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 C n , there is a differential operator P ( s ) and a polynomial b ( s ) satisfying P ( s ) f s +1 = b ( s ) f s , 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 C n D X = D : the sheaf of holomorphic linear partial differen- tial operators of finite order on X O X = O : the sheaf of holomorphic functions on X D [ s ] = D ⊗ C [ s ]. C x = ( x 1 , . . . , x n ) n For f ∈ O , f i = ∂f � ∂x i , A = O f i i =1 J ( s ) = { P ( s ) ∈ D [ s ] | P ( s ) f s = 0 } .

invariant L ( f ) For a linear partial differential operator s j P j ( z, ∂ ) ∈ D [ s ] , � P ( s ) = ord T ( P ( s )) = max j ( j +ord P j ( z, ∂ )) is called the total order of P ( s ). There exists operators of the form ℓ s ℓ − j P j ( z, ∂ ) � P ( s ) = j =0 in J ( s ) s uch that ord T P ( s ) = ℓ , P 0 ( z, ∂ ) = 1. L ( f ) denotes the minimum of ord T P ( s ) for P ( s ) ∈ J ( s ) of 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 = z q 1 + z p 2 + z 1 z p − 1 with p, q ∈ N , q ≥ 4 and 2 p ≥ q + 1. 2 + z 1 z p − 1 The hypersurface z q 1 + z p = 0 in C 2 defines a 2 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 aumen , holomorphe Differentialformen auf komplexen R¨ Math. Zeitchr., 101 (1967), 269–284. ).

1 + z p 2 + z 1 z p − 1 Theorem Let f = z 4 with p ∈ N and p ≥ 5. 2 Then L ( f ) = 2 . Proof. Let P ( s ) be the following differential operator; (( p − 1) 3 z p − 4 − 4 p 3 ) s 2 2 − p 2 (4 p 2 − 7 p + 12) − ( p − 4) p 3 + 3( p − 4) p 2 +( p − 1) 2 (7 p − 16) � � ∂ ∂ z p − 4 + z 2 z 2 s 2 p − 1 ∂z 1 p − 1 ∂z 2 4 p − 1 � ∂ 2 ( − ( p − 1) 3 + p 3 1 + ( p − 4) p 3 � 4( p − 1) z 1 z 2 + ( p − 4)( p − 1)( p + 4) z p − 4 z p − 2 4 ) z 2 + 2 2 ∂z 2 4 2 4 2 1 1 + ( − 3( p − 1) 2 + ( p − 4) p 2 ∂ 2 � − ( p − 4)( p + 4) + ( p + 4) pz 2 ) z 1 − ( p − 4)( p − 1) � z p − 3 z p − 2 z 2 z 2 + 2 2 2 4 8 2 p − 1 ∂z 1 ∂z 2 � ∂ 2 � 3( p − 4) 1 − ( p − 4)(5 p + 4) z 1 z 2 − 9( p − 1) + ( p + 8) p z p − 2 z 2 p − 1 z 2 + 2 2 ∂z 2 2 4( p − 1) 4 2 � ∂ 2 + p 2 (5 p 2 − 8 p + 12) ( − ( p − 1) 2 (8 p − 17) + p 3 ( p − 4) � ) z 1 − 3( p − 1)( p − 2)( p − 4) z p − 4 z p − 3 + 4( p − 1) z 2 2 2 16 4( p − 1) 8 ∂z 1 � ∂ + p (15 p 2 − 16 p + 64) � − 3 p ( p − 4)( p + 2) z 1 − 3( p − 1)(7 p − 13) z p − 3 + z 2 . 2 4( p − 1) 16 4( p − 1) ∂z 2

By the definition, the total order of the operator is 2. One can check that the operator P ( s ) annihialtes f s . 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 ] f s is a D [ t, s ]-module with actions of t and s given by t : P ( s ) �→ P ( s + 1) f, s : P ( s ) �→ P ( s ) s Put ˜ M = N /t N , M = ( s + 1) M Then, M = D [ s ] / ( J ( s ) + D [ s ] f ) ˜ M = D [ s ] / ( J ( s ) + D [ s ]( A + O f )) b-function is the minimal polynomial of the action s to M s : M → M , P ( s ) f s �→ sP ( s ) f s .

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, ˜ ˜ M . b ( s ) is the minimal polynomial of the action s to Thus, the determination of b-function is ˜ M . reduced to the study of

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 H om D ( · , B pt ) • Compute the representation matrix of s • Compute the minimal polynomial of the matrix Here, B pt = D δ , δ :delta function.

L ( f ) = 1 case. Assume that f is quasihomogeneous function. Then, M ∼ ˜ = D / DA . ˜ M has the presentation ( f i ) − D n . − ˜ 0 ← M ← − D ← Applying the functor H om D ( · , B pt ), we have → B n 0 − → F − → B pt − pt where F = H om D ( ˜ M , B pt ). Here, F = { η ∈ B pt | gη = 0 , g ∈ A}

In quasihomogeneous case, since s − X 0 ∈ J ( s ) holds with the euler operator X 0 satisfying X 0 f = f , P ( s ) = � s j P j ( x, D ) ∈ D [ s ] and P j ( x, D ) X j 0 are congru- ent modulo J ( s ). Thus the action of s in F is X 0 . This action compute the weighted degree of each classes in F .

Local cohomology classes η ∈ B pt can be denoted c λ [ 1 1 � � η = x λ ] = c ( ℓ 1 ,...,ℓ n ) [ ] x ℓ 1 1 x ℓ 2 2 · · · x ℓ n n λ ( ℓ 1 ,...,ℓ n ) For the weight ( w 1 , . . . , w n ), we define the weighted degree of the cohomology class η by n � w η = − max { w i ℓ i | c λ � = 0 } . i =1

Let f = x 4 + y 5 . A basis of the dual space of Milnor algebra is given by the algebraic local cohomology classes of the form [ 1 x i y j ] , 1 ≤ i ≤ 3 , 1 ≤ j ≤ 4 . We denote the class [ 1 x i y j ] by � i, j � for simplicity. The operator X 0 is given by X 0 = 5 20 x ∂ ∂x + 4 20 y ∂ ∂y. Applying the operator X 0 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 � − 20 X 0 � i, 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 = x 4 + y 5 b ( s ) = ( s + 1)(20 s + 9)(20 s + 13)(20 s + 17) (20 s + 19)(20 s + 21)(20 s + 23)(20 s + 27) (20 s + 31)(10 s + 7)(10 s + 9)(10 s + 11) (10 s + 13)

L ( f ) = 2 case There are non-constant functions in ideal quotient A : f . Let a ν ( ν = 1 , . . . , r ) be the generators of A : f . Let a ν,i ( x ) ∈ O X ( i = 1 , . . . , n ) be functions satisfying n � a ν ( x ) f + a ν,i ( x ) f i = 0 i =1 i =1 a ν,i ( x ) ∂ ν = � n for each a ν ( x ). Set a ′ . ∂x i

˜ A representation of M is given by   f i 0 � �   1 f 0     a ′ ν a ν s − D 2 D n + r +1 0 ← ˜ M ← ← − , X X   f i 0 � � f 1 . . . f 1 f a ′ 1 . . . a ′ r  = t where f 0 .   0 . . . 0 0 a 1 . . . a r  a ′ ν a ν Applying functor H om D X ( · , B pt ), we have 0 → F → B 2 pt → B n + r +1 pt with F = H om D X ( ˜ M , B pt ).

Let F 1 = { u ∈ B pt | ( A + O X f ) u = 0 } and F 2 = { v ∈ B pt | ( A : f ) v = 0 } . Set µ 1 = dim F 1 and µ 2 = dim F 2 . Then µ 1 = dim O X / ( A + O X f ), µ 2 = dim O X / ( A : f ) and thus µ 1 + µ 2 = µ := dim O X / A holds. Since F 2 ⊂ F 1 , for a basis ( u 1 , . . . , u µ 2 ) of F 2 , we can take a basis of F 1 as ( u 1 , . . . , u µ 2 , u µ 2 +1 , . . . , u µ 1 ). For each u i ∈ F 1 , there exists algebraic local cohomology class v i so that a ν ( x ) v = − a ′ ν ( x, D ) u mod F 2 . � � � � 0 u i Then , i = 1 , . . . , µ 2 and , i = 1 , . . . , µ 1 form u i v i the basis of F .

A first order differential operator A ∈ D X and a second order differential operator B ∈ D X exist so that s 2 + As + B ∈ J ( s ) . The action of s on F is represented by � � � � � � u 0 1 u �→ s : . v − B − A 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 of F .

Let f = x 4 + y 5 + xy 4 . 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 ∈ B pt | f j η = 0 , j = 1 , . . . , n } of O X / A as a vector space: � 1 , 1 � ( − 9 20 ) η 10 = � 1 , 2 � ( − 13 20 ) η 9 = � 2 , 1 � ( − 7 10 ) η 8 = � 1 , 3 � ( − 17 20 ) η 7 = � 2 , 2 � ( − 9 10 ) η 6 = � 3 , 1 � ( − 19 20 ) η 5 = � 1 , 4 � ( − 21 20 ) η 4 = � 2 , 3 � ( − 11 10 ) η 3 = � 3 , 2 � ( − 23 20 ) η 2 = � 2 , 4 � − 4 5 � 1 , 5 � + 1 ( − 13 ( − 27 5 � 4 , 1 � 10 ) η 1 = � 3 , 3 � 20 ) and � 3 , 4 � − 4 5 � 2 , 5 � + 16 25 � 1 , 6 � + 1 5 � 5 , 1 � − 4 ( − 31 25 � 4 , 2 � 20 ). The number on the right hand side is the weighted degree of each cohomology class.