MONODROMY OF HYPERSURFACE SINGULARITIES

MATHIAS SCHULZE

Abstract. We describe algorithmic methods for the Gauss-Manin connection of an isolated hypersurface singularity based on the mi- crolocal structure of the Brieskorn lattice. They lead to algorithms for computing invariants like the monodromy, the spectrum, the spectral pairs, and M. Saito’s matrices A0 and A1. These algo- rithms use a normal form algorithm for the Brieskorn lattice, stan- dard basis methods for power series rings, and univariate factoriza- tion. We give a detailed description of the algorithm to compute the monodromy.

Contents 1. Introduction 1 2. Monodromy and Gauss-Manin Connection 3 3. V-Filtration and Brieskorn Lattice 4 4. Saturation and Resonance 6 5. Microlocal Structure and Algorithms 7 6. Examples 9 References 10

1. Introduction We consider a germ of a holomorphic map f :(Cn+1, 0) / (C, 0) with isolated critical point and Milnor number µ. J. Milnor [Mil68] first studied this situation by differential geometry. The regular fibres of a good representative over a punctured disc form a C ∞ fibre bundle with fibres of homotopy type of a bouquet of µ n-spheres. The coho- mology of the fibres form a flat vector bundle and there is an associated flat connection on the corresponding sheaf of holomorphic sections, the Gauss-Manin connection. Moreover, there is a monodromy represen- tation of the fundamental group of the base in the cohomology of the general fibre. A counterclockwise generator acts via the monodromy which is an automorphism defined over the integers. Using the De Rham isomorphism, the cohomology of the fibres can be described in terms of holomorphic differential forms. E. Brieskorn [Bri70] first gave an algorithm to compute the com- plex monodromy based on this idea. The original Brieskorn algorithm 1 2 MATHIAS SCHULZE

Figure 1. The Milnor fibration

M

0 X X t

0 t T

was first implemented by P.F.M. Nacken [Nac90] in the computer al- gebra system Maple V. A later implementation [Sch99, Sch01c] by the author in the computer algebra system Singular [GPS01] using standard basis methods turned out to be more powerful. In section 2 and 3, we briefly introduce the monodromy, the Gauss- Manin connection, the V-filtration, and the Brieskorn lattice, and sum- marize properties and results which are relevant for the computation of the monodromy. This will lead us to the notions of saturation and resonnance of lattices in section 4. In section 5, we describe algorith- mic methods for the Gauss-Manin connection based on the microlocal structure of the Brieskorn lattice and the Fourier-Laplace transform [Sch00, SS01]. We use standard basis methods, univariate factoriza- tion, and a normal form algorithm for the microlocal structure of the Brieskorn lattice, the latter of which is not published yet. These meth- ods lead to algorithms [Sch02, Sch01a, Sch01b] to compute Hodge- theoretic invariants like the spectrum, the spectral pairs, and M. Saito’s matrices A0 and A1 [Sai89]. We describe an algorithm to compute the complex monodromy based on the above ideas. This is also implemeted in Singular [Sch01b] and is much faster then the original Brieskorn algorithm [Sch01c]. In section 6, we explain how to use the Singular implementation by computing an example with a Jordan block of size 3 × 3 of the monodromy. MONODROMY OF HYPERSURFACE SINGULARITIES 3

2. Monodromy and Gauss-Manin Connection Let f :(Cn+1, 0) / (C, 0) be an isolated hypersurface singu- n+1 larity. We choose local coordinates x = (x0, . . . , xn) at 0 ∈ C and C t at 0 ∈ and set ∂ := (∂0, . . . , ∂n) where ∂j := ∂xj . We denote by

µ := dimC C{x}/h∂(f)i < ∞ the Milnor number of f. We choose an (n + 1)-ball B centered at 0 ∈ Cn+1 and a disc T centered at 0 ∈ C such that f B ∩ f −1(0) =: X / T

 i is a Milnor representative [Mil68]. Let T 0 := T \{0} / T be the inclusion of the punctured disc. Then the restriction

f X\f −1(0) =: X0 / T 0

∞ −1 is a C fibre bundle, the Milnor fibration. The fibres Xt := f (T ) have homotopy type of a bouquet of µ n-sheres and hence the coho- mology of the general fibre is given by (Zµ k ∼ , k = n, He (Xt, Z) = 0, else. The n-th cohomology bundle n n [ n H := R f∗CX0 = H (Xt, C) t∈T 0 is a locally constant sheaf. The Gauss-Manin connection is the associated flat connection

n ∇ 1 n / 0 H ΩT ⊗OT 0 H n n 0 C on the sheaf of holomorphic sections H := OT ⊗ T 0 H which is defined by n ∇(g ⊗ v) := dg ⊗ v, g ∈ OT 0 , v ∈ H . n / n We denote by ∂t := ∇∂t : H H its covariant derivative with 0 respect to ∂t. Lifting paths in T along flat sections defines an action

0 n
π1(T , t) / Aut H (Xt, Z)

0 of the fundamental group π1(T , t) on the n-th cohomology of the gen- 0 eral fibre. A counterclockwise generator of π1(T , t) acts by the mon- odromy n
M ∈ Aut H (Xt, Z) . Theorem 2.1 (Monodromy Theorem). The eigenvalues of the mon- odromy are roots of unity and its Jordan blocks are of size at most (n + 1) × (n + 1). 4 MATHIAS SCHULZE

Let u : T ∞ / T , u(τ) := exp(2πiτ), be the universal covering of T 0 where τ is a coordinate on T ∞. The canonical Milnor fibre is defined to be the pullback ∞ 0 ∞ X := X ×T 0 T ∼ ∞  ∞ to the universal covering. The natural maps Xu(τ) = Xτ / X are homotopy equivalences. We consider A ∈ Hn(X∞, C) as a global flat multivalued section A(t) in H n. Note that

∂tA(t) = 0, M(A)(τ) = A(τ + 1).

3. V-Filtration and Brieskorn Lattice

Let M = MsMu be the decompostion of the monodromy into semisim- ple and unipotent part and set

N := log Mu. Note that N n+1 = 0 by the monodromy theorem. Let n ∞ M n ∞ n ∞ H (X , C) = H (X , C)λ, H (X , C)λ := ker(Ms − λ), λ be the decomposition into generalized eigenspaces of M and

n ∞ Mλ := M|H (X ,C)λ . n ∞ For A ∈ H (X , C)λ, λ = exp(−2πiα), α ∈ Q, we set  N  ψ (A)(t) := tα exp − log t A(t). α 2πi n Then ψα(A) is monodromy invariant and hence a global section in H .

Definition 3.1. We call the C{t}[∂t]-module n ∞ n G := (i∗ψα(A))0 α ∈ Q,A ∈ H (X , C)exp(−2πiα) ⊂ (i∗H )0 OT,0 the Gauss-Manin connection of f. For all α ∈ Z, the map n ∞  ψα :H (X , C)λ / G α is an inclusion with image C := im ψα. The following lemma shows the correspondence between the monodromy action on Hn(X∞, C) and the C{t}[∂t]-module structure on G via the maps ψα. Lemma 3.2.

(1) t ◦ ψα = ψα+1 N
(2) ∂t ◦ ψα = ψα−1 ◦ α − 2πi N
(3) (t∂t − α) ◦ ψα = ψα ◦ − 2πi (4) exp(−2πit∂t) ◦ ψα = ψα ◦ Mλ. α n+1 (5) C = ker(t∂t − α) MONODROMY OF HYPERSURFACE SINGULARITIES 5

(6) t : Cα / Cα+1 is bijective. α α−1 (7) ∂t : C / C is bijective for α 6= 0. α The generalized eigenspaces C of the operator t∂t define a filtration on G . Definition 3.3. The V-filtration V on G is the decreasing filtration by C{t}-modules X X V α := C{t}Cβ,V >α := C{t}Cβ. α≤β α<β Note that V α/V >α ∼= Cα. There is not only the C{t}-structure on −1 α G . For α > −1, the action of ∂t on V extends to a structure over −1 a power series ring C{{∂t }}. This structure is the key to powerful algorithms. Definition 3.4. The ring of microdifferential operators with con- stant coefficients is defined by   −1 X −k −1 X ak k C{{∂t }} := ak∂ ∈ C[[∂ ]] t ∈ C{t} . k! k≥0 k≥0

−1 α Note that C{{∂t }} is a discrete valuation ring and t C{t}, α ∈ Q, −1 is a free C{{∂t }}-module of rank 1. Together with lemma 3.2, this implies the following proposition. Proposition 3.5. (1) For all α ∈ Q, V α is a free C{t}-module of rank µ. (2) G is a µ-dimensional C{t}[t−1]-vector space. α −1 (3) For α > −1, V is a free C{{∂t }}-module of rank µ.

There is a lattice in G on which the action of ∂t can be computed. Definition 3.6. 00 n+1 n−1 H := ΩX,0 /df ∧ dΩX,0 . is called the Brieskorn lattice. −1 The Brieskorn lattice is embedded in G and is a C{t}- and C{{∂t }}- lattice. There is an explicit formula for the action of ∂t in terms of differential forms. We summarize these well known properties of the Brieskorn lattice in the following theorem. Theorem 3.7. [Seb70, Bri70, Pha77, Mal74] (1) H 00 is a free C{t}-module of rank µ. 00  /
R ω (2) s : H G , s [ω] (t) := df X

t (3) ∂ts [df ∧ η] = s [dη] 00 −1 (4) H is a free C{{∂t }}-module of rank µ. (5) V −1 ⊃ H 00 ⊃ V n−1 6 MATHIAS SCHULZE

00 The action of ∂t may have a pole of order up to n + 1 on H . Since the monodromy is related to the action of t∂t, we consider t∂t-invariant lattices in the next section.

4. Saturation and Resonance Since C{t} is a discrete valuation ring, for any two C{t}-lattices L , L 0 ⊂ G , there is a k ∈ Z such that tkL ⊂ L 0. Hence, for any C{t}-lattice L , V α2 ⊂ L ⊂ V >α1 for some α1, α2 ∈ Q.

Definition 4.1. Let L ⊂ G be a C{t}-lattice. If t∂tL ⊂ L then L is called saturated and the induced endomorphism t∂t ∈ EndC(L /tL ) is called the residue of L . If t∂t has non-zero integer differences of eigenvalues then L is called resonant. Let L ⊂ G be a C{t}-lattice with V α2 ⊂ L ⊂ V >α1 for some α1 α1, α2 ∈ Q. By the Leibnitz rule and since V is saturated,

L0 := L , Lk+1 := Lk + t∂tLk, defines an increasing sequence of C{t}-lattices

α2 α1 V ⊂ L0 ⊂ L1 ⊂ · · · ⊂ V . Since V α1 is noetherian, this sequence is stationary and [ L∞ := Lk. k≥0 is a saturated C{t}-lattice.

Definition 4.2. L∞ is called the saturation of L . The following proposition is not difficult to prove using lemma 3.2.

Proposition 4.3. [GL73] Lµ−1 = L∞ If L is saturated then  M  L = L ∩ Cα ⊕ V α2 .

α1<α<α2 Together with lemma 3.2, this implies the following proposition. Proposition 4.4. Let L ⊂ G be a saturated C{t}-lattice with residue t∂t ∈ EndC(L /tL ).

(1) The eigenvalues of exp(−2πit∂t) are the eigenvalues of the mon- odromy. (2) If L is non-resonant then exp(−2πit∂t) is conjugate to the monodromy. MONODROMY OF HYPERSURFACE SINGULARITIES 7

5. Microlocal Structure and Algorithms • • We abbreviate Ω := ΩX,0. The microlocal structure extends to a C{{s}}[∂s]-module structure by the Fourier-Laplace transform −1 2 −2 s := ∂t , ∂s := ∂t t = s t. Since 2 −1 2 −1 [∂s, s] = [∂t t, ∂t ] = ∂t t∂t − ∂tt = 1, α 00 2 2 V for α > −1 and H are C{{s}}[s ∂s]-modules. Note that t = s ∂s is a C{{s}}-derivation. Since −1 ∂tt = s t = s∂s, the saturation L∞ of a C{t}- and C{{s}}-lattice L is a saturated C{t}- and C{{s}}-lattice. Note that this holds for H 00. By the finite determinacy theorem, we may assume that f ∈ C[x] is a polynomial. A C-basis of H 00/sH 00 = Ωn+1/df ∧ Ωn ∼= C{x}/h∂(f)i represents a C{{s}}-basis of H 00. If g is a standard basis of h∂(f)i with respect to a local monomial ordering then the monomials which are not contained in the leading ideal hlead gi = hlead(∂(f))i form a monomial C-basis of C{x}/h∂fi. Hence, one can compute a monomial m1 . C{{s}}-basis m =  .  of H 00. mµ P k We define the m-matrix A = A(s) = k≥0 Aks of t by tm =: Am. Then the m-basis representation of t on H 00 is given by 2
tgm = gA + s ∂s(g) m. If U is a C{{s}}[s−1]-basis transformation and A0 the Um-matrix of t then 0 2
−1 A = UA + s ∂s(U) U is the basis transformation formula for U. Let Ωb resp. Hc00 be the hxi-adic resp. hsi-adic completion of Ω resp. 00 C / n+1 H . The isomorphism dx : [[x]] ∼ Ωb induces an isomorphism  n X
∼ 00 C[[x, s]] ∂j(f) − s∂j C[[x, s]] =C[[s]] Hc j=0 and Hc00 is a differential deformation of the Jacobian algebra C[[x]]/h∂(f)i in the sense of [Sch01d]. Using the normal form algorithm in PK k [Sch01d], one can compute any K-jet A≤K = A≤K (s) = k=0 Aks of A. 8 MATHIAS SCHULZE

00 00 The m-basis representation H∞ of the saturation H∞ can be com- puted recursively by 00 C µ H0 := {{s}} , 00 00 −1 00 00
Hk+1 := Hk + s Hk A≤k + s∂sHk . Note that only finite jets of A are involved. We use a local monomial 00 degree ordering. By computing a normal form of Hk+1 with respect to 00 Hk , one can check when the sequence 00 00 00 H0 ⊂ H1 ⊂ H2 ⊂ · · · 00 becomes stationary and compute generators of the saturation H∞ of 00 H0 . m1 . By Nakayama’s lemma, a minimal standard basis M =  .  of mµ 00 C 0 0 P 0 k H∞ is a {{s}}-basis. The Mm-matrix A = A (s) = k≥0 Aks of t is defined by 2 0 MA + s ∂sM =: A M. We set  i
k
i k δ(M) := max ord mj − ord ml mj 6= 0 6= ml ≤ µ − 1 such that 2
0 MA≤K+δ(M) + s ∂sM ≤K =: A≤K M. for any K ≥ 0. Note that only finite jets of A are involved. Hence, one 0 0 PK 0 k 0 can compute any K-jet A≤K = A≤K (s) = k=0 Aks of A . Note that 00 the Mm-basis representation of t on H∞ is given by 0 2
tgMm = gA + s ∂s(g) Mm.

−1 00 K 00 Hence, the Mm-basis representation of ∂tt = s t on H∞/s H∞ is given by −1 0
∂ttgMm = s gA≤K + s∂s(g) Mm −1 0 0 and s A≤1 = A1 is the Mm-basis representation of the residue of 00 0 H∞. Note that the eigenvalues of A1 are rational by the monodromy 0 theorem and can be computed using univariate factorisation. If A1 is 0 non-resonant then exp(−2πiA1) is a monodromy matrix. Otherwise, we proceed as follows. Let δ(A0) > 0 be the maximal 0 0 integer difference of A1. First we compute A1+δ(A0) from A1+δ(M)+δ(A0) as before. After a C-linear coordinate transformation, we may assume that A01,1 A01,2 A0 = A02,1 A02,2 MONODROMY OF HYPERSURFACE SINGULARITIES 9

01,2 02,1 0 01,1 with A 1 = 0, A 1 = 0, A0 = 0, the eigenvalues of A are minimal in their class modulo Z, and the eigenvalues of A02,2 are non-minimal in their class modulo Z. Then the C{{s}}[s−1]-coordinate transformation

s 0 U := 0 1 gives

A001,1 A001,2 A01,1 + s sA01,2 A00 = := UA0 + s2∂ (U)
U −1 = . A002,1 A002,2 s s−1A02,1 A02,2

00 00 0 00 Note that A0 = 0, δ(A ) ≤ δ(A ) − 1, and that AK depends only on 0 0 0 A≤K+1. With δ(A ) of these transformations we decrease δ(A ) to zero 0 such that exp(−2πiA1) is a monodromy matrix as before. The above methods lead to algorithms [Sch02, Sch01a, Sch01b] to compute Hodge-theoretic invariants like the spectrum, the spectral pairs, and M. Saito’s matrices A0 and A1 [Sai89].

6. Examples The algorithms described above is implemented in the computer alge- bra system Singular [GPS01] in the library gaussman.lib [Sch01a]. We use this implementation to compute an example. First, we have to load the library: > LIB "gaussman.lib";

Then we define the ring R := Q[x, y, z](x,y,z) and the polynomial f = x2y2z2 + x7 + y7 + z7 ∈ R: > ring R=0,(x,y,z),ds; > poly f=x2y2z2+x7+y7+z7; Finally, we compute the Jordan data of the monodromy of the singu- larity defined by f at the origin. > spprint(monodromy(f)); ((1/2,1),18),((1/2,3),1),((9/14,1),15),((9/14,2),3), ((11/14,1),15),((11/14,2),3),((6/7,1),3),((13/14,1),15), ((13/14,2),3),((1,2),1),((15/14,1),15),((15/14,2),3), ((8/7,1),3),((17/14,1),15),((17/14,2),3),((9/7,1),3), ((19/14,1),15),((19/14,2),3),((10/7,1),3),((11/7,1),3), ((12/7,1),3) The computation takes about 2 minutes in a Pentium III 800.A Jordan block of the monodromy of size s with eigenvalue exp(−2πiα) occuring with multiplicity m is denoted by ((α, s), m). Note that there is a Jordan block of size 3 with eigenvalue −1 which is the maximum possible size. 10 MATHIAS SCHULZE

References [AGZV88] V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko, Singularities of differentiable maps, vol. II, Birkh¨auser, 1988. [Bri70] E. Brieskorn, Die Monodromie der isolierten Singularit¨atenvon Hy- perfl¨achen, Manuscr. Math. 2 (1970), 103–161. [GL73] R. G´erardand A.H.M. Levelt, Invariants m´esurant l’irr´egularit´een un point singulier des syst`emesd’´equationsdiff´erentielles lin´eaires, Ann. Inst. Fourier, Grenoble 23 (1973), no. 1, 157–195. [GPS01] G.-M. Greuel, G. Pfister, and H. Sch¨onemann, Singular 2.0, A Computer Algebra System for Polynomial Computations, Cen- tre for Computer Algebra, University of Kaiserslautern, 2001, http://www.singular.uni-kl.de. [Mal74] B. Malgrange, Int´egrales asymptotiques et monodromie, Ann. scient. Ec. Norm. Sup. 7 (1974), 405–430. [Mil68] J. Milnor, Singular points on complex hypersurfaces, Ann. Math. Stud., vol. 61, Princeton University Press, 1968. [Nac90] P.F.M. Nacken, A computer program for the computation of the mon- odromy of an isolated singularity, predoctoral thesis, Department of Mathematics, Catholic University, Toernooiveld, 6525 ED Nijmegen, The Netherlands, 1990. [Pha77] F. Pham, Caustiques, phase stationnaire et microfonctions, Acta. Math. Vietn. 2 (1977), 35–101. [Pha79] , Singularit´esdes syst`emesde Gauss-Manin, Progr. in Math., vol. 2, Birkh¨auser, 1979. [Sai89] M. Saito, On the structure of Brieskorn lattices, Ann. Inst. Fourier Grenoble 39 (1989), 27–72. [Sch96] H. Sch¨onemann, Algorithms in Singular, Reports on Computeralge- bra 2, Centre for Computer Algebra, University of Kaiserslautern, 1996. [Sch99] M. Schulze, Computation of the monodromy of an isolated hyper- surface singularity, Diplomarbeit, Universit¨at Kaiserslautern, 1999, http://www.mathematik.uni-kl.de/ mschulze. [Sch00] , Algorithms to compute the singularity spec- trum, Master Class thesis, Universiteit Utrecht, 2000, http://www.mathematik.uni-kl.de/ mschulze. [Sch01a] , Algorithms for the Gauss-Manin connection, to appear in Jour- nal of Symbolic Computation, 2001. [Sch01b] , gaussman.lib, Singular 2.0 library, Centre for Computer Algebra, University of Kaiserslautern, 2001, http://www.singular.uni-kl.de. [Sch01c] , mondromy.lib, Singular 2.0 library, Centre for Computer Algebra, University of Kaiserslautern, 2001, http://www.singular.uni-kl.de. [Sch01d] , A normal form algorithm for differential deformations, math.CV/0108144, 2001. [Sch02] , Algorithmic Gauss-Manin connection, Ph.D. thesis, University of Kaiserslautern, 2002. [Seb70] M. Sebastiani, Preuve d’une conjecture de Brieskorn, Manuscr. Math. 2 (1970), 301–308. [SS85] J. Scherk and J.H.M. Steenbrink, On the mixed Hodge structure on the cohomology of the Milnor fibre, Math. Ann. 271 (1985), 641–655. [SS01] M. Schulze and J.H.M. Steenbrink, Computing Hodge-theoretic in- variants of singularities, New Developments in Singularity Theory MONODROMY OF HYPERSURFACE SINGULARITIES 11

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Mathias Schulze, Department of Mathematics, D-67653 Kaiserslautern E-mail address: [email protected]