INNER RATES OF FINITE MORPHISMS YENNI CHERIK Abstract. Let X0 be a complex analytic surface germ embedded in Cn0 with an isolated singularity

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INNER RATES OF FINITE MORPHISMS

YENNI CHERIK

Abstract. Let (X, 0) be a complex analytic surface germ embedded in (Cn,0) with an isolated singularity

and Φ = (g, f):(X, 0) −→ (C2,0) be a ﬁnite morphism. We deﬁne a family of analytic invariants of the

morphism Φ, called inner rates of Φ. By means of the inner rates we study the polar curve associated to the

morphism Φ when ﬁxing the topological data of the curve (gf )−1(0) and the surface germ (X, 0), allowing

to address a problem called polar exploration. We also use the inner rates to study the geometry of the

Milnor ﬁbers of a non constant holomorphic function f: (X, 0) −→ (C,0). The main result is a formula

which involves the inner rates and the polar curve alongside topological invariants of the surface germ (X, 0)

and the curve (gf )−1(0).

Introduction

Let (X, 0) be a complex analytic surface germ with an isolated singularity and Φ = (g, f) : (X, 0) −→

(C2,0) be a ﬁnite morphism. The polar curve of the morphism Φ is the curve ΠΦdeﬁned as the topological

closure of the ramiﬁcation locus of Φ. Polar curves play an important role in the study of the geometry and

the topology of germs of singular complex varieties, see, e.g.[Tei82,GBT99a,HP03,BNP14,BdSFP22a].

In this paper we introduce and study a family of analytic invariants of the morphism Φ that we call inner

rates of Φ, generalizing the notion of inner rates of a complex analytic surface germ (X, 0) ﬁrst introduced by

Birbrair, Neumann and Pichon in [BNP14] as metric invariants, and later deﬁned in [BdSFP22a] by Belotto,

Fantini and Pichon. Our main result (Theorem A) establishes a formula, the inner-rates formula, which

relates the inner rates with analytical data of the polar curve ΠΦand, in particular, provides a concrete

way to compute the inner rates. It has an equivalent version in terms of the laplacian of a certain graph

(Corollary 5.7) which is a broad generalization of the Laplacian formula [BdSFP22a, Theorem 4.3]. An

important motivation of our result concerns the study of Milnor ﬁbers. Consider a non-constant holomorphic

function f: (X, 0) →(C,0) and a generic linear form (Deﬁnition 10.7)ℓ: (X, 0) →(C,0). We provide an

interpretation of the inner rates of the morphism Φ = (ℓ, f) : (X, 0) →(C2,0) in terms of metric properties

of the Milnor ﬁber, Theorem D.

As an application of our methods, we study the problem of polar exploration associated to Φ, following

[BdSFP22a,BdSFNP22]. Roughly speaking, it is the study of the relative position on (X, 0) of the two curves

ΠΦand (gf)−1(0) i.e., of the relative positions of their strict transform by a resolution of (X, 0). More precisely

it is the problem of determining the embedded topological type (Deﬁnition 9.1) of the union ΠΦ∪(gf)−1(0)

from that of the curve (gf)−1(0). This is a natural problem which has been studied by many authors such

as Merle, Garc´ıa Barroso, Delgado, Maugendre, Kuo, Parusi´nski, Michel, Belotto, Fantini, N´emethi, Pichon,

among others (see, e.g.,[Mer77,KP04,GB00,DM03,Mic08,MM20,DM21,BdSFNP22,BdSFP22b]). An

important contribution in the general case was made by Michel in [Mic08] via the Hironaka quotients of

the morphism Φ (Deﬁnition 7.1). New techniques and results involving inner rates were recently developed

in [BdSFNP22] in the case where Φ is a generic linear projection of (X, 0), and a complete answer to polar

exploration was given in [BdSFP22b] in the case where (X, 0) is a Lipschitz normally embedded surface germ.

In the present paper, we address the case of a general ﬁnite morphism Φ. We show the relation between the

inner rates and the Hironaka quotients (Theorem B(ii)), and we give an alternative proof of [Mic08, Theorem

4.9] based on our inner rates formula. Finally, we give a family of examples where the inner rates formula

provides tighter restrictions on the relative position of the polar curve than the ones obtained from previous

methods (Proposition C).

We now present our results in a sharp form. Let π: (Xπ, E)−→ (X, 0) be a good resolution of singularities

of (X, 0), that is, a proper bimeromorphic map which is an isomorphism on the complementary of a simple

normal crossing divisor π−1(0) = E, called the exceptional divisor. Let Evbe an irreducible component of

E. We call curvette of Eva smooth curve germ which intersects transversely Evat a smooth point of E.

Throughout the paper, we use the big-Theta asymptotic notation of Bachmann–Landau in the following

form: given two function germs h1, h2:[0,∞),0→[0,∞),0we say that h1is a big-Theta of h2and we

2010 Mathematics Subject Classiﬁcation. Primary 32S25, 57M27 Secondary 14B05; 32S55.

Key words and phrases. Complex Surface Singularities, Resolution of singularities, Polar Curves, Milnor Fibers.

arXiv:2210.02949v6 [math.AG] 11 Mar 2024

2 YENNI CHERIK

write h1(t)=Θh2(t)if there exists real numbers η > 0 and K > 0 such that for all t∈[0, η) we have

K−1h2(t)≤h1(t)≤Kh2(t).

Let (u1, u2) = (g, f) be the coordinates of C2. We prove (Proposition 2.3) the existence of a rational

number qf

g,v such that for any smooth point pof Ein Evwhich does not meet the strict transforms of

f−1(0), g−1(0) or the polar curve ΠΦand for any pair of disjoint curvettes γ∗

1and γ∗

2of Evpassing through

points of Eclose enough to p,

d(γ1∩ {u2=ϵ}, γ2∩ {u2=ϵ}) = Θ(ϵqf

g,v ),

where γ1= Φ ◦π(γ∗

1), γ2= Φ ◦π(γ∗

2) and d is the standard hermitian metric of C2. We call the number qf

g,v

inner rate of fwith respect to galong Ev.

As already mentioned, the notion of inner rates associated to a ﬁnite morphism generalizes the inner rates

of [BdSFP22a]. Indeed, suppose that (X, 0) is embedded in Cn. Assume that πis a good resolution which

factors through the blowup of the maximal ideal and the Nash transform (see e.g [Spi90, Introduction] for

the deﬁnition of the Nash transform) and that Φ is a ”generic” linear projection in the sense of [BdSFP22a,

Subsection 2.2]. Then, the inner rates associated to Φ coincide with the inner rates of the surface germ (X, 0)

(see [BdSFP22a, Deﬁnition 3.3]), which are metric invariants of (X, 0).

Let us now state the main result of this paper, which is proved in section 3. Let Γπbe the dual graph of

the good resolution π, that is, the graph whose vertices are in bijection with the irreducible components of

Eand such that the edges between the vertices vand v′corresponding to Evand Ev′are in bijection with

Ev∩Ev′. Each vertex vof this graph is weighted with the self intersection number E2

vand the genus gv

of the corresponding complex curve Ev. Let valΓπ(v) := Pi∈V(Γπ),i̸=vEi·Evbe the valency of v. We

denote by V(Γπ) the set of vertices of Γπand E(Γπ) the set of edges. Let us denote by mv(f) the order of

vanishing of the function f◦πalong the irreducible component Evof Eand by f∗the strict transform of

the curve f−1(0) by π.

Theorem A (The inner rates formula).Let (X, 0) be a complex surface germ with an isolated singularity

and let π: (Xπ, E)−→ (X, 0) be a good resolution of (X, 0). Let g, f : (X, 0) −→ (C,0) be two holomorphic

functions on Xsuch that the morphism Φ=(g, f):(X, 0) −→ (C2,0) is ﬁnite. Let Ev1, Ev2, . . . , Evnbe

the irreducible components of E. Let Mπ= (Evi·Evj)i,j∈{1,2,...,n}be the intersection matrix of the dual

graph Γπ. Consider the four following vectors: af

g,π := (mv1(f)qf

g,v1, . . . , mvn(f)qf

g,vn),Kπ:= (valΓπ(v1) +

2gv1−2,...,valΓπ(vn) + 2gvn−2), the F-vector Fπ= (f∗·Ev1, . . . , f∗·Evn)and the P-vector Pπ=

(Π∗

Φ·Ev1,...,Π∗

Φ·Evn). Then we have:

Mπ.af

g,π =Kπ+Fπ−Pπ.

Equivalently, for each irreducible component Evof Ewe have the following:



X

i∈V(Γπ)

mi(f)qf

g,iEi

·Ev= valΓπ(v) + f∗·Ev−Π∗

Φ·Ev+ 2gv−2,

where ”·” denotes the intersection number between curves.

The idea of the proof is to relate the inner rates to the canonical divisor of the complex surface Xπand then

apply the classical adjunction formula on the irreducible components of the exceptional divisor E=π−1(0).

This proof is quite shorter and diﬀerent from the one provided in [BdSFP22a] which relies on topological

tools.

Not only does theorem Agives us a concrete way to compute the inner rates in terms of a good resolution

πof (X, 0) just by computing the polar curve (see Example 9.2), but, since the intersection matrix Mπis

negative deﬁnite by a result of Mumford [Mum61,§1], it also proves that given the dual graph Γπtogether

with the F-vector, the inner rates qf

g,v determines and are determined by the P-vector (Π∗

Φ·E1,...,Π∗

Φ·En).

This fact allows us to study polar curves by means of the inner rates, more speciﬁcally, to address the

problem of polar exploration, which we now describe. Suppose that π: (Xπ, E)−→ (X, 0) is the minimal

good resolution of (X, 0) and (gf)−1(0), that is, the minimal good resolution of (X, 0) such that f∗∪g∗is a

disjoint union of curvettes of E. Following [BdSFP22a], polar exploration consists in answering the following

question: is it possible to determine the P-vector from the data of the dual graph Γπ, the F-vector and the

G-vector?

An important contribution to this problem was made by Michel ([Mic08, Theorem 4.9]) by means of the

Hironaka quotients. The Hironaka quotient of the morphism Φ associated to an irreducible component Ev

of Eis the rational number hf

g,v =mv(g)

mv(f), v ∈V(Γπ). In this paper we will improve this result of Michel by

INNER RATES OF FINITE MORPHISMS 3

using the inner rates instead of the Hironaka quotients. The reason why the inner rates are more eﬃcient

than the Hironaka quotients for polar exploration comes from the following result.

Theorem B. Let (X, 0) be a complex surface germ and let Φ = (g, f) : (X, 0) −→ (C2,0) be a ﬁnite

morphism. Let π: (Xπ, E)−→ (X, 0) be a good resolution of (X, 0). There exists a subgraph Aπof Γπsuch

that:

(i) Call f-node any vertex wof Γπsuch that f∗·Ew̸= 0. Let vbe a vertex of Γπ. There exists a path

from an f-node to vin Γπalong which the inner rates are strictly increasing.

(ii) The inner rates and the Hironaka quotients of Φcoincide on Aπ.

(iii) The Hironaka quotients are constant on Γπ\Aπ.

Points (i) and (ii) are proved in the sections 6and 7, while the point (iii) comes from [MM20, Theorem

1].

In Section 8, we show that [Mic08, Theorem 4.9] can be obtained as a consequence of Theorems Aand B.

We state it here taking account of [MM20, Theorem 1]:

Theorem ([Mic08, Theorem 4.9]).Let (X, 0) be a complex surface germ with an isolated singularity and let

g, f : (X, 0) −→ (C,0) be two holomorphic functions on Xsuch that the morphism Φ = (g, f) : (X, 0) −→

(C2,0) is ﬁnite. Let π: (Xπ, E)−→ (X, 0) be a good resolution of (X, 0). Let Aπbe a subgraph of Γπas in the

statement of Theorem B. Let Zbe a connected component of Γπ\Aπor a single vertex on the complementary

of Γπ\Aπ, then :

v∈Z

mv(f)Π∗

Φ·Ev=− X

v∈Z

mv(f)χ′

v!.

where χ′

v:= 2 −2gv−val(v)−f∗·Ev−g∗·Ev.

It is now natural to ask: can we get a better restriction for the value of the P-vector when using the inner

rates and their properties? We give a positive answer to this question via the following result (see proposition

9.4 for a more precise statement).

Proposition C. There exists a sequence of graphs with arrows (Γn)n≥2such that for each n:

(i) There exists a complex surface singularity (Xn,0) and a ﬁnite morphism Φn= (gn, fn):(Xn,0) −→

(C2,0) such that Γnis the dual graph of the minimal good resolution of (Xn,0) and (gnfn)−1(0);

(ii) The P-vector of any such morphism Φnbelongs to a set of n+ 5 elements.

If one performs a polar exploration on this family of examples using only [Mic08, Theorem 4.9], one obtains

an exponential bound for the number of P-vectors (see Remark 9.8), while Proposition Cprovides a linear

bound.

Another important motivation to study the inner rates is to provide analytic invariants of the Milnor

ﬁbration. Let (X, 0) be a complex analytic surface germ embedded in Cnand let f: (X, 0) −→ (C,0) be a

non constant holomorphic function. Let π: (Xπ, E)−→ (X, 0) be a good resolution which factors through

the blowup of the maximal ideal and the Nash transform relative to f(Deﬁnition 10.1). Let ℓbe a generic

linear form with respect to π(Deﬁnition 10.7). The inner rates qf

ℓ,v, v ∈V(Γπ) associated to the morphism

(ℓ, f) do not depend of the choice of the generic linear form ℓ, we then denote them qf

v, v ∈V(Γπ). In this

case the inner rates qf

vgives informations on the inner metric of the Milnor ﬁbers:

Theorem D (Theorem 10.8).Let (X, 0) ⊂(Cn,0) be a complex surface germ with an isolated singularity at

the origin of Cnand let f: (X, 0) −→ (C,0) be a non constant holomorphic function. Let π: (Xπ, E)−→

(X, 0) be a good resolution which factors through the Nash transform of Xrelative to fand the blowup of the

maximal ideal.

Let γ∗

1and γ∗

2be two curvettes of an irreducible component Evof the exceptional divisor Esuch that

γ∗

i∩f∗=∅for i∈ {1,2}. Then there exists qf

v∈Q>0such that

dϵ(γ1∩f−1(ϵ), γ2∩f−1(ϵ)) = Θ(ϵqf

where γ1=π(γ∗

1),γ2=π(γ∗

2)and dϵis the Riemanian metric induced by Cnon the Milnor ﬁber f−1(ϵ).

Furthermore we have qf

v=qf

ℓ,v whenever ℓis a generic linear form with respect to fand π.

Theorem Dis a relative version (with respect to f) and a generalization of [BdSFP22a, Lemma 3.2]. Indeed,

one obtains [BdSFP22a, Lemma 3.2] by taking fa generic linear form in the sense of [BdSFP22a, Subsection

2.2].

As an application of Theorems Aand Dwe give a generalization of a result of Garc´ıa Barroso and

Teissier on the asymptotic behavior of the integral of the Lipschitz-Killing curvature (Deﬁnition 11.1) along

4 YENNI CHERIK

Milnor ﬁbers. Let π: (Xπ, E)−→ (X, 0) be a good resolution which factors through the blowup of the

maximal ideal and the Nash transform relative to f. Let Evbe an irreducible component of Eand let

N(Ev, ϵ), ϵ > 0 be a family of tubular neighborhoods of Evin Xπsuch that lim

ϵ→0N(Ev, ϵ) = Evand such

that Horn(ϵ, v) := π(N(Ev, ϵ)) is included in Bϵ. Consider the set Fv

ϵ,t =f−1(t)∩Horn(ϵ, v) and let δℓ

ϵ>0

be such that for any complex number twith |t| ≤ δℓ

ϵ, we have Card{Fv

ϵ,t ∩Πℓ}=mv(f)Π∗

ℓ·Evwhere ℓis a

generic linear form with respect to π. Set δϵ= minℓ{δℓ

ϵ}.

Theorem E (Theorem 11.5).Let (X, 0) be a complex surface germ with an isolated singularity embedded in

(Cn,0) and let f: (X, 0) −→ (C,0) be a non constant holomorphic function germ. Let π: (Xπ, E)−→ (X, 0)

be a good resolution of (X, 0) which factors through the Nash transform of Xrelative to fand through the

blowup of the maximal ideal of (X, 0). Let vbe a vertex of Γπ, then:

lim

ϵ→0,|t|<δϵZp∈Fv

ϵ,t

KFv

ϵ,t (p)dV=πω2

2ω2n−1

Vol(Gn−1(Cn))Cf,

where

Cf=mv(f)

2gv−2 + ValΓπ(v) + f∗·Ev−X

i∈V(Γπ)

mi(f)qf

iEi·Ev

.

and ωiis the volume of the unit sphere Si.

Theorem Egeneralizes the work of Garc´ıa Barosso and Teissier in [GBT99b] which treat the case of germs

of holomorphic functions at the origin of C2.

Acknowledgments. I would like to express my deep gratitude to my thesis advisors Andr´e Belotto and

Anne Pichon for their help and enthusiastic encouragements during the preparation of this paper. I would

also like to thank Patricio Almir´on for fruitful conversations about polar curves and in particular for pointing

out Example 9.2. This work has been supported by the Centre National de la Recherche Scientiﬁque (CNRS)

which funds my PhD scholarship.

Contents

Introduction 1

Acknowledgments 4

1. Resolution of curves and surfaces 4

2. Inner rates of a ﬁnite morphism 5

3. Inner rates formula 8

4. Non-archimedean link and metric dual graph 10

5. The inner rates function 12

5.1. Laplacian formula on the metric dual graph of a good resolution 15

6. Growth behaviour of the inner rates function 16

7. Proof of points (ii) and (iii) of Theorem B 17

8. A ﬁrst application of the inner rates formula 19

9. Polar exploration 20

10. Geometric interpretation of the inner rates 24

10.1. Generic linear form and generic polar curve 24

10.2. Inner rates of a Milnor ﬁbration 26

11. Application to Lipchitz-Killing curvature 26

11.1. Lipchitz-Killing curvature 27

11.2. The Lipshitz-Killing curvature of a Milnor ﬁber 27

References 28

1. Resolution of curves and surfaces

In this section we introduce some classical materials as they are presented in the introductions of [MM20],[Mic08]

and [BdSFP22a].

Deﬁnition 1.1. Let (X, 0) be a complex surface germ with an isolated singularity. A resolution of (X, 0)

is a proper bimeromorphic map π: (Xπ, E)−→ (X, 0) such that Xπis a smooth complex surface and the

restricted function π|Xπ\E:Xπ\E−→ X\0is a biholomorphism. The curve E=π−1(0) is called the

exceptional divisor.

INNER RATES OF FINITE MORPHISMS 5

The resolution π: (Xπ, E)−→ (X, 0) is good if Eis a simple normal crossing divisor i.e., it has smooth

compact irreducible components and the singular points of Eare transversal double points. Let Evbe an

irreducible component of E. A curvette of Evis a smooth curve germ which intersects Evtransversely at a

smooth point of E.

Deﬁnition 1.2. Let π: (Xπ, E)−→ (X, 0) be a resolution of (X, 0) and let (C, 0) be a curve germ in

(X, 0). The strict transform of Cby πis the curve C∗in Xπdeﬁned as the topological closure of the set

π−1(C\{0}). Let E1, E2, . . . , Enbe the irreducible components of Eand h: (X, 0) −→ (C,0) be a holomorphic

function. The total transform of hby πis the principal divisor (h◦π)on Xπ, i.e.,

(h◦π) =

i=1

mi(h)Ei+h∗

where mi(h)is the order of vanishing of the holomorphic function h◦πon the irreducible component Eiof

Eand h∗is the strict transform of the curve h−1(0).

Proposition 1.3 ([Lau71, Theorem 2.6] or [N´

99, 2.6] for a topological proof).Let π: (Xπ, E)−→ (X, 0) be

a resolution of (X, 0). Let h: (X, 0) −→ (C,0) be a holomorphic function, then we have the following:

(h◦π)·Ev= 0,∀v∈V(Γπ).

where ”·” denote the intersection multiplicity between curves.

Deﬁnition 1.4. Let (C, 0) be a curve germ in (X, 0). A proper bimeromorphic map π: (Xπ, E)−→ (X, 0)

is a good resolution of (X, 0) and (C, 0) if it is a good resolution of (X, 0) such that the strict transform

C∗is a disjoint union of curvettes.

Deﬁnition 1.5. The dual graph of a good resolution π: (Xπ, E)−→ (X, 0) of (X, 0) is the graph Γπ

whose vertices are in bijection with the irreducible components of Eand such that the edges between the

vertices vand v′corresponding to Evand Ev′are in bijection with Ev∩Ev′, each vertex vof this graph is

weighted with the self intersection number E2

vand the genus gvof the corresponding curve Ev. We denote

by V(Γπ)the set of vertices of Γπand E(Γπ)the set of edges. The valency of a vertex vis the number

valΓπ(v) := Pi∈V(Γπ),i̸=vEi·Ev. Let Φ = (g, f) : (X, 0) −→ (C,0) be a ﬁnite morphism and let Ev⊂E

be an irreducible component which meets the strict transform g∗(resp. f∗), following notations of [MM20],

we attach to the vertex vcorresponding to Eva going-out arrow (resp. a going-in arrow) weighted with the

intersection number g∗·Ev(resp. f∗·Ev).

Example 1.6. Consider the ﬁnite morphism Φ=(g, f ):(C2,0) −→ (C2,0), where g(x, y) = x+yand

f(x, y) = y5−x12. The minimal good resolution π: (Xπ, E)−→ (C2,0) of the curve (gf)−1(0) has the

following dual graph: all the irreducible components of Ehave genus 0.

(1) (1)

−2−4−2−1

(1,5) (1,10) (3,35) (5,60)

(2,24)

(1,12)

−3

−2

Figure 1. The numbers between parenthesis are the orders of vanishing (mv(g), mv(f)) of

the functions g◦πand f◦πalong the irreducible components of E, these numbers can be

determined from the dual graph using Proposition 1.3.

2. Inner rates of a finite morphism

Let (X, 0) be a complex surface germ with an isolated singularity and g, f : (X, 0) −→ (C,0) two holomor-

phic functions such that the morphism Φ = (g, f) : (X, 0) −→ (C2,0) is ﬁnite. The aim of this section is to

deﬁne the notion of inner rates associated to the morphism Φ = (g, f). This deﬁnition is a generalization of

the inner rates of a complex surface germ with an isolated singularity ﬁrst introduced in [BNP14]

and later in [BdSFP22a, Deﬁnition 3.3].

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INNERRATESOFFINITEMORPHISMSYENNICHERIKAbstract.Let(X,0)beacomplexanalyticsurfacegermembeddedin(Cn,0)withanisolatedsingularityandΦ=(g,f):(X,0)−→(C2,0)beafinitemorphism.WedefineafamilyofanalyticinvariantsofthemorphismΦ,calledinnerratesofΦ.Bymeansoftheinnerrateswestudythepolarcurveassociatedtothemorphi...

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INNER RATES OF FINITE MORPHISMS YENNI CHERIK Abstract. Let X0 be a complex analytic surface germ embedded in Cn0 with an isolated singularity

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