key: cord-0620734-xfkuu54d
authors: Almir'on, Patricio; Schulze, Mathias
title: Limit spectral distribution for non-degenerate hypersurface singularities
date: 2020-12-11
journal: nan
DOI: nan
sha: 8c1c323252a33c3b629a3cd0f5b99c45ec6ea2e5
doc_id: 620734
cord_uid: xfkuu54d

We establish Kyoji Saito's continuous limit distribution for the spectrum of Newton non-degenerate hypersurface singularities. Investigating Saito's notion of dominant value in the case of irreducible plane curve singularities, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. We show that this bound is asymptotically sharp.

Let f : (C n+1 , 0) → (C, 0) be the germ of a holomorphic function with isolated critical point 0 and Milnor number µ. Its spectrum is a discrete invariant formed by µ rational spectral numbers (see [Kul98, II.8 .1]) α 1 , . . . , α µ ∈ Q ∩ (0, n + 1).

They are certain logarithms of the eigenvalues of the monodromy on the middle cohomology of the Milnor fibre which correspond to the equivariant Hodge numbers of Steenbrink's mixed Hodge structure. In the context of Poincaré polynomials it is convenient to consider the spectrum as a polynomial

K. Saito [Sai83a] was the first to study the asymptotic distribution of the spectrum. He considered the normalized spectrum of f ,

as the Fourier transform of the discrete probability density on the interval (0, n + 1), 1 µ where δ(s) is Dirac's delta function. In the case of Brieskorn-Pham singularities (see Example 4.1) he identified a continuous limit probability distribution N n+1 defined by N n+1 (s)ds :=

where ϕ is the indicator function of the unit interval [0, 1],

Under the Fourier transform F, N n+1 corresponds to the power

K. Saito [Sai83a, (2.5) i)] suggested to find singularities for which χ f converges to N n+1 . Our main result establishes his limit spectral distribution for Newton non-degenerate singularities.

Theorem 1.1. For a fixed Newton diagram Γ, consider the Newton diagrams ̟Γ obtained from Γ by scaling with the factor ̟. Then we have

where f ̟ is any Newton non-degenerate function germ of n + 1 variables with Newton diagram ̟Γ.

The proof is given in §4. Combined with the following remark, our result generates new cases where Saito's limit distribution is valid for a suitably chosen limit as in (1.2). The general choice of limit is unclear.

Remark 1.2. K. Saito [Sai83a, (3.7), (3.9), (3.2.1)] proved the following facts. (a) For quasihomogeneous f of degree 1 with respect to weights w 0 , . . . , w n , (1.2) holds, even with lim ̟→∞ replaced by lim w 0 ,...,wn→0 . (b) For irreducible plane curve singularities f with Puiseux pairs (n 1 , l 1 ), . . . , (n g , l g ),

(c) The join f + g of two functions in disjoint sets of variables satisfies

Therefore (1.2) is compatible with joins by (1.1).

K. Saito [Sai83a, (2.5) ii), (2.8) i)] further suggested to describe up to what extent the spectral distribution is bounded by N n+1 and introduced the notion of (weakly) dominating values. Consider the function 

from the case of surface singularities.

Question 1.3. Is 1 a dominating value for all n ≥ 2? In other words, for f in n + 1 variables, is the geometric genus bounded by p g < µ (n + 1)! ?

Kerner and Nemethi [KN17] give a positive answer for Newton nondegenerate singularities with Newton diagram ̟Γ for sufficiently large ̟.

As opposed to Question 1.3, Hertling's variance conjecture [Her01, Conj. 6.7] addresses the distribution of the spectrum in its full range.

Conjecture 1.4 (Hertling's Variance Conjecture). The variance of spectral numbers α 1 ≤ · · · ≤ α µ is bounded by

It was confirmed by Brélivet [Bré02; Bré04] for Newton non-degenerate singularities and for plane curves. We refer to Brélivet and Hertling [BH20] for refined investigations in this direction.

In §3, we investigate (the extremal) spectral numbers below 1 for their dominance in the case n = 1 of irreducible plane curve singularities C = f −1 (0). For a single Puiseux pair (p, q) we describe these spectral values in terms of the value semigroup S = p, q of C (see (3.5)). This can be used to visualize the graph of Φ f as a difference (see Figure 1 ).

One can write the smallest spectral number, the log canonical threshold, as 1 p + 1 q and the largest below 1 as 1 − 1 pq . For these extremal spectral numbers we prove the following Theorem 1.6. For any irreducible plane curve singularity C = f −1 (0) with value semigroup different from 2, 3 and 2, 5 , we have Φ f ( 1

In other words, the squared log canonical threshold is bounded by

In particular, Theorem 1.6 provides a quite surprising constraint on the first Puiseux pair of an irreducible plane curve singularity with a given Milnor number.

Acknowledgements. The first named author wants to thank the second named author for his kindness and facilities for hosting him at TU Kaiserslautern in a pleasant working atmosphere during his research stay in September-November 2020 despite the difficulties of travels and face to face work due to the COVID19 pandemic.

Suppose that f : (C n+1 , 0) → (C, 0) is Newton non-degenerate. This means that there are local coordinates z 0 , . . . , z n such that

is a Newton non-degenerate convenient power series (see [Kou76, 1.19 Def.] and [Kul98, II.8.5]). Let Γ denote the Newton diagram of f . We write σ ∈ Γ to indicate that σ is a face of Γ. For σ, τ ∈ Γ, write τ ≤ σ if τ is a face of σ. By g σ we denote the polynomial obtained from the power series g ∈ O by restricting the monomial support to the cone of σ.

There is a (decreasing) Newton filtration N defined by Γ on O. For σ ∈ Γ let A σ be the corresponding graded subring of A and denote by

The Brieskorn module (see [MP83; Sai88] )

also carries a Newton filtration which is induced by the inclusion

z 0 ···zn dz 0 ∧···∧dzn M. Saito [Sai88] and Varchenko-Khovanskiȋ [VK85] identified the Poincaré series of Ω f with the singularity spectrum of f defined by Steenbrink [Ste77] . 

where α 1 , . . . , α µ are the spectral numbers of f .

Based on results of Kouchnirenko [Kou76] (and Hochster [Hoc72] ) Steenbrink [Ste77, (5.7)] gave a formula for for Newton non-degenerate f decomposing p H f = Sp f with respect to faces of the Newton diagram: For a face σ ∈ Γ he first writes the Poincaré series of the subspace of A σ / F 0,σ , . . . , F n,σ corresponding to the interior of the cone Q ≥0 σ of σ as

Denote the minimal dimension of a coordinate space containing σ ∈ Γ by k(σ) := min{k ∈ Z | ∃i 1 , . . . , i k ∈ {0, . . . , n} : σ ⊂ Qe i 1 + · · · + Qe i k }.

Then Steenbrinks formula is given by Theorem 2.2 (Steenbrink). For Newton non-degenerate f in n + 1 variables, the Poincaré series of H f can be written as

In this section, we elaborate on the case n = 1 where f defines an irreducible plane curve singularity C = f −1 (0). We first consider the case of a single Puiseux pair and prove Proposition 1.5, then move on to the general case and prove Theorem 1.6.

Suppose first that C as a single Puiseux pair (p, q). Then f is Newton non-degenerate with Newton diagram Γ consisting of a single line segment [(p, 0), (q, 0)] and defines an irreducible plane curve singularity C = f −1 (0). The function f is semiquasihomogeneous of weighted degree 1 with respect to weights

on variables z 0 , z 1 and can be written explicitly as

The normalizationC ։ C is given by

The valuation ν :Õ C → N, ν(t) = 1, defines the value semigroup

Due to the finiteness of the normalization S has a finite set of gaps N \ S, which yields k + N ⊂ S for k ≫ 0. The minimal such k is the conductor of S and equals the Milnor number (see [BG80, Prop.1.2.1.1) 

The Gorenstein property of C is reflected by the symmetry between elements and gaps (see [Kun70] )

The normalized valuation ν/d induces the filtration O C defined by weights w = (w 0 , w 1 ) on z 0 , z 1 . By assumption, this is the Newton filtration N . Factoring (3.2) as

C{t}/ t d and using (2.1) yields a Newton filtered inclusion

This identifies the corresponding ranges of spectral numbers and of values in the semigroup by means of (3.5)

The smallest spectral number w 0 + w 1 corresponds to 0 ∈ S, and the gap µ − 1 of S defining the Gorenstein symmetry (3.4) corresponds to the nonspectral number 1. It follows that (1.3) can be written explicitly as

Under (3.4) the gap 1 ∈ N \ S is the mirror of µ − 2 ∈ S and corresponds to the largest spectral number 1 − w 0 w 1 < 1 by (3.5).

After these preparations we are ready to give the Proof of Proposition 1.5.

which tends to 0 for p → ∞. If p ≥ 4 and q ≥ 5, then (3.7) is positive since

If p = 3, then (3.7) becomes

which is positive if q ≥ 4. Finally, if p = 2, then (3.7) becomes

which is positive if q ≥ 6, but negative if q ∈ {3, 5}. (b) Using (3.6) and (3.1) we compute

which tends to 0 for p → ∞.

Consider now an irreducible plane curve singularity C = f −1 (0) with arbitrary number g of Puiseux pairs. To prepare the proof of Theorem 1.6, we review some standard integer invariants (see [Zar06,  Ch. II, §1-3]): Let β 0 < β 1 < · · · < β g denote the minimal generators of the value semigroup of C and set (3.8) e i := gcd(β 0 , β 1 , . . . , β i ), n i := e i−1 e i , q i := β i e i for i = 0, . . . , g. These greatest common divisors form a strictly decreasing sequence (3.9) β 0 = e 0 > e 1 > · · · > e g = 1.

Moreover, the minimal generators of the value semigroup satisfy inequalities (3.10) n i−1 β i−1 < β i for i = 1, . . . , g. The characteristic Puiseux exponents of C are defined recursively by

for i = 2, . . . , g. By (3.10) they form a strictly increasing sequence

The Milnor number of f can be written as (see [Zar06,  Ch.II, §3, (3.14)])

On the other hand, A'Campo showed that (see [ACa73, Thm. 3 

Proof of Theorem 1.6. The case where g = 1 is covered by Proposition 1.5.(a). Using (3.9) and (3.12), we find a lower bound

Suppose first that g ≥ 3. Using (3.13) and (3.15), we compute

It follows that

Suppose now that g = 2. By (3.14), (3.8), (3.9) and (3.10), e 1 µ 1 = (n 1 − 1)(β 1 − e 1 ) = n 1 β 1 − β 1 − e 0 + e 1 < n 1 β 1 ≤ β 2 − 1 = q 2 − 1 and hence µ − e 2 1 µ 1 = e 1 µ 1 + e 2 µ 2 − e 2 1 µ 1 (3.16) > e 1 (1 + e 2 (n 2 − 1) − e 1 )µ 1 = e 1 (1 + e 1 − e 2 − e 1 )µ 1 = e 1 (1 − e 2 )µ 1 = 0.

If (n 1 , q 1 ) / ∈ {(2, 3), (2, 5)}, then by (3.8), Proposition 1.5 and (3.16)

Otherwise, we have n 1 = 2 ≤ e 1 = n 2 and (3.10) yields q 2 > 2e 1 q 1 . Using (3.14) it follows that µ = (e 1 − 1)(q 2 − 1) + e 1 (q 1 − 1)

> (e 1 − 1)(2e 1 q 1 − 1) + e 1 (q 1 − 1) = 2e 1 q 1 (e 1 − 1) + e 1 (q 1 − 2) + 1 = 2q 1 e 2 1 − (q 1 + 2)e 1 + 1 and hence (2 + q 1 ) 2 µ − 8e 2 1 q 2 1 = 78e 2 1 − 125e 1 + 25 if q = 3, 290e 2 1 − 343e 1 + 49 if q = 5.

In both cases e 1 ≥ 2 implies

which tends to 0 for n g → ∞ since this entails e 1 → ∞ by (3.9) and µ → ∞ by (3.14).

In this section we return to the general setup of §2 and prove our main result Theorem 1.1. Our approach is to subdivide the Newton diagram and mimic an argument of K. Saito (see [Sai83a, (2. 2), (3.7)]). We begin with his motivating Example 4.1 (Brieskorn-Pham type singularities). Suppose first that n = 0 and f = f (z) = z d is quasihomogeneous of degree 1 with respect to the weight w = 1/d on z with Milnor number µ = d − 1. By (2.2), H = z ⊂ C{z}/ z d and hence

By Theorem 2.2, using L'Hôpital's rule in the second step,

.

j , which is quasihomogeneous of degree 1 with respect to weights w 0 = 1/d 0 , . . . , w n = 1/d n on the variables z 0 , . . . , z n with Milnor number µ = µ f = n j=0 µ f j . Then H = H f = H f 0 ⊗ C · · · ⊗ C H fn and hence, by the first part and (1.1), lim w 0 ,...,wn→0

In this sense the normalized spectrum converges in distribution to the continuous probability distribution N n+1 .

For our purpose we adapt the calculation (4.1) as follows.

Lemma 4.2. lim w→0 w 1−T 1−T w = F(ϕ)(t). Proof. Using L'Hôpital's rule in the second step and (4.1), we compute

For the subdivision of the Newton diagram we rely on the following general result. The basis of a rational pointed cone σ are the irreducible integral vectors α 0 , . . . , α k on its rays. If it extends to a lattice basis, then σ is called regular. In this case σ is a simplicial cone and the convex hull of {0, α 0 , . . . , α k } has k-dimensional volume 1 (see [ Theorem 4.4 (Kouchnirenko) . The Milnor number of any Newton nondegenerate f in n + 1 variables can be written as

where V k is the sum of k-dimensional volumes of the intersection of the convex hull of Γ ∪ {0} with the k-dimensional coordinate planes.

We are now ready for the Proof of Theorem 1.1. By Theorem 4.3, Γ has a subdivisionΓ corresponding to a regular subdivision of its fan of cones. For any τ ∈Γ let w τ 0 , . . . , w τ k be the weights of the basis of Q ≥0 τ ∩ Z. Then (1 − t w τ j ) .

Substituting into Steenbrink's formula from Theorem 2.2 yields

.

Passing to ̟Γ, w τ j is preplaced by εw τ j where ε̟ = 1 and hence

. The summand in (4.2) indexed by τ is then computed using (4.5), Lemma 4.2 and (4.3):

The claim now follows by substituting into (4.2) and applying (4.4).

Sur la monodromie des singularités isolées d'hypersurfaces complexes

The Milnor number and deformations of complex curve singularities

Bernoulli moments of spectral numbers and Hodge numbers

Variance of the spectral numbers and Newton polygons

The Hertling conjecture in dimension 2

Frobenius manifolds and variance of the spectral numbers

Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes

On the first terms of certain asymptotic expansions

Newton polyhedra (resolution of singularities)

Durfee-type bound for some non-degenerate complete intersection singularities

Polyèdres de Newton et nombres de Milnor

Mixed Hodge structures and singularities

The value-semigroup of a one-dimensional Gorenstein ring

On log canonical thresholds of reducible plane curves

Milnor number of complete intersections and Newton polygons

The zeroes of characteristic function χ f for the exponents of a hypersurface isolated singular point

On the exponents and the geometric genus of an isolated hypersurface singularity

Exponents and Newton polyhedra of isolated hypersurface singularities

Mixed Hodge structure on the vanishing cohomology

Asymptotic behavior of integrals over vanishing cycles and the Newton polyhedron

With an appendix by Bernard Teissier, Translated from the 1973 French original by Ben Lichtin

Departamento deÁlgebra, Geometría y Topología, Facultad de Ciencias Matemáticas

67663 Kaiserslautern, Germany Email address: mschulze@mathematik.uni-kl