8 + 3 500η9 − 3 625η10

• ,
• η7

− 9

10η7 + 1 50η8 − 2 625η9 + 8 3125η10

• ,
• η6

−19

20η6 + 1 100η7 − 1 125η8 + 4 3125η9 − 16 15625η10

• ,
• η5

−21

20η5 − 1 100η6 + 1 125η7 − 4 625η8 + 16 15625η9 − 64 78125η10

• ,
• η4

−11

10η4 + 3 100η5 + 3 500η6 − 3 625η7 + 12 3125η8 − 48 78125η9 + 192 390625η10

• ,
• η3

−23

20η3 + 1 50η4 − 2 125η5 − 2 625η6 + 8 3125η7 − 32 15625η8 + 128 390625η9 − 512 1953125η10

• ,

   η2 −13

10η2 + 9 500η3 − 9 625η4 + 36 3125η5 + 36 15625η6 − 144 78125η7 + 576 390625η8 − 2304 9765625η9

+

9216 48828125η10

  ,    η1 −27

20η1 + 3 100η2 − 3 625η3 + 12 3125η4 − 48 15625η5 − 48 78125η6 + 192 390625η7 − 768 1953125η8

+

3072 48828125η9 − 12288 244140625η10

  

We have in J (s) a second order annihilator P = s2+As+B where A = − 1

8000 1 1− 16

125y(−500y∂x + 300y∂y + 1216y − 7700),

B = − 1

8000 1 1− 16

125y(((−64y + 500)x2 + 125yx + 36y3)∂2

x

+ (−36x2 + (−96y2 + 720y)x − 32y3 + 100y2)∂x∂y + ((−368y + 2425)x − 72y2 + 125y)∂x + (24x2 − 29yx − 36y3 + 260y2)∂2

y

+ (−105x − 264y2 + 1795y)∂y).

Then the upper ten minors of representation matrix of s on the basis above is triangular whose diagonal elements are the weighted degrees of the corresponding algebraic local coho- mology classes. Thus we only need to compute the minimal polynomial of the lower two minors. For the cohomology classes, e1 =

• 1, 1
• , e2 =
• 1, 1
• , we have
• 1

−B −A

• e1 = −( 9

20 + 11 20)e1 + e2,

• 1

−B −A

• e2 = − 9

20 11 20e1.

Thus, the representation matrix of the action of s on (e2 e1) is given by

• 1

− 99

400 −( 9 20 + 11 20)

• . The characteristic poly-

nomial of this matrix is (λ + 9

20)(λ + 11 20). Then we have

b(s) = (s+1)(10s+7)(10s+9)(10s+11)(10s+13)(20s+9)(20s+11) (20s+13)(20s+17)(20s+19)(20s+21)(20s+23)(20s+27).

η10 = 1, 2 (−13

20) η9 = 2, 1 (− 7 10)

η8 = 1, 3 (−17

20) η7 = 2, 2 (− 9 10) η6 = 3, 1 (−19 20)

η5 = 1, 4 (−21

20) η4 = 2, 3 (−11 10) η3 = 3, 2 (−23 20)

η2 = 2, 4 − 4

51, 5 + 1 54, 1

(−13

10)

η1 = 3, 3 (−27

20)

1, 1 (− 9

20),

3, 4 − 4

52, 5 + 16 251, 6 + 1 55, 1 − 4 254, 2

(−31

20).

b(s) = (s + 1)(20s + 9)(20s + 11) (20s + 13)(10s + 7)(20s + 17)(10s + 9)(20s + 19) (20s + 21)(10s + 11)(20s + 23)(10s + 13)(20s + 27).

For f = z4

1 + zp 2 + z1zp−1 2

, a basis of the vector space {η ∈ Bpt | f1η = f2η = 0} is given by 3(p − 1) local cohomology classes, [

1 zℓ1

1 zℓ2 2

] with 1 ≤ ℓ1 ≤ 3 and 1 ≤ ℓ2 ≤ p − 2, [

1 z1zp−1

2

] and the classes of the form [ 1 z2

1zp−1 2

] + (−p − 1 p )k[ 1 z1zp

2

] − 1 4(−p − 1 p )[ 1 z4

1z2

] and [ 1 z3

1zp−1 2

] + (−p − 1 p )[ 1 z2

1zp 2

] + (−p − 1 p )2[ 1 z1zp+1

2

] −1 4(−p − 1 p )[ 1 z5

1z2

] − 1 4(−p − 1 p )2[ 1 z4

1z2 2

]

Since the upper 3(p−1)−2 minors of representation matrix

• f s on the basis of F is triangular whose diagonal elements

are the weighted degrees of the corresponding algebraic lo- cal cohomology classes. Thus we only need to compute the minimal polynomial of the lower two minors. For the coho- mology classes, e1 =

• 1, 1
• , e2 =
• 1, 1
• , we have
• 1

−B −A

• e1 = −(3p − 4

4p + p + 4 4p )e1 + e2,

• 1

−B −A

• e2 = −3p − 4

4p · p + 4 4p e1.

Thus, the representation matrix of the action of s on (e2 e1) is given by

• 1

− 99

400 −(3p−4 4p + p+4 4p )

• .

The characteristic polynomial of this matrix is (λ + 3p−4

4p )(λ + p+4 4p ).

Theorem Let f = z4

1 + zp 2 + z1zp−1 2

with p ∈ N, p ≥ 5, (4, p) = 1. Then, the roots of the b-function of f are given by −pℓ1 + 4ℓ2 4p (1 ≤ ℓ1 ≤ 3, 1 ≤ ℓ2 ≤ p − 2), −5p − 4 4p , −6p − 4 4p , −3p − 4 4p .

参考文献

[1] F. J. Carderon Moreno, D. Mond, L. Nar- vaez Macarro, F.J.Castro Jimenez , Logarith- mic cohomology of the complement of a plane curve,

• Comment. Math. Helv. 77 (2002), 1, 24–38.

[2] H.-J. Reiffen, Das Lemma von Poincar´ e f¨ ur holo- morphe Differentialformen auf komplexen R¨ aumen,

• Math. Zeitchr., 101 (1967), 269–284.

[3]

• T. Yano, On the Theory of b-functions, Publ.

RIMS, 14 (1978), 111-202.