Monodromy conjecture for semi-quasihomogeneous hypersurfaces

We give a proof the monodromy conjecture relating the poles of motivic zeta functions with roots of b-functions for isolated quasihomogeneous hypersurfaces, and more generally for semi-quasihomogeneous hypersurfaces. We also give a strange generalization allowing a twist by certain differential forms.


Introduction
The strong monodromy conjecture of Igusa and Denef-Loeser predicts that the order of a pole of the motivic zeta function Z mot f (s) of a nonconstant polynomial f ∈ C[x 0 , . . ., x n ] is less than or equal to its multiplicity as a root of the b-function b f (s) of f .The conjecture is open even in the case f has isolated singularities.In this note we prove it for a subclass of isolated hypersurface singularities: Theorem 1.1.The strong monodromy conjecture holds for semi-quasihomogeneous hypersurface singularities.
Recall that a germ of holomorphic function on a complex manifold is said to define a quasihomogeneous hypersurface singularity if it is analytically isomorphic to the germ at the origin of a weighted homogeneous polynomial.A hypersurface singularity is semi-quasihomogeneous if it is analytically isomorphic to the germ at the origin of a polynomial f = f d + f >d where f d is a weighted homogeneous polynomial of degree d with an isolated singularity at the origin, and f >d is a finite linear combination of monomials of weighted degree > d.We call such polynomials f semi-weighted homogeneous of initial degree d.
Theorem 1.1 follows from the next one, which we prove using the main result of [L+20] allowing computations of motivic zeta functions from partial embedded resolutions: Theorem 1.2.Let w 0 , . . ., w n ∈ Z n+1 >0 be a weight vector.Let f ∈ C[x 0 , . . ., x n ] be a semiweighted homogeneous polynomial of initial degree d with respect to these weights.Assume f d is irreducible (this is automatic if n > 1).Then the poles of Z mot f (s) are of order at most one and are contained in {−1, − w 0 +...+wn d } if w 0 + . . .+ w n = d, otherwise −1 is the only pole of Z mot f (s) and it has order at most two.A similar result holds if C is replaced by a field of characteristic zero, see Remark 2.3.1.The case n = 1 and f d is reducible is also easy to deal with but it depends on some classification, see Remark 2.2.1.
Both results, even for the isolated weighted homogeneous case, do not seem recorded in the literature.
The version of Theorem 1.2 for Igusa's local zeta functions is [Z01, Theorem 3.5].It is known that motivic zeta functions specialize to Igusa's p-adic local zeta functions.Thus Theorem 1.2 implies the characteristic-zero case of [Z01, Theorem 3.5], giving it a different proof.The version of Theorem 1.1 for Igusa local zeta functions of semi-weighted homogeneous polynomials with an additional non-degeneracy assumption is [Z01, Corollary 3.6].
Remark 1.1.A homogeneous polynomial with an isolated singularity does not have to be Newton nondegenerate, e.g.(x + y) 2 + xz + z 2 , from [Ko76,1.21].Thus the existing results on motivic zeta functions for nondegenerate polynomials do not suffice to prove any of the two theorems from above.
We also give a strange generalization of Theorem 1.1.Let g(x 0 , . . ., x n ) be another polynomial.One has the twisted motivic zeta function Z mot f,g (s) obtained by replacing the algebraic top-form dx with gdx, see (3).One also has the twisted b-function b f,g (s) obtained by replacing f s with gf s , see (5).We denote by (∂f ) be the Jacobian ideal of f in the ring O = C{x 0 , . . ., x n } of convergent power series, that is, the ideal generated by the first order partial derivatives of f .
Then the order of a pole of Z mot f,g (s) is less than or equal to its multiplicity as a root of b f,g (s).This provides the first cases for which a question of Mustat ¸ȃ [M10] has a positive answer.Theorem 1.3 follows from properties of the microlocal V -filtration together with the following (not strange) generalization of Theorem 1.2: Theorem 1.4.With the assumptions of Theorem 1.2, let g = x β with β ∈ N n+1 be a monomial satisfying (1).Then the poles of Z mot f,g (s) are of order at most one and are contained in {−1, −l(β)} if l(β) = 1, otherwise −1 is the only pole of Z mot f,g (s) and has order at most two.
Remark 1.2.We explain why Theorem 1.3 is a strange generalization of the strong monodromy conjecture.A twisted generalization of Theorem 1.1 relating the poles of Z mot f,g (s) to the roots of b f,g (s) is not true.For example, Theorem 1.3 is not true for arbitrary monomials: (i) Take f = y 2 − x 3 with weight vector w = (2, 3) for (x, y), and let g = y.Then −l(β) = − 8 6 is a pole of Z mot f,g (s), but it is not a root of b f,g (s) = (s+1)(s+ 11 6 )(s+ 13 6 ).Here g ∈ (∂f ) fails (2).(ii) Take f = y 3 − x 7 + x 5 y with weight vector w = (3, 7), and let g = x 6 .Then −l(β) = − 28 21 is a pole of Z mot f,g (s), but it is not a root of b f,g (s), since one can check that − 29 21 is the biggest root of b f,g (s) s+1 .Here g fails (2) in a more subtle way since x 4 y ∈ g + (∂f ) and l(x 4 y) = 29 21 > 28 21 .Remark 1.3.A (not strange) generalization of the strong monodromy conjecture was posed in [Bu15]: for any polynomials f and g, the poles of Z f,g (s) should be roots of the monic polynomial b(s) generating the specialization of the Bernstein-Sato ideal B (f,g) ⊂ C[s 1 , s 2 ] to (s 1 , s 2 ) = (s, 1).(In the example (i) from Remark 1.2, b(s) = k=6,8,10,11,13,14,16 (6s + k) so − 8 6 is a root.)In fact, it is more generally conjectured in [Bu15] that the polar locus of the multi-variable motivic zeta function Z mot F (s 1 , . . ., s r ) is contained in the zero locus in C r of the Bernstein-Sato ideal B F ⊂ C[s 1 , . . ., s r ] for any tuple of polynomials F = (f 1 , . . ., f r ).
Remark 1.4.Condition (2) on g = x β implies that l(β) is a spectral number of f , see 3.3 and [J+19, 1.6].More generally, we say that g ∈ C[x 0 , . . ., x n+1 ] achieves the spectral number α > 0 of a polynomial f with an isolated singularity if the class of gdx is nonzero in Gr α V Ω n+1 f , see 3.3.Then one can view the above results as partial confirmation of: ] be a semi-weighted homogeneous polynomial.Let α be a spectral number of f at the origin.Does there exist g ∈ C[x 0 , . . ., x n+1 ] achieving the spectral number α such that the only non-integral pole of Z mot f,g (s) is −α?Remark 1.5.
(i) The eigenvalue version of the question is true for all polynomials f with an isolated singularity: if λ is an eigenvalue of the monodromy of f at the origin, there exists g ∈ C[x 0 , . . ., x n+1 ] such that Z mot f,g (s) with only one non-integral pole −α such that e 2πiα = λ by [CV17].
(ii) The b-function version of the question is not true for all polynomials f with an isolated singularity: Here f is not semi-weighted homogeneous and 6 13 is not a spectral number of f .
(iii) The spectral version, namely Question 1.1, is not true for all polynomials f with an isolated singularity: take f = (y 2 − x 3 ) 2 − x 5 y, then 5 12 is a spectral number at the origin.Here 5 12 is also the log canonical threshold lct(f ).It is known, with the same proof as for g = 1, that the maximal pole of Z mot f,g (s) is the negative of where J (f α ) are the multiplier ideals of f , cf. [M10], [DM20].Thus the only g with Z mot f,g (s) having − 5 12 as a pole must satisfy that lct g (f ) = lct(f ).Therefore g(0) = 0 and hence Z mot f,g (s) has the same poles as Z mot f (s).Since f is an irreducible plane curve with 2 Puiseux pairs, one can compute that − 5 12 and − 11 26 are the only non-integral poles of Z mot f (s).
Remark 1.6.It is known that s+1 is the minimal polynomial of the action of s on H′′ f /s H′′ f , where H′′ f is the saturation of the Brieskorn lattice, by a result of Malgrange and Pham, see [Sa94].In light of the strong monodromy conjecture, a natural question is if the canonical splitting of the class of [dx] in H′′ f /s H′′ f is a linear combination of classes ω α ∈ H′′ f /s H′′ f , such that α are the non-trivial poles of Z mot f (s) and ω α are eigenvectors for s with eigenvalue α.While this is true in the isolated weighted homogeneous case, it is not true in general: example (iii) from Remark 1.5 is a counterexample.postdoctoral fellowship.N.B. was supported by the grants Methusalem METH/15/026 from KU Leuven and G097819N from FWO. R.v.d.V. was supported by an FWO PhD fellowship.

Motivic zeta functions
2.1.Motivic zeta functions.Consider two regular functions f, g : X → C on a smooth complex algebraic variety X, with f non-invertible.Let µ : Y → X be an embedded resolution of f g.Let E i with i ∈ J be the irreducible components of the pullback µ * (div(f )) of the divisor of f , where The smallest set Ω of rational numbers − ν N with multiplicities such that Z mot f,g (s) is a rational function in 1 − L −(N s+ν) with ν N ∈ Ω (with pole orders given by the multiplicities) over the ring be the w-weighted blowup of the origin.Let E be the exceptional divisor and H the strict transform of {f = 0}.We show first that µ is an embedded Q-resolution of f , see [L+20, §1.4].By definition this means that E ∪ H has Q-normal crossings, that is, it is locally analytically isomorphic to the quotient of a union of coordinate hyperplanes among those given by a local system of coordinates t 0 , . . ., t n with a diagonal action of a finite abelian subgroup of G of GL n+1 (C), i.e. locally The exceptional divisor E is isomorphic to the w-weighted projective n-space of the action of the w 0 -roots of unity λ defined by (x 1 , . . ., x n ) → (λ w 1 x 1 , . . ., λ wn x n ).
A similar description holds for the other charts.Write f = f d +f >d , where f d is the degree d term, so that the origin is an isolated singularity of f and f d .Since we assumed f d is irreducible, f is also since their singularity is isolated.
In the chart Y 0 , the exceptional divisor E is given by {x 0 = 0}.The pullback of f is given by f (x w 0 0 , x w 1 0 u 1 , . . ., x wn We check this with the Jacobian criterion. together with the fact that the origin is the only singular point of f d .Note that E ∩ H is given in Y 0 by the image under φ 0 of the zero locus of the ideal (x 0 , g).Since a similar description holds in the other charts, E ∩ H has abelian quotient singularities since φ 0 is a quotient map.
Next we note that where |w| = w 0 + . . .+ w n .A similar description holds in the other charts.
We have now all the information needed to apply the formula of [L+20] computing the motivic zeta function of f .Since µ is an embedded Q-resolution of f , Y is a disjoint union of strata S k characterised by the existence of data (G k , N k , ν k ) satisfying the following conditions.Locally around a generic point of S k , Y is analytically isomorphic to C n+1 /G k for some finite abelian group G k , acting diagonally on the coordinates t 0 , . . ., t n of C n+1 and small (i.e.not containing rotations around the hyperplanes other than the identity); f • µ is given by t ; and, the relative canonical divisor of µ is given by t where T k (s) has no poles.We have showed that candidate poles from the product term in this formula contributed by S k are the zeros of 1, ds + |w|, s + 1, (s + 1)(ds + |w|) respectively.This finishes the proof.
Remark 2.2.1.If n = 1 and f d is not irreducible, then one has a classification up to a change of holomorphic coordinates of all possible cases for f d in [K00, Lemmas 3.3 and 3.4].
2.3.Proof of Theorem 1.4.In the proof of Theorem 1.2 one has . By [L+20, Theorem 4], the zeta function Z mot f,x β (s) is as in (4) but with the relative canonical divisor replaced by the above form.Running the proof of Theorem 1.2, we note that everything works similarly.The assumption on x β implies that is smooth in the variables u and hence its tangent cone at {x 0 = u 1 = . . .= u n = 0} must be a linear combination of the u i with 1 ≤ i ≤ n.
Remark 2.3.1.All the results from the introduction admit a slight generalization by replacing in the definition (3) of the motivic zeta function the category of C-varieties with that of k-varieties.This holds since all the morphisms, including the group actions, in the above proofs concerning motivic zeta functions are defined over k.

Bernstein-Sato polynomials
3.1.b-functions.Let f, g : (X, 0) → (C, 0) be germs of holomorphic functions on a complex manifold with f (0) = 0. We set O = O X,0 and D = D X,0 , the ring of germs of analytic functions, respectively analytic linear differential operators.We denote by b f,g (s) the nonzero monic generator b f,g (s) of the ideal of polynomials b(s) ∈ C[s] of minimal degree satisfying If f, g : X → C are regular functions on a smooth complex affine variety, one can apply the same definitions with D replaced by the ring of global algebraic linear differential operators, and the resulting b-function is the lowest common multiple (well-defined due to finiteness of b-constant strata) of the local b-functions defined above for germs at points along f −1 (0).

Microlocal b-functions.
Let f : (X, 0) → (C, 0) be the germ of a holomorphic function on a complex manifold with f (0 and similarly for R.
Let M be a D-module.Define the stalk of the D-module direct image i + M at (0, 0) in X × C. Denoting m ⊗ ∂ i t by m∂ i t δ for a local section m of M, the left R-module structure on M f is defined by t δ for ξ a local vector field on (X, 0).Equivalently, δ the delta function of t − f .Define the algebraic microlocalization Then Mf is a left R-module with the action defined as above.
Assume from now that M is holonomic.Let u be a local section of M f (resp.Mf ).The b-function b u (s) (resp.microlocal b-function bu (s)) is the minimal polynomial of the action of s . This is a well-defined polynomial by [Ka78], [KK80], [L87] (with bu (s) defined in terms of the usual microlocalization , but this definition can be shown to be equivalent to the one above, see [Sa94, 1.4]).If M has quasi-unipotent local monodromy on subsets forming a suitable Whitney stratification, this polynomial has rational roots.
It is known that δ can be identified with f s and b mδ (s) is the monic generator of the ideal of polynomials b(s (iii) In particular, if g ∈ O = M and g/f is not holomorphic, then b f,g (s) is divisible by s + 1 and the reduced b-function bf,g (s) is the microlocal b-function bgδ (s).

Proof. (i) Setting
(ii) The proof of [Sa94, Lemma 1.6] gives without using any of the two assumptions on m that there exists P ∈ ∂ −1 t V 0 R such that bmδ (s)mδ = P mδ.By multiplying by s + 1 = −t∂ t one obtains that (s + 1) bmδ (s) is divisible by b mδ (s).
Conversely, we show bmδ (s) divides by b mδ (s)/(s + 1).One has Q(s)f = tQ ′ (s) for some Since M is holonomic, there exists m ∈ M with Dm = M locally.The filtration on M f (resp.Mf ) defined by Mf ) indexed discretely by α ∈ C with a fixed total order on C (if M has quasi-unipotent local monodromy one can take α ∈ Q), using the decomposition of the action by s on quotients V p /V q with p < q, see [Sa93, §1], [Sa94,§2].The existence of the V -filtration is equivalent to the existence of b-functions (resp.microlocal b-functions).The V -filtration (resp.microlocal V -filtration) is uniquely characterized by: (i) One has has all roots ≤ −α} by [S87], [Sa93,Cor. 1.7].The same proof, relying on the unique characterization from above, and using that 3.3.Brieskorn lattices.Let f : (X, 0) → (C, 0) be the germ of a holomorphic function on a complex manifold of dimension n + 1 with f (0) = 0 and f having an isolated singularity at 0. There is a diagram where the lower map is an isomorphism of C-vector spaces, the vertical map is surjective, and the top map is an inclusion.Here O = O X,0 , (∂f ) is the ideal generated by the first order partial derivatives of f , Ω p consists of the germs of the holomorphic p-forms at the origin, O/(∂f ) is called the Milnor algebra, H ′′ f is called the Brieskorn lattice, and G f is called the Gauss-Manin system.The Brieskorn lattice is a free module of rank equal to the Milnor number µ f = dim C O/(∂f ) over C{t} and also over C{{∂ −1 t }}, where the action of t is given by multiplication by f and the action of -module of rank µ f with an action of t, and it is a regular holonomic D-module.Consequently it admits the rational V -filtration such that ∂ t t − α is nilpotent on Gr α V G f .The induces a V -filtration on H ′′ f and on the quotient Ω n+1 f .See [Br70,Seb70,Sa89].On the other hand, the microlocal V -filtration Ṽ α O defined above induces a filtration on the quotient.These two filtrations are the same:  Then −α is the biggest root of bgδ (s).
Proof.The assumption is equivalent in general with [gdx] = 0 in Gr α V Ω n+1 f , by Proposition 3.4.1.It follows from Proposition 3.3.1 that gδ is non-zero in Gr α V Mf for M = O in the notation of 3.2.This implies that the maximal root of bgδ (s) is −α by Proposition 3.2.1.
3.5.Proof of Theorem 1.1.Let h ∈ C[x 0 , . . ., x n ] define a semi-quasihomogeneous hypersurface singularity locally analytically isomorphic to f as in Theorem 1.2.The proof of Theorem 1.2 only uses local analytic properties of f , hence it also holds for h.Now, the local b-function of h is an analytic invariant and thus equals that of f .By Corollary 3.4.1 for the monomial g = 1, one has that (s + 1)(s + |w|/d) divides the local b-function of f .This finishes the proof for the case n > 1.If n = 1 one can analyze directly the classification of f d following Remark 2.2.1 (this is also a particular case of result [Lo88] for general plane curves.)3.6.Proof of Theorem 1.3.It follows from Theorem 1.4 together with Corollary 3.4.1 and Lemma 3.2.1 (iii).The equivalency of the assumption in the case f = f d follows by considering the decomposition into weighted homogeneous terms, since (∂f ) is generated by weighted homogeneous polynomials in this case.x0 , x1 ∈ C{x 0 , x 1 } such that f = x0 x1 .The claim then follows from b x 0 x 1 (s) = (s + 1) 2 .This finishes the proof of Theorem 1.3.
Note that Theorem 1.3 implies Theorem 1.1.The mistake pointed above does not really affect the proof of Theorem 1.1 in 3.5: one knows (without invoking the problematic Corollary 3.4.1 or the new Corollary 3.6.1)that (s + 1)(s + |w|/d) divides the local b-function of f by the constancy of the minimal exponent in µ-constant deformations and by the computation of b-functions of weighted homogeneous isolated hypersurfaces.
gf s = P gf s+1 for some P ∈ D[s].It is a non-trivial well-known result that b f,g (s) is well-defined.If g/f is not holomorphic, then b f,g (s) is divisible by s + 1 by Lemma 3.2.1.Define in this case the reduced b-function bf,g (s) = b f,g (s) s + 1 .When g = 1, b f,g (s) (resp.bf,g (s)) is denoted b f (s) (resp.bf (s)), the usual b-function (resp.reduced b-function) or Bernstein-Sato polynomial of f .

If g = 1 ,
then by [Sa94, Prop.0.3] the microlocal b-function is the reduced b-function: bf (s) = bfδ (s).The proof can be easily adapted to yield a more general result: Lemma 3.2.1.Let M be a holonomic D-module and m ∈ M a local section.(i) If f −1 m ∈ Dm then b mδ (s) is divisible by s + 1. (ii) If in addition f is injective on Dm, then b mδ (s) = (s + 1) bmδ (s).
Thus the strict transform H is irreducible.The intersection E ∩H is the hypersurface defined by f d in P n That proof also implies our claim about µ being an embedded Q-resolution, as we show next.By definition Y ⊂ C n+1 × P n w and µ is the restriction of the projection onto the first factor.The chart Y 0 [St77,St77,  §4], the intersection E ∩ H has, like E ≃ P n w , only abelian quotient singularities.
using (i).It is enough to show that t is injective on the algebraic microlocalization(Dm)[∂ t , ∂ −1 t ], since this implies that b mδ (s) s + 1 − ∂ −1 t Q ′ mδ = 0by the invertibility of ∂ t .Since (Dm)[∂ t , ∂ −1 t ] is exhausted by the filtration F p = ⊕ i≤p Dm∂ −i t δ, it is enough to show that t is injective on Gr F p for all p ∈ Z.This is equivalent to f being injective on Dm.