vanishing cycles of polynomial functions

Reader

Vanishing cycles of polynomial functions for the course of the Summer school on

Singularities: algebra, geometry and applications (Ukraine, Kiev, August 8-19, 2011)

Lecturers:

Dirk Siersma (University Utrecht, The Netherlands), Mihai Tibăr (Université de Lille 1, France)

The main goal is to read the paper: Betti Bounds for Polynomials (to appear in Moscow Mathematical Journal 2011) by M. Tibăr and D. Siersma. This paper can be downloaded separately. In this reader can find parts of papers where some prerequisites, background or enrichment for this paper are treated. Only those pages are reproduced, which possibly play a role in the course. Vanishing cycles and Special Fibres (from Singulariy Theory and Applications, Warwick 1989, Part I; Springer Lecture Notes in Mathematics nr 1462 (1991)) by D. Siersma. Isolated Line Singularities (Proceedings of Symposia in Pure Mathematics Volume 40 (1983) part 2) by D. Siersma. The Vanishing Topology of Non Isolated Singularities. (New Developments in Singularity Theory; Proceedings of Advanced Study Institute in Cambridge 2000) by D. Siersma. Lectures on the topology of polynomial functions ans singularities at infinity (Singularities in Geometry and Topology; Proceedings of the Trieste Summer School and Workshop 2005) by M. Tibăr and D. Siersma.

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Warwick Proceedings

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Arcata Symposium

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The Vanishing Topology of Non Isolated Singularities

Dirk SiersmaMathematisch Instituut Universiteit Utrechtsiersmamathuunl

Introduction

We consider holomorphic function germs f Cj n O Cj andallow arbitrary singularities isolated or nonisolated We are interested in the topology of this situation especially the so called vanishinghomologyWe rst recall the denition of the Milnor bration For small

enough there exist an ball B in Cj n and an disc D in Cj suchthat the restriction

f E fD B D

is a locally trivial bre bundle overD nfg The bres mostly denotedby F are called Milnor bres The groups HE F are called thevanishing homology groupsThe Milnor bre F its homotopy type and homology are interesting

topological objects So is the monodromy operator

T HF HF

of the brationWell known facts are

The Milnor bre is ndimensional and has the homotopy type ofan ndimensional CWcomplex Milnor

The Milnor bre is n s connected where s is equal to thedimension of the singular locus of f KatoMatsumoto

If f has an isolated singularity then the Milnor bre has the homotopy type of a bouquet of ndimensional spheres The number ofthese spheres is called the Milnor number of the isolated singularityMilnor

the eigenvalues of the monodromy operator are roots of unity SeeGriths for references to four dierent proofs

c Kluwer Academic Publishers Printed in the Netherlands

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Dirk Siersma

Isolated singularities have been studied in great detail during the last years They have wonderful properties which relate dierent aspectsof the singularity In this paper we want to discuss especially nonisolated singularities Although the properties eg the topological typeof the Milnor bre are more complicated than for isolated singularitiesthere is a lot of interesting structure availableLet f be the singular locus of f For every point of we

can do the Milnor construction So for every x we have a localMilnor bration eg a space Ex the local Milnor bre Fx and aMilnor monodromy Tx We want to investigate the relation betweenthese objects for all x This is near to the study of the sheaf ofvanishing cycles To be more precise one could try to dene a stratication of in

such a way that two points of are in the same stratum if they canbe joined by a continuous path such that there exits a continuousfamily of Milnor brations of constant bration typeAccording to a result of Massey the constancy of Le numbers for

denition see the contribution of Ganey in this Volume impliesconstancy of the bration type under certain dimension conditionsmore precisely s n for the homotopytype and s n forthe dieomorphismtype We refer to Masseys monograph fordetails and for many other related factsLet us suppose that we end up with a situation where we have

stratied according to the above principle

k

where j n j is jdimensional and smooth For every connectedcomponent of j n j we have a monodromy representation of itsfundamental group on the homology groups of a typical Milnor bre ata general point on the statum

j n j x Aut HFx

We call these monodromiesvertical The vertical monodromies contain a lot of extra information about the singularity The Milnor monodromy is called horizontal and commutes with the vertical monodromiesWe intend to discuss this situation in several examples paying most

attention to the situation where is dimensional where the stratication is rather simple We also intend to treat some examples of higherdimension

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Non Isolated Singularities

This paper is organized as follows In section we recall somefacts about isolated singularities In particular we discuss the relationbetween variation mapping monodromy and intersection formIn section we treat singularities with a dimensional critical set

We follow rst and treat several examples where vertical andhorizontal monodromy play a role and focus at the end on bouquetdecompositions of the Milnor bre These seem to occur as soon as westay near to the case of isolated singularitiesSection is about singular sets of higher dimension We discuss and

summarize recent work

About isolated singularities

The theory of variation mappings plays an important role in our discussions We rst repeat some of the well known facts about isolated singularities They can be found in the literature on several places eg Milnor Lamotke ArnoldGusein ZadeVarchenko Stevens

In the isolated singularity case there exists a geometric monodromyh F F such that hjF is the identityLet Tq h HqF HqF be the algebraic monodromy The mapTq I HqF HqF factors over

VARq HqF F HqF

which is dened by

VARqx hx x

We have a commutative diagram

HqF TqIHqF

j

VAR

j

HqF F TqIHqF F

LEMMA VARq HqF F HqF is an isomorphism if q Proof Consider the exact sequence of the pair Sn F and the

following isomorphism

HqSn F HqF F HI I HqF F

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This gives the following exact variation sequence

HqSn HqF F

VARq HqF HqS

n

The lemma now follows from the fact that HqSn for q n

PROPOSITION For isolated singularities Ker j Ker TnIProof Let E be the total space of the Milnor bration f E D

The diagram above relates the variation mapping and j to the Wangsequence of the bration

HnE HnF TnI HnF HnE

REMARK The intersection form S onHnF is related by Poincareduality to j by

Sx y jx y

So Thas eigenvalue S is degenerateLet K fO B A related fact is that for n

K is a topological sphere detTn I

Because Eh SnnK and by duality

HnE HnK HnK

HnE HnK HnK

the Wang sequence tells

K is a homology sphere detTn I

For the step from homology sphere to topological sphere we refer toMilnor

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One dimensional singular locus

Introduction

In this section we consider singularities with a dimensional criticallocus for short isolated singularities and study the vanishing homology in a full neighbourhood of the origin In this case the vanishinghomology is concentrated on the dimensional set We can write

r

where each i is an irreducible curveAt the origin we consider the Milnor bre F of f and on each

i fOg a local system of transversal singularitiesTake at any x ifOg the germ of a generic transversal section Thisgives an isolated singularity whose class is welldened We denote atypical Milnor bre of this transversal singularity by F

i On the levelof homology we get in this way a local system with bre HnF

i

More precisely we consider in the isolated case the following data

The Milnor bre F The vanishing homology is concentrated in dimensions n and n

HnF ZZn which is freeHnF which can have torsion

The Milnor monodromy acts on the bre F

Tn HnF HnF

Tn HnF HnF

The transversal Milnor bres F i The vanishing homology is con

centrated in dimension n

HnFi ZZi which is free

On this group there act two dierent monodromies

the vertical monodromy or local system monodromy

Ai HnFi HnF

i

which is the characteristic mapping of the local system over thepunctured disc i fOg

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the horizontal monodromy or Milnor monodromy

Ti HnFi

HnFi

which is the Milnor bration monodromy when we restrict f to atransversal slice through x i fg

In fact Ai and Ti are dened over i fOg S which is homotopyequivalent to a torus So they commute

AiTi TiAi

One of the topics of this section is to show how the above data enterinto a good description of the topology of a isolated singularity Fordetails we refer to

EXAMPLE Dsingularity f xy z is given by y z and is a smooth line The transversal type isA

It is known that F is homotopy equivalent to S cf One can show that

HF ZZ T I HF HF

ZZ T I and A I

EXAMPLE Tsingularity f xyz

and consists of the three coordinate axes in Cj

The transversal type is again A

It is known that F is homotopy equivalent to the torus S S

cf One can show that

HF ZZ T IHF ZZ ZZ T IHF

ZZ Ti I and Ai I i

EXAMPLE Let f be isolated and homogeneous of degree d Inthis case one has the relation

Ai Tdi

We can assume that all the is are straight lines through O We cansuppose that i is the xaxis the formula follows from fsx x xn sdfx s

x sxn cf

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Series of singularities

Let again f Cj n O Cj be a germ of an analytic function Letf have a dimensional critical locus f One considers for eachN IN the series of functions

fN f xN

where x is an admissible linear form which means that f fx g has an isolated singularity One calls this series of function germsa Yomdin series of the hypersurface singularity f Under the abovecondition all members of the Yomdin series have isolated singularitiesMoreover their Milnor numbers can be computed using the socalledLeYomdin formula

f xN nf nf Ne

Here n resp n are the corresponding Bettinumbers of the Milnorbre F of the nonisolated singularity f and eo is the intersectionmultiplicity of and x The formula holds for all N sucientlylarge Moreover e

Pdi

i where di is the intersection multiplicity

of i with reduced structure and xThe following formula relates the characteristic polynomials of the

monodromies of f and fN Other ingredients are the horizontal andvertical monodromies The eigenvalues of the monodromy satisfy Steenbrinks spectrum conjecture cf This conjecture was later provedby M Saito using his theory of Mixed Hodge Modules

THEOREM Let f Cj n O Cj have dimensional critical locus r irreducible components Let x be anadmissible linear form Let Mf be the alternating product of thecharacteristic polynomials of the monodromy T of f in dimensions nand n Let Mf xN be the characteristic polynomial of themonodromy of f xN in dimensions n For all N suciently large

Mf xN MfYdetNdiI AiT

Ndii

where Ai and Ti are the vertical and horizontal monodromy along thebranch i

Proof The idea behind the proof is to use polar methods and toconsider the map germ

f x Cj n Cj Cj

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! Dirk Siersma

The Milnor bres F of f resp FN of fN occur as inverse imagesunder of the sets ff tg resp ff xN tg Next one constructsvia a stratied isotopy an embedding

F FN

From the corresponding homology sequence one gets the following term exact sequence

HnF HnFN HnF

N F HnF

The dierence FN nF is by excision and homotopy equivalence relatedto the part of FN located near the di intersection points of i and F

N One obtains

HqFN F r

i Ndik

HnFik

where each Fik is a copy of the Milnor bre of the transversal singularity F

i From this one gets

bnF bnF bnFN N

Xdi

i

To obtain the monodromy statement one constructs a geometricmonodromy which acts on the term sequence One uses Les carrouselmethod The monodromy on FN respects the distance function jxj asnearly as possible The geometric monodromy gets an xcomponentwhich gives rise to the appearance of the vertical monodromy Ai indetNdiI AiT

Ndii For details cf

REMARK M f is related to Zf the zeta function of the mon

odromy which is dened by Zf t Q

qdetI tTq cf

For homogeneous singularities of degree d the formula Zt

tdF d is well known and valid in all generality without assumptions

on the dimension of the critical set cf eg

REMARK The theorem can be used in two ways computingmononodromies for isolated singularities in the series but also forcomputing monodromies of certain nonisolated singularities with onedimensional singular setsIn the case of a homogeneous polynomial one can compute almost

all ingredients in the formula by taking N as the degree d of f

One gets in this way the formula Zf t td

Freg d

P

i where

the i are transversal Milnor numbers and Freg ndn

the Euler characteristic of an isolated singularity of degree d Also

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Siersma

Note

Unmarked set by Siersma

Dirk Siersma

discriminant spaces of Coxeter arrangements In both cases there isalso combinatorial and geometric structure around For the homologyof the Milnor bre of an arrangement we refer to OrlikTerao ! Thezeta function is just Zt tdM

where M is the complementof the arrangement modulo the natural Cj action The zeta functionof the discriminant hypersurface of a Coxeter arrangement is studiedin geometric terms by

References

A G Aleksandrov The Milnor numbers of nonisolated Saito singularitiesRussian Funkts Anal i Prilozhen

A G Aleksandrov Nonisolated Saito singularities Russian Mat Sb NS Translation in Math USSRSb

V I Arnold S M Gusein Zade and A N Varchenko Singularities ofDierentiable Maps I and II Birkhauser and

A Artal Bartolo P CassouNogues I Luengo and A Melle HernandezMonodromy conjecture for some surface singularities preprint September

D Barlet Interaction de strates consecutives pour les cycles evanescentsComptes Rendus Acad Sci Paris ser I

D Barlet Interaction de strates consecutives pour les cycles evanescents AnnSci Ecole Norm Sup

G Barthel A Dimca On complex projective hypersurfaces which are homology IPns Singularities ed JP Brasselet London Math Soc Lect Notes Cambridge Univ Press

E Brieskorn Die monodromie der isolierten Singularitaten von HyperachenManuscripta Math

J Damon Nonlinear sections of nonisolated complete intersections thisvolume

P Deligne et al Groupes de monodromie en geometrie algebrique IISeminaire de Geometrie Algebrique du BoisMarie SGA IISpringer lecture notes in Math esp pp

J Denef F Loeser Geometry on arc spaces of algebraic varieties Proceedingsof the Third European Congress of Mathematics Barcelona to appear

A Dimca On the Milnor bration of weighted homogeneous polynomialsCompositio Math

A Dimca Singularities and topology of hypersurfaces Universitext SpringerVerlag New York

A Dimca and A Nemethi On the monodromy of complex polynomialspreprint MathAG

J Denef and F Loeser Character sums associated to nite Coxeter groupsTrans Amer Math Soc

A Dold Lectures on Algebraic Topology Grundlehren der Math Wissenschaften SpringerVerlag

T Ganey The theory of integral closure of ideals and modules applicationsand new developments this volume

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V Grandjean Residual discriminant of a function germ singular along an Eulerfree divisor preprint Rennes

P A Griths Periods of integrals on algebraic manifolds summary of mainresults and discussion of open problems Bull Amer Math Soc

I N Iomdin Local topological properties of complex algebraic sets SibirskMat Z

G Jiang Functions with nonisolated singularities on singular spaces Dissertation University of Utrecht httpwwwmathuunlpeoplesiersma

G Jiang Algebraic descriptions of nonisolated singularities Hokkaido MathJ

G Jiang and M Tibar Splitting of Singularities preprint No TokyoMetropolitan University MathAG

T de Jong Some classes of line singularities Math Zeits T de Jong and D van Straten Deformations of the normalization of

hypersurfaces Math Ann M Kato and Y Matsumoto On the connectivity of the Milnor bre of a

holomorphic function at a critical point Manifolds Tokyo University ofTokyo Press pp

K Lamotke Die Homologie isolierter Singularitaten Mat Zeits

D T Le Ensembles analytiques complexes avec lieu singulier de dimensionun dapres I N Jomdin Seminaire sur les Singularites Publ Math UnivParis VII pp

D T Le F Michel and C Weber Courbes polaires et topologie des courbesplanes Ann Sci cole Norm Sup

D Massey A quick use of the perversity of the vanishing cycles preprintNortheastern University Boston MA

D Massey Le cycles and hypersurface singularities Springer Lecture Notes inMath

D Massey and D Siersma Deformation of polar methods Ann Inst Fourier

J Milnor Singular points of complex hypersurfaces Ann Math StudiesPrinceton Univ Press

D Mond Some remarks on the geometry and classication of germs of mapsfrom surfaces to space Topology

D Mond Vanishing cycles for analytic maps Singularity Theory and its Applications Warwick Part I eds D Mond and J Montaldi SpringerLecture Notes in Math

A Nemethi The Milnor bre and the zeta function of the singularities of thetype f P h g Compositio Math

A Nemethi Hypersurface singularities with dimensional critical locus JLondon Math Soc

P Orlik H Terao Arrangements and Milnor bers Math Ann

R Pellikaan Finite determinacy of functions with nonisolated singularitiesProc London Math Soc

R Pellikaan Series of isolated singularities Singularities ed R RandellContemp Math Amer Math Soc Providence RI pp

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Dirk Siersma

R Randell On the topology of nonisolated singularities Proc GeorgiaTopology Conference

M Saito On Steenbrinks Conjecture Math Ann H Samelson On Poincare duality Jour dAnalyse Math M Shubladze Hyperplane singularities of transversal type Ak Russian

Soobshch Akad Nauk Gruzin SSR D Siersma Isolated line singularities Singularities ed Peter Orlik Proc

Symp pure math Amer Math Soc pp D Siersma Hypersurfaces with singular locus a plane curve and transversal

type A Singularities Banach Center Publ Warsaw pp D Siersma Singularities with critical locus a dimensional complete inter

section and transversal type A Topology and its Applications

D Siersma Quasihomogeneous singularities with transversal type A Contemporary Math

D Siersma Vanishing cycles and special bres Singularity Theory and itsApplications Warwick Part I eds D Mond and J Montaldi SpringerLecture Notes in Math

D Siersma The monodromy of a series of hypersurface singularities CommentMath Helvet

D Siersma Variation mappings on singularities with a dimensional criticallocus Topology

D Siersma A bouquet theorem for the Milnor bre J Alg Geom

D Siersma and M Tibar Is the polar relative monodromy of nite orderAn example Chinese Quart J Math Proc workshop onTopology and Geometry Zhanjiang

J H M Steenbrink The spectrum of hypersurface singularities Actes duColloque de Thorie de Hodge Luminy Asterisque

J Stevens Periodicity of branched cyclic covers of manifolds with open bookdecomposition Math Ann

D van Straten On the Betti numbers of the Milnor bre of a certain class ofhypersurface singularities Singularities representation of algebras and vectorbundles Lambrecht Springer Lecture Notes in Math

M Tibar Bouquet decomposition of the Milnor bre Topology

M Tibar Embedding nonisolated singularities into isolated singularities Singularities the Brieskorn anniversary volume eds V I Arnold GM Greueland J H M Steenbrink Progress in Math Birkhauser pp

A Zaharia A study about singularities with nonisolated critical locus thesisRijksuniversiteit Utrecht

A Zaharia Topological properties of certain singularities with critical locus adimensional complete intersection Topology and Appl

Address for Oprints D Siersma Mathematisch Instituut Universiteit UtrechtPOBox TA Utrecht The Netherlands

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TWO LECTURES ON THE TOPOLOGY OF POLYNOMIALFUNCTIONS AND SINGULARITIES AT INFINITY

DIRK SIERSMA AND MIHAI TIBAR

Abstract. In the last 15 years there has been an increasing interest for the study ofthe global topology of polynomial functions, especially in connection with the asymptoticbehaviour of fibres. This stream of research is closely related to the affine geometry andto dynamical systems on non-compact spaces. We present here several aspects of thistopic.

Contents

1. Introduction 22. Topology of the fibres of a polynomial 33. Regularity conditions at infinity 74. Polar curves and regularity conditions 125. The case n = 2 156. Families of complex polynomials with singularities at infinity 197. Singularity exchange at infinity 20References 25

Date: December 5, 2005.2000 Mathematics Subject Classification. 14H10, 57R27, 14D99, 32S30, 14D06, 53C65.Key words and phrases. affine hypersurfaces, singularities at infinity, exchange of singularities, real

and complex curves.1

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2 DIRK SIERSMA AND MIHAI TIBAR

1. Introduction

One of the first authors who studied the topology of polynomial functions was Broughton[Br1]. In the same time Pham [Ph] was interested in regularity conditions under whicha polynomial has good behaviour at infinity, and Ha and Le [HaL] proved a criterion fordetecting atypical values in two complex variables.

Some evidence for the crucial importance of singularities at infinity in understanding thebehaviour of polynomials is the Jacobian Conjecture. In C2, an equivalent formulation ofthis conjecture is the following, see [LW, ST1]: If f : C2 → C has no critical points but hassingularities at infinity then, for any polynomial h : C2 → C, the critical locus Z(Jac(f, h))is not empty. Indeed, if the polynomial f has no critical points and no singularities atinfinity then Corollary 2.11 will show that all the fibres of f are CW-complexes withtrivial homotopy groups, hence contractible. In this case the Abhyankar-Moh theoremtells that f is linearisable. The open case is therefore the one of singularities at infinity.

In general, let f : Kn → K be a polynomial function, where K is C or R. We assumeonce and for all in this paper that n ≥ 2 and that the polynomial is not a constant. Avalue a ∈ K is called typical for f if the fibre f−1(a) is nonsingular and the function fis a locally trivial topological fibration at a. Otherwise a is called atypical. The set ofatypical values is known to be finite [Th]1.

The first natural problem is to identify the set of noncritical atypical values of f andto describe how the topology of fibres changes at such a value. There is yet no solutionin whole generality. One can solve the problem in case n = 2 over the complex [HaL] andover the reals [TZ], but in case n > 2 there have been proved such criteria only when thesingularities which occur are in some sense isolated and K = C: [ST1, Pa1, Pa2, Ti3].This includes the situation of “no singularity at infinity” [Br2], [NZ] which had beentreated earlier.

These notes report on a few topics which have been advanced in the last years andthey are essentially collected from the following papers and manuscripts of the authors:[ST1, Ti4, TZ, BT, ST3, ST5]. They do not cover several others topics like monodromy,Hodge theory, equisingularity, curvature etc. In order to keep a reasonable length, anumber of proofs are skipped; on the other hand we give many examples and some ofthem are largely commented. The forthcoming book [Ti6] will include all this materialand a lot more on polynomials and their vanishing cycles.

The contents of this survey are as follows. We discuss in §2 the topology of the fibresof polynomials f : Cn → C. We first define W-singularities at infinity, where W refersto a certain Whitney stratification on the space X which is the union of the compactifiedfibres of f . We show that the general fibre of a polynomial with isolated W-singularitiesat infinity has the homotopy type of a bouquet of spheres and that the number of thesespheres is µf + λf , where µf is the total Milnor number of f (i.e. the sum of Milnornumbers of the affine singularities) and λf is the sum of Le-Milnor numbers at infinity.

We study in §3 several regularity conditions at infinity which insure topological trivialityand we study their mutual relations. We introduce in §4 the polar curve, a key tool

1for a proof, see e.g. [Ph, Appendix], which uses resolution of singularities, or [Ve], [ST1] and Corollary3.11 in this paper, where statification theory is used.

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SINGULARITIES AT INFINITY 3

in exploring topological properties of polynomial functions, and explain its relation toregularity conditions.

We look closer in §5 to the case of two variables, in the complex and in the real setting,and completely characterise the topological triviality of fibres.

In §6 we introduce the problem of global topological equivalence of polynomial func-tions. We explain a recent result which gives a numerical criterion for topological trivialityin a family of polynomial functions.

A special phenomenon which one encounters in families of polynomial functions is the“singularity exchange at infinity”, when singular points escape from the space Cn andproduce “virtual” singularities (i.e. singularities at infinity) of the limit polynomial. Thetotal quantity of singularity involved in this phenomenon may not be conserved, as itis in the case of local deformations of holomorphic functions with isolated singularities.We discuss in §7 semi-continuity results which enable us to find rules of the exchangephenomenon.

These two lectures were delivered at the Singularities School of the ICTP Trieste, inAugust 2005. We own special thanks to the organisers and to Le Dung Trang, head ofthe Mathematics section at ICTP.

2. Topology of the fibres of a polynomial

2.1. Singularities at infinity. Let f : Cn → C be a polynomial of degree d and letf(x, x0) be the homogenized of f by the new variable x0. One replaces f : Cn → C bya proper mapping τ : X → C which depends on the chosen system of coordinates onCn, as follows (see [Br2]). Consider the closure in Pn × C of the graph of f , that is thehypersurface

X := ((x; x0), t) ∈ Pn × C | F := f(x, x0) − txd

0 = 0,

which fits into the commuting diagram

Cn i

−→ X

f ց ւτ

C

,

where i denotes the inclusion x 7→ (x, f(x)) and τ is the projection on the second factor.Let H∞ denote the hyperplane at infinity x0 = 0 ⊂ Pn. The singularities of X are

contained in the part “at infinity” X∞ := X ∩ (H∞ × C), namely:

Xsing := Σ × C, where Σ := ∂fd

∂x1= · · · =

∂fd

∂xn= 0, fd−1 = 0 ⊂ H∞.

The singular set of X∞ is:

X∞sing := W × C, where W :=

∂fd

∂x1

= · · · =∂fd

∂xn

= 0 ⊂ H∞.

We have Σ ⊂W .The singularities of f , i.e. the affine set Sing f := Z( ∂f

∂x1, · · · , ∂f

∂xn), can be identified,

by the above diagram, with the singularities of τ on X \ X∞. One can prove, by an easy

computation, that Sing f ∩H∞ ⊂ Σ, where Sing f denotes the closure of Sing f in Pn. Inparticular we get dim Sing f ≤ 1 + dim Σ.

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Definition 2.1. We consider the following classes of polynomials:

(i) f is a F-type polynomial if its compactified fibres and their restrictions to thehyperplane at infinity have at most isolated singularities.

(ii) f is a B-type polynomial if its compactified fibres have at most isolated singularities.(iii) f is a W-type polynomial if its proper extension τ has only isolated singularities

in a stratified sense ([ST1]) as defined below.

Let us recall that, for an analytic function ψ on a complex space Y endowed with acomplex Whitney stratification C, one has the well defined notion of stratified singularityof ψ (alternatively: singularity of ψ with respect to C). Namely, the stratified singularitiesof ψ, denoted by Singψ, are the union ∪Ci∈CSingψ|Ci

(see e.g. [GM], [Le2]). We introducethe following definitions:

Definition 2.2. (Canonical stratification at infinity)Let W be the least fine Whitney stratification of X that contains the strata X \ X∞ andX∞ \ Xsing. This is a canonical Whitney stratification (see [Ma, §4]) with two imposedstrata instead of the smooth open X \ Xsing. We may call W the canonical Whitneystratification at infinity of X.

Let Sing τ be the singularities of τ : X → C with respect to the canonical Whitneystratification at infinity and denote Sing ∞f := Sing τ ∩ X∞. Using §2.1, one then getsthe following equality:

Sing τ = Sing f ∪ Sing ∞f.

Let us also remark that Sing t ∩ (X∞ \ Xsing) = ∅ and that dim Sing ∞f ≤ dim Σ.The class of polynomials we want to focus at is defined, by making use of Le’s definition

of isolated singularities [Le2, Definition 1.1], as follows.

Definition 2.3. (Isolated W-singularities at infinity)We say that the polynomial f : Cn → C has isolated W-singularities at infinity if theprojection τ : X → C has isolated singularities with respect to the stratification W(equivalently: dim Sing τ ≤ 0).

We have: F-class ⊂ B-class ⊂ W-class. The first inclusion is clear from the definitionand the second one is proved in [ST1]. In 2 variables, if f has isolated singularities inC

2, then it is automatically of F-type. Deformations inside the F-class were introducedin [ST3] under the name FISI deformations. Broughton [Br2] considered for the firsttime B-type polynomials and studied the topology of their general fibers. The W-class ofpolynomials appears in [ST1].

Remark 2.4. From the above definition and the expressions of the singular loci we havethe following characterisation:

(i) f is a B-type polynomial ⇔ dim Sing f ≤ 0 and dim Σ ≤ 0,(ii) f is a F-type polynomial ⇔ dim Sing f ≤ 0 and dimW ≤ 0.

Remark 2.5. “Isolated W-singularities at infinity” implies that the polynomial f hasisolated singularities in Cn (in the usual sense), which fact does not depend on coordinates.

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Examples 2.6. (a) If the polynomial f has isolated singularities in Cn and dim Σ ≤ 0then f has isolated W-singularities at infinity. This is the case for all reduced planecurves.

(b) h = x3y+ x+ z2 : C3 → C has isolated W-singularities at infinity (with respect toW ′) but dim Σ = 1.

(c) The polynomial g := x2y+x : C3 → C has non-isolated W-singularities at infinity(namely in the fibre g−1(0)). It turns out that it has nonisolated W-singularitiesat infinity in any coordinates, see also our next Remark.

2.2. A bouquet theorem. The next result may be viewed as a global version of thelocal bouquet theorems of J. Milnor [Mi, Theorem 6.5] and Le D.T. [Le2, Theorem 5.1].

Theorem 2.7. [ST1] Let f : Cn → C be a polynomial with isolated W-singularities atinfinity. Then the general fibre of f is homotopy equivalent to a bouquet of spheres of realdimension n− 1.

Remark 2.8. As in the local case, if f has nonisolated W-singularities at infinity, thenone cannot expect to get a bouquet of the type in the Theorem above. The Example2.6(c) shows that even if the polynomial has smooth fibres (but some nonisolated W-singularity contained in X∞), the result above is no more true: the general fibre of g is acircle (whereas it should have been a bouquet of spheres of dimension 2).

Proof of Theorem 2.7.Step 1. We prove first that the reduced homology of a general fibre is concentrated indimension n − 1. Relatively to the stratification W, the projection τ : X → C has onlyisolated singularities, namely a finite number of points situated on X∞ and another finiteset on X \ X∞ which corresponds to the set Sing f . Let R be the set of critical values ofτ .

For each b ∈ R, let δb be a small enough disc centred at b. Then τ : X∩τ−1(C\∪b∈Rδb) →C \ ∪b∈Rδb is a stratified topological fibration (with respect to W), hence its restrictionto f−1(C \ ∪b∈Rδb) is a locally trivial topological fibration, by Thom first isotopy lemma.

Let XS := τ−1(S), FS := f−1(S), for some S ⊂ C. Let us fix c ∈ C\∪b∈Rδb and cb ∈ ∂δb.We get, as usually by deformation retraction and excision, the following splitting:

(1) Hi(Fc) = Hi+1(Cn, Fc) = ⊕b∈RHi+1(Fδb

, Fcb),

which actually holds in full generality, for any polynomial f , whatever its singularitiesare.

We stick to such a term Hi+1(Fδb, Fcb

). For simplicity of notations, let D be one of thediscs δb and fix some d ∈ ∂D. We have, according to [Br2, Proposition 5.2]:

H•(FD, Fd) ∼= H2n−•(XD,Xd).

It remains to prove that H•(XD,Xd) is concentrated. Let b be the centre of D. Thesingularity of τ|XD

are on Xb, let those be denoted by a1, . . . , ak. We may choose a goodneighbourhood of ai, say of the formBi∩XD, where Bi is a small enough closed ball in somelocal chart and also suppose D small enough such that the restriction τ : Bi ∩XD → D isa Milnor representative of the germ τ : (XD, ai) → (C, b). Since this germ is an isolatedsingularity with respect to the induced stratification, it follows that the fibres τ−1(u),

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∀u ∈ D, are transversal to a certain semi-algebraic Whitney stratification of ∂Bi ∩ XD,constructed as in the proof of [Le1, Theorem 1.1] or [Le2, Theorem 1.3]. Thus there is atrivial topological fibration:

τ : XD \ ∪i=1,kBi → D

and, by an excision, we get the isomorphism:

H•(XD,Xd) = ⊕i=1,kH•(Bi ∩ XD, Bi ∩ Xd) = ⊕i=1,kH

•−1(Bi ∩ Xd),

where Bi ∩ Xd is the local Milnor fibre of the germ of τ at ai.We may conclude our proof by applying a theorem due to Le D.T. (see e.g. [Le2,

Theorem 5.1] for a more general result) which says that the Milnor fibre of an isolatedsingularity function germ on a hypersurface of pure dimension n has the homotopy typeof a bouquet of spheres of dimension n− 1.

Step 2. To get the homotopy result, we would like to replace in the above proof thehomology excision by the homotopy excision. By inspecting the proof, it easily appearsthat one can do this until the local situation. For instance, since the trivial topologicalfibration τ : XD \ ∪i=1,kBi → D is a stratified one, it restricts to the trivial fibrationτ : (XD \ X∞) \ ∪i=1,kBi → D.

It remains to manage the local situation. Let first ai ∈ X∞ be a singular point of τ .This fits into a statement due to Hamm and Le [HmL, Theorem 4.2.1, Corollary 4.2.2]:the conditions are obviously fulfilled, namely τ has isolated singularities with respect toW and rHd(X \ X∞) ≥ n (since X \ X∞ is smooth). The ingredients in the proof arehomotopy excision (Blakers-Massey theorem) and stratified Morse theory.

Now by a very slight modification of the above cited result of Hamm and Le (i.e. byusing cylindrical neighbourhoods, which are conical by [GM, p. 165]), we get that thepair

(Bi ∩ XD \ X∞, Bi ∩ Xd \ X

∞)

is (n− 1)-connected.We may apply [Sw, Prop. 6.13] to conclude that Bi ∩ XD \ X∞ is obtained from

Bi ∩ Xd \ X∞ by adding cells of dimensions ≥ n.

A similar (actually better) situation is encountered on the affine piece: if aj ∈ Xb \X∞

is a singularity of τ|XDthen it is well known that Bj ∩ FD is obtained from Bj ∩ Fd by

attaching n-cells.For global fibres, it follows that FD is obtained (up to homotopy) by adding a finite

number of cells of dimension ≥ n to Fd.Finally, the whole space Cn = FC is obtained, up to homotopy, by attaching a finite

number of cells of dimensions ≥ n to a general fibre Fc. Since Fc has the homotopy type

of a n-dimensional CW-complex, we get Fcht≃ ∨γS

n−1, by Whitehead’s theorem.

Note 2.9. Improvements of Theorem 2.2, by considering a sharper type of singularities,were proved in [Pa2] and [Ti3]; see also the forthcoming book [Ti6] for an updated proof.

Our bouquet statement applies to complex polynomials of two variables with irreduciblegeneric fibre and to n-variables complex polynomials with isolated singularities such thatare ρ-regular at all points y ∈ X∞. The latter fact follows from the definition of ρ-regularity, see §3, and its proof is left to the reader. Our result also extends the one

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for some classes of polynomials with “good behaviour at infinity”: tame [Br2], quasi-tame [Ne], M-tame [NZ] (which coincides with ρE-regular, see §3). We recall that “tame”implies “quasi-tame”, which implies in its turn “M-tame” [Ne], [NZ]. It turns out fromCorollary 2.11(a) that if such a polynomial has isolated W-singularities at infinity thenactually it has no W-singularities at infinity.

The homology counterpart of our Theorem 2.2 (i.e. Step 1 of its proof) was provedunder the additional hypothesis dim Σ = 0, by Broughton himself [Br2, Theorem 5.2]; healso got the particular case of point (a) of the following Corollary 2.11.

More recently, Hamm [Hm] studied the cohomology of fibres. He proves the semicon-tinuity of the ranks of certain cohomology groups and of stalks of direct image sheaves.This contributes to understanding why and how occur the “jumps” in the topology of thefibres.

Definition 2.10. We denote by λa the number of spheres in the Milnor fibre of the germτ : (X, a) → (C, b) and call it the Milnor number at infinity, at a.

Corollary 2.11. Let f be a polynomial with isolated W-singularities at infinity. Then:

(a) The number γ of spheres in a general fibre is equal to the sum µf +λf , where µf isthe total Milnor number of f and λf is the sum of the Milnor numbers at infinity.In particular λf is invariant under diffeomorphisms of Cn.

(b) Let µFbdenote the sum of the Milnor numbers of all the singularities of the fibre

Fb and let λFbdenote the sum of all Milnor numbers at infinity at Xb ∩X∞. Then

χ(Fu) − χ(Fb) = (−1)n−1(λFb+ µFb

),

where Fu is a general fibre of f .

Corollary 2.12. (General connectivity estimation) Let f : Cn → C be any polyno-mial. Then its general fibre Fu is at least (n− 2 − dim(Σ ∪ Sing f))-connected.

Note 2.13. For the proof we refer to [ST1]. This result improves, in case dim Sing f ≤dim Σ the connectivity estimations by Kato and by Dimca. One should remark that ourstatement is only true for general fibres. More precisely, it was proved in [Ti3] that theestimation for the connectivity of atypical fibres can be at most 1 less than the aboveestimation for general fibres (see for instance the example f = x2y + x : C

2 → C). Animproved connectivity estimation was shown more recently by Libgober and Tibar [LT],which superseded another improvement by Dimca and Paunescu [DP].

3. Regularity conditions at infinity

For proving topological triviality at infinity (i.e. on Kn \ K, where K is some largecompact) one would try to produce a foliation which is transversal to the fibres of f . A

natural attempt is to integrate the vector field grad f (or grad f/‖ grad f‖2, which is alift by f of the vector field ∂/∂t on K). The resulting foliation may have leaves that“disappear” at infinity (since grad f may tend to 0 along some non-bounded sequence ofpoints), hence it is not of the kind we want.2

2Such a foliation was used by L. Fourrier [Fo1], [Fo2] to characterise, in two complex variables, thetopological right-equivalence at infinity of polynomials. See also §6 for the general problem of globaltopological triviality of families of polynomials.

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In order to construct a “good” foliation, one needs some regularity conditions on theasymptotic behaviour of the fibres of f . Keeping in mind the idea of using controlledvector fields (rather than just grad f), we introduce two regularity conditions at infinity:t-regularity and ρ-regularity. The former depends on the compactification of f , but allowsone to apply algebro-geometric tools, more effectively in the complex case. It alreadyappeared in [ST1] (with a different definition) and in [Ti2, Ti4]. The latter conditiondoes not depend on any extension, but on the choice of a proper non negative C1-functionρ which defines a codimension one foliation.

These regularity conditions correspond to the two main strategies used up to now in thestudy of affine functions, which were regarded as parallel methods. Let us very roughlydescribe them first. One method is to “compactify” in some way the function (i.e. toextend it to a proper one). This gives the advantage of having the “infinity” as a subspaceof the total space Y , but in the same time creates the problem of getting rid of it in theend. The space Y has singularities (usually nonisolated ones) exactly “at infinity”; onemay either resolve those singularities or endow Y with a stratification. The second mainstrategy is to stay within the affine and to use for instance Milnor-type methods.

We prove that t-regularity implies ρ-regularity, which fact provides a link between thetwo aforementioned methods. We consider also the real case, which is more and moreexplored in the last time (Kurdyka, Szafraniek, Parusinski, D’Acunto, Grandjean, Jelonekand others). Our approach relies on the study of the limits at infinity of tangents to thelevels of f and their behaviour with respect to the divisor at infinity.3

3.1. ρ-regularity and t-regularity. Let f : Kn → K be a polynomial function and let:

XK := f(x, x0) − txd0 = 0 ⊂ P

nK× K,

be the closure in PnK×K of its graph, as defined in 2.1. Let τ : XK → K be the projection

to K. Then the restriction τ is a proper function and if we identify Kn with the graph of

f , then the restriction τ|Kn is just f . Let us denote by X∞K

:= XK ∩ x0 = 0 the divisorat infinity. For simplicity, we shall often suppress the lowercase K.

This proper extension of f has been used several times in the study of polynomialfunctions at infinity [Br1], [Ph], [Pa1], [ST1], [Ti3], etc.

Definition 3.1. We say that f is topologically trivial at infinity at t0 if there is aneighbourhood D of t0 ∈ K and a compact set K ⊂ Kn such that the restrictionf| : (X \ K) ∩ f−1(D) → D is a topologically trivial fibration. We say that f is lo-cally trivial at y ∈ X∞ if there is a fundamental system of neighbourhoods Ui of y in X

and, for each i, some small enough neighbourhood Di of τ(y), such that the restrictionf| : Ui ∩ (X \ X∞) ∩ f−1(Di) → Di is a topologically trivial fibration.

Remark 3.2. Local triviality at all points y ∈ X∞ ∩ τ−1(t0) does not imply topologicaltriviality at infinity at t0. To be able to glue together a finite number of locally trivialnonproper fibrations, one needs a global control over these, which is not available ingeneral. See Example 3.13. This should be contrasted to the regularity conditions wedefine in the next.

3We have already used this point of view in investigating the topology “at infinity” of complex poly-nomial functions, see [ST1], [Ti3], [Ti4], [Ti5] .

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Definition 3.3. (ρ-regularity) Let ρ : Kn \ K → R≥0 be a proper C1-submersion,where K ⊂ Kn is some compact set. We say that f is ρ-regular at y ∈ X∞ if there is aneighbourhood U of y in X \K such that f is transversal to ρ at all points of U ∩ Kn.

We say that the fibre f−1(t0) is ρ-regular at infinity if f is ρ-regular at all pointsy ∈ X∞ ∩ τ−1(t0).

Remark 3.4. The definition of ρ-regularity at infinity of a fibre f−1(t0) does not dependon any proper extension of f , since it is equivalent to the following: for any sequence(xk)k∈N ⊂ Kn, |xk| → ∞, f(xk) → t0, there exists some k0 = k0((xk)k∈N) such that, ifk ≥ k0 then f is transversal to ρ at xk. It also follows from the definition that if f−1(t0)is ρ-regular at infinity then this fibre has at most isolated singularities.

The transversality of the fibres of f to the levels of ρ is a “Milnor type” condition. Incase ρ is the Euclidean norm, denoted in this paper by ρE, this condition has been used byJohn Milnor in the local study of singular functions [Mi, §4,5]. For complex polynomialfunctions, transversality to big spheres (i.e. ρE-regularity, in our definition) was used in[Br2, pag. 229] and later in [NZ], where it is called M-tameness.

Example 3.5. ρ : Kn → R≥0, ρ(x) = (∑n

i=1 |xi|2pi)1/2p, where (w1, . . . , wn) ∈ Nn, p =

lcmw1, . . . , wn and wipi = p, ∀i. This function is “adapted” to polynomials which arequasihomogeneous of type (w1, . . . , wn). By using it, one can show that a value c ∈ K isatypical for such a polynomial if and only if c is a critical value of f (hence only the value0 can be atypical). Namely, let Er := x ∈ Kn | ρ(x) < r for some r > 0. Then the localMilnor fibre of f at 0 ∈ Kn (i.e. f−1(c)∩Eε, for some small enough ε and 0 < |c| ≪ ε) isdiffeomorphic to the global fibre f−1(c), since f−1(c) is transversal to ∂Er, ∀r ≥ ε.

The first pleasant property of ρ-regularity is that it implies topological triviality, moreprecisely we prove the following:

Proposition 3.6. If the fibre f−1(t0) is ρ-regular at infinity then f is topologically trivialat infinity at t0.

Proof. If f is ρ-regular at y ∈ τ−1(t0)∩X∞ then one can lift the (real or complex) vectorfield ∂/∂t defined in a neighbourhood D of t0 ∈ K to a (real or complex) vector field ξon U ∩ K

n tangent to the levels ρ =constant. If f−1(t0) is ρ-regular at infinity then wemay glue these local vector fields by a partition of unity and get a vector field definedin some neighbourhood of X∞ without X∞. This is a controlled vector field which canbe integrated to yield a topologically trivial fibration f| : (Kn \ K) ∩ f−1(D) → D, asshown by Verdier in his proof [Ve, Theorem 4.14] of Thom-Mather isotopy theorem [Th],[Ma]. (Notice however that Thom-Mather theorem does not directly apply since f is notproper.) We use here ρ as “fonction tapissante”, a notion introduced by Thom in [Th].In case n = 2, a similar procedure was used by Ha H.V. and Le D.T. [HaL].

To introduce the second regularity condition we need some preliminaries. First, letus define the relative conormal, following [Te2], [HMS] then state some technical resultswhich we need. Let X ⊂ K

N be a K-analytic variety. In the real case, assume that Xcontains at least a regular point. Let U ⊂ KN be an open set and let g : X ∩ U → K beK-analytic and nonconstant. The relative conormal T ∗

g|X∩U is a subspace of T ∗(KN )|X∩U

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defined as follows:

T ∗g|X∩U := closure(y, ξ) ∈ T ∗(KN ) | y ∈ X 0 ∩ U, ξ(Ty(g−1(g(y))) = 0 ⊂ T ∗(KN )|X∩U ,

where X 0 ⊂ X is the open dense subset of regular points of X where g is a submersion.The relative conormal is conical (i.e. (y, ξ) ∈ T ∗

g|X∩U ⇒ (y, λξ) ∈ T ∗g|X∩U , ∀λ ∈ K

∗). Thecanonical projection T ∗

g|X∩U → X ∩ U will be denoted by π.

Lemma 3.7. [Ti3] Let (X , x) ⊂ (KN , x) be a germ of an analytic space and let g :(X , x) → (K, 0) be a nonconstant analytic function. Let γ : X → K be analytic such thatγ(x) 6= 0 and denote by W a neighbourhood of x in KN . Then (T ∗

g|X∩W )x = (T ∗γg|X∩W )x.

Definition 3.8. Let Ui = xi 6= 0, for 0 < i ≤ n, be an affine chart of Pn and lety ∈ (Ui ×K)∩X∩x0 = 0. The relative conormal T ∗

x0|X∩Ui×Kis well defined. We denote

by (C∞K

)y the fibre π−1(y) and call it the space of characteristic covectors at infinity, at y.It follows from Lemma 3.7 that (C∞)y does not depend on the choice of affine chart Ui.

Definition 3.9. (t-regularity) We say that f (or that the fibre f−1(t0)) is t-regular aty ∈ X∞ ∩ τ−1(t0) if (y, dt) 6∈ (C∞

K)y.

We also say that f−1(t0) is t-regular at infinity if this fibre is t-regular at all its pointsat infinity.

Proposition 3.10. If f is t-regular at y ∈ X∞ then f is ρE-regular at y, where ρE is the

Euclidean norm.

Proof. The mapping d∞ : XK → R, defined by:

d∞(x, f(x)) = 1/ρ2E(x), for x ∈ Kn

d∞(y) = 0, for y ∈ X∞

is analytic and defines X∞. In the real case we have (T ∗d∞|X)y = (T ∗

|x0|2|X)y, by Lemma

3.7, and the latter is in turn equal to (T ∗x0|X

)y = (C∞R

)y. The ρE-regularity at y ∈ X∞ is

certainly implied by (y, dt) 6∈ (T ∗d∞|X)y, which is just t-regularity, by the above equalities.

This finishes the proof in the real case.Now the complex case. Let us first introduce the map ι : PT ∗(R2n) → PT ∗(Cn)

between the real and the complex projectivised conormal boundles (where R2n is the realunderlying space of Cn) defined as follows: if ξ is conormal to a hyperplane H ⊂ R2n thenι([ξ]) is the conormal to the unique complex hyperplane included in H . This is clearly acontinuous map. We then have the following equality:

P(C∞C

)y = ι(P(T ∗|x0|2|X

)y),

since the complex tangent space Txx0 =constant is exactly the unique complex hy-perplane contained into the real tangent space Tx|x0|

2 =constant. The equality fol-lows by the fact that ι commutes with taking limits. Now (y, dt) 6∈ (C∞

C)y implies

(y, ι−1([dt]) 6∈ P(T ∗|x0|2|X

)y, which in turn implies ρE-regularity at y since, as above in

the real case, we still have (T ∗d∞|X)y = (T ∗

|x0|2|X)y.

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Corollary 3.11. Let f : Cn → C be a complex polynomial. Then the set of values t0 suchthat f−1(t0) is not ρE-regular at infinity is a finite set. In particular, the set of atypicalvalues of f is finite.

Proof. Take a Whitney stratification W = Wii of XC with a finite number of strataand with Cn as a stratum. It turns out from [ST1, Lemma 4.2] or [Ti3, Theorem 2.9]that any pair of strata (Cn,Wi) with Wi ∈ X

∞ has Thom property with respect to thefunction x0, in any local chart. If τ−1(t0) is transversal to a stratum Wi ⊂ X∞, thenf−1(t0) is t-regular at infinity. Now the restriction of the projection τ : XC → C to astratum contained in X∞ has a finite number of critical values. These imply that thevalues t0 such that f−1(t0) is not t-regular at infinity are finitely many. The conclusionfollows by Proposition 3.10.

Remark 3.12. The t-regularity at some point y ∈ X∞ is implied by the transversality oft = t0 to the strata of W. In particular, if the polynomial f has isolated W-singularities,then these singularities are the only points where the corresponding fibres of f might benot t-regular. Indeed, this follows from the proof of Corollary 3.11 above, and was alsomentioned in [ST1, Remark 5.2].

Nevertheless, the ρE-regularity is really weaker than t-regularity. We show by the nextexample that the converse of Proposition 3.10 is not true. This makes the ρ-regularityinteresting and raises questions concerning real methods and their interplay with thecomplex ones.

Example 3.13. Let f be the following complex polynomial function f := x + x2y :C3 → C in 3 variables x, y, z. We show below that f is not t-regular at a whole lineL := x0 = x = t = 0 within X∞

C, hence has a nonisolated t-singularity.

On the other hand, there is a single point in X∞C

at which f is not ρE-regular. Thereforethe singularity is isolated in this sense. Here follow the computations.

Let F := xx20+x2y−tx3

0. The t-regularity in the chart y 6= 0 is equivalent to |y|·‖∂f∂x‖ 6→

0, as ‖x, y‖ → ∞, cf. [ST1] p. 780. But in our example |y| · ‖1 + 2xy‖ tends to 0, forinstance if y → ∞, x = 1/y3 − 1/(2y), z = ay. The limit points in X∞ are the 1-dimensional set L.

We now find the set of points (p, t) ∈ X∞ where f is not ρE-regular. This amounts tofinding the solutions of the equation grad f = (λx, λy, λz). One can assume λ 6= 0. Itfollows ‖x‖2x = y + 2x‖y‖2, z = 0. The solution is a 2-dimensional real algebraic set Aand the set we are looking for is A ∩ X∞. If we work in two variables instead, i.e. withg := x+ x2y : C2 → C, then the single point where g is not ρE-regular is x0 = x = t = 0.In our case (3 variables), one intersects with z = 0, hence A ∩ X

∞ consists of a singlepoint.

3.2. The relation to Malgrange condition. In the complex case, F. Pham formulatesa regularity condition which had been found by B. Malgrange [Ph, 2.1]. We give below adefinition which works also in the real case, together with a localised version at infinity.

Definition 3.14. We consider sequences of points xi ∈ Kn and the following properties:

(L1) ‖xi‖ → ∞ and f(xi) → t0, as i→ ∞.(L2) xi → y ∈ X∞

K, as i→ ∞.

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One says that the fibre f−1(t0) verifies Malgrange condition if there is δ > 0 such that,for any sequence of points with property (L1) one has

(M) ‖xi‖ · ‖ grad f(xi)‖ > δ.

We say that f verifies Malgrange condition at y ∈ X∞K

if there is δ > 0 such that one has(M), for any sequence of points with property (L2).

Note 3.15. It clearly follows from the definition that f−1(t0) verifies Malgrange conditionif and only if f verifies Malgrange condition at any y = (z, t0) ∈ X∞

K. We have proved in

[ST1, Proposition 5.5] that t-regularity implies the Malgrange condition, both at a pointor at a fibre (the proof works in the complex case as well as in the real one). Reciprocally,Malgrange condition implies t-regularity, as more recently proved in the complex case byParusinski [Pa2, Theorem 1.3]. In fact the same proof works over the reals. Briefly, wehave the following relations:

(2) Malgrange condition ⇐⇒ t-regularity =⇒ ρE-regularity.

The Malgrange condition at y ∈ X∞K

is equivalent to saying that the Lojasiewicz numberLy(f) at y is ≥ −1, where Ly(f) is defined as the smallest exponent θ ∈ R such that, forsome neighbourhood U of y and some constant C > 0 one has:

| grad f(x)| ≥ C|x|θ, ∀x ∈ U ∩ Cn.

There are two other regularity conditions used in the literature which are similar toMalgrange condition but clearly stronger than that: Fedoryuk’s condition (or tameness,see Proposition 5.1) [Fe], [Br1], [Br2] and Parusinski’s condition [Pa1].

4. Polar curves and regularity conditions

We further explain, in the complex case, the relation between t-regularity and affinepolar curves associated to f . Local-at-infinity and affine polar curves were used in severalpapers [ST1, Ti3, ST2, ST3] to get information on the topology of the fibres of f .

Given a polynomial f : Kn → K and a linear function l : Kn → K, one denotes byΓ(l, f) the closure in K

N of the set Crt(l, f) \Crtf , where Crt(l, f) is the critical locus ofthe map (l, f) : KN → K2. A basic and useful result is that Γ(l, f) is a curve (or empty)if l is general enough. We give below the precise statement (a particular case of [Ti3,Lemma 1.4]). For some hyperplane H ∈ P

n−1, one denotes by lH : Kn → K the unique

linear form (up to multiplication by a constant) which defines H .

Polar Curve Theorem 4.1. [Ti3]There exists an open dense set Ωf ⊂ Pn−1 (Zariski-open in the complex case) such that,for any H ∈ Ωf , the critical set Γ(lH , f) is a curve or it is empty.

Definition 4.2. For H ∈ Ωf , we call Γ(lH , f) the affine polar curve of f with respectto lH . A system of coordinates (x1, . . . , xn) in Kn is called generic with respect to f iffxi = 0 ∈ Ωf , ∀i.

It follows from Theorem 4.1 that such systems are generic among the systems of coor-dinates.

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We first relate the affine polar curves to the local polar curves at infinity, as follows.Let us consider the map germ:

(τ, x0) : (X, p) → (K2, 0),

for some p ∈ X∞, with critical locus denoted by Crtp(τ, x0). The local (nongeneric) polarlocus Γp(τ, x0) is defined as the closure of Crtp(τ, x0) \ X∞ in X.

Let p = ([0 : p1 : · · · : pn], α), where pi 6= 0, for some fixed i. In the chart Ui × K, thepolar locus Γp(τ, x0) is the germ at p of the analytic set Gi ⊂ X, where

Gi := ([x0 : x], t) ∈ X \ X∞ |

∂f (i)

∂x1= · · · =

ˆ∂f (i)

∂xi= · · · =

∂f (i)

∂xn= 0,

and f (i) := f(x0, x1, . . . , xi−1, 1, xi+1, . . . , xn).

On the intersection of charts (U0 ∩ Ui) × K, the function ∂f(i)

∂xjdiffers from ∂f(0)

∂xj, by a

nowhere zero factor, for j 6= 0, i.Thus the germ of Gi at p is the germ at p of the following algebraic subset of Pn × K:

closure(x, f(x)) ∈ Kn × K |

∂f

∂x1

= · · · =∂f

∂xi−1

=∂f

∂xi+1

· · · =∂f

∂xn

= 0,

which is equal to Γ(xi, f), where:

Γ(l, f) := closure(x, t) ∈ Kn × K | x ∈ Γ(l, f), t = f(x) ⊂ P

n × K.

One can now easily prove the following finiteness result:

Lemma 4.3. If the system of coordinates (x1, . . . , xn) is generic with respect to f , thenthere is a finite number of points p ∈ X∞ for which the polar locus Γp(τ, x0) is non-emptyand at such a point Γp(τ, x0) is a curve.

Proof. Fix a generic system of coordinates. By Theorem 4.1, the set Γ(xi, f) is a curve(or empty) and therefore Γ(xi, f) ⊂ X is a curve too (or empty), ∀i ∈ 1, . . . , n. We haveshown above that, for a point p ∈ X∞ with pi 6= 0, one has the equality of germs Γp(τ, x0) =

Γ(xi, f)p. Then the assertion follows by the fact that the intersection X∞ ∩ (∪n

i=1Γ(xi, f)is a finite set.

In the complex case, the affine polar curves are closely related to the isolated t-singularities, defined as follows.

Definition 4.4. We say that f has isolated t-singularities at the fibre f−1(t0) if this fibrehas isolated singularities and the set p ∈ X∞ | f−1(t0) is not t-regular at p is a finiteset.

Let us also remind that if a complex polynomial f has no W-singularity at some pointp ∈ X∞ then f is t-regular at p. This was indicated in the proof of Corollary 3.11.

Proposition 4.5. Let f : Kn → K be a polynomial function and let p ∈ X∞ ∩ τ−1(t0).Then the following are equivalent:

(a) Γp(t, x0) 6= ∅.

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(b) Γ(xi, f) ∋ p for some i.

If K = C and f has isolated t-singularities at f−1(t0) then these conditions are moreoverequivalent to the following ones:

(c) p is a t-singularity.(d) λp 6= 0, where λp is the Milnor number at infinity at p, as defined in [ST1, Definition

3.4].

Proof. The equivalence (a) ⇐⇒ (b) follows from the already proved equality Γp(t, x0) =Γ(xi, f)p.

The equivalence (a) ⇐⇒ (c) is Proposition 5.3 in [ST1] and (c) ⇐⇒ (d) is a consequenceof [ST1, Proposition 4.5] and [Ti3], see Remarks 4 bellow.

In the case of a nonisolated t-singularity at p ∈ X∞, the general affine polar curve mightbe empty at p. See Example 3.13 for a discussion of such a case.

On the Milnor number at infinity. In case of a complex polynomial with isolatedt-singularities, we prove in [ST1, Corollary 3.5 (b)] the following formula, which shows inparticular that the fibre Fa := f−1(a) is atypical if and only if its Euler characteristic isdifferent from the one of a general fibre Fu := f−1(u):

(3) χ(Fa) − χ(Fu) = (−1)n(µFa+ λFa

),

where µFais the sum of the Milnor numbers at the isolated singularities on the fibre Fa

and λFais the sum of the so-called Milnor numbers at infinity at the t-singularities on

X∞ ∩ τ−1(a).In the more general case of a polynomial with isolated singularities in the affine space,

the t-singularities at infinity may be non-isolated and one does not have numbers λp

anymore. However, one can still give a meaning to the number λFaby taking the relation

(3) as its definition:

Definition 4.6. Let f : Cn → C be a polynomial with isolated singularities. We callEuler-Milnor characteristic at infinity of the fibre Fa the following number:

λFa:= (−1)n−1(χ(Fu) − χ(Fa)) − µFa

.

If λFa= 0 then Fa is a typical fibre, whereas the converse is not true in general.

However, one has the following interesting statement, which gives an extension, at theEuler characteristics level, of [ST1, Corollary 3.5(a)]:

Proposition 4.7. If f : Cn → C is a polynomial with isolated singularities then

χ(Fu) = 1 + (−1)n−1∑

a∈Λ

(λFa+ µFa

),

where Λ is the set of atypical values of f and u ∈ Λ.

Proof. Let δa be a small disc centered at a ∈ C, which does not contain any other atypicalvalue. Let γa, a ∈ Λ, be suitable paths (non self-intersecting, etc.) from a to a typicalvalue u. Then 1 = χ(Cn) = χ(

⋃

a∈Λ(f−1(δa) ∪ f−1(γa)) = χ(Fu) +∑

a∈Λ[χ(Fa) − χ(Fu)],by retraction and by a standard Mayer-Vietoris argument.

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Example 7.14. fs = x4 + sz4 + z3 + y.This is a deformation inside the B-class with constant µ+ λ, which is not cgst at infinity(Definition 7.4). We have λ = µ = 0 for all s. Next, Ys,t is singular only at p := ([0 :1 : 0], 0) and the singularities of Y∞

0,t change from a single smooth line x4 = 0 with

a special point p on it into the isolated point p which is a E7 singularity of Y∞s,t. We

use the notation ⊕ for the Thom-Sebastiani sum of two types of singularities in separatevariables. We have:s = 0: the generic type of Y0,t at infinity is A3 ⊕ E7 with Milnor number 21, andχ(Y∞

0,t) = 2.s 6= 0: the generic type of Ys,t at infinity is A3⊕E6 with Milnor number 18, and χ(Y∞

s,t) =5.

The jumps of +3 and −3 compensate each other in the formula (7).

Example 7.15. fs = x4 + sz4 + z2y + z.This is a µ + λ constant B-type family, with two different singular points of Y0,t atinfinity, and where the change in one point interacts with the other. It is locally cgst inone point, but not in the other. We have that λ = 3 and µ = 0 for all s, Ys,t is singular inp := ([0 : 1 : 0], 0) for all s (see types below) and in q := ([1 : 0 : 0], 0) for s = 0 with typeA3. The singularities of Y∞

s,t change from a single smooth line x4 = 0 into the isolated

point p with E7 singularity.For the point p we have for all s the generic type A3 ⊕ D5 if t 6= 0, which jumps to

A3 ⊕D6 if t = 0. This causes λ = 3.At q, the A3-singularity for s = 0 gets smoothed (independently of t) and here the

deformation is not locally cgst. The change on the level of χ(Y∞s,t) is from 2 to 5, so

∆χ∞ = −3, which compensates the disappearance of the A3-singularity from Y0,t to Ys,t.

Example 7.16. fs = x2y + x + z2 + sz3.This is a cgst B-type family, where µ+ λ is not constant. Notice that fs is F-type for alls 6= 0, whereas f0 is not F-type (but still B-type). The generic type at infinity is D4 forall s and there is a jump D4 → D5 for t = 0 and all s. For s 6= 0 a second jump D4 → D5

occurs for t = c/s2, for some constant c.There are no affine critical points, i.e. µ(s) = 0 for all s, but λ(s) = 2 if s 6= 0 and

λ(0) = 1. We have that Λ(fs) = 0, c/s2 for all s 6= 0, and that χ∞ changes from 3 ifs = 0 to 2 if s 6= 0, so ∆χ∞ = +1.

There is a persistent λ-singularity in the fibre over t = 0 and there is a branch of thecritical locus CritΨ which is asymptotic to t = ∞.

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[SS] D. Siersma, J. Smeltink, Classification of singularities at infinity of polynomials of degree 4 in

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Mathematisch Instituut, Universiteit Utrecht, PO Box 80010, 3508 TA Utrecht The

Netherlands.

E-mail address: [email protected]

Mathematiques, UMR 8524 CNRS, Universite des Sciences et Technologies de Lille,

59655 Villeneuve d’Ascq, France.

E-mail address: [email protected]

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Vanishing cycles of polynomial functions

Lecturers: Dirk Siersma (University Utrecht, The Netherlands) , Mihai Tibăr (Université de Lille 1, France) Outline of the course: We intend to study the behaviour of polynomial functions from complex n-space to the complex numbers. The topology of a general fibre is related to the singularities of the polynomial function. These singularities can occur locally in the affine plane or at infinity. Local case: We will start with defining singular points,the Milnor fibration and its mononodromy and study some properties of the Milnor fibre. In case of an isolated singularity we will show the bouquet theorem for the Milnor fibre and later we will extend the proof to the global case. We next will discuss some properties of non-isolated singularities; especially those with a 1-dimensional singular set. An important tool will be the Lê attaching method, where one uses slicing for computing the Euler characteristics of the Milnor fibres. Global case: We will show a bouquet theorem for (global) polynomial functions with isolated singularities both in the affine space and at infinity (to be defined carefully). The number of spheres can be computed in two ways (by counting the number of vanishing cycles or with information about the boundary singularities). Next we will focus on recent results on Betti Bounds of polynomials. Given degree and number of variables there is a maximum of the top Betti number of a general fibre. We will study those functions where the top Betti number is near to this maximum. We will describe a range in which the polynomial must have isolated singularities and another range where it may have at most non-isolated singularities of a very special kind: line singularities of Morse transversal type. Our method uses deformations into particular pencils with non-isolated singularities. References: (attached to the website) 1. Betti bounds for Polynomiials: To appear in Moscow Mathematical Journal 2011 2. Reader for this course.

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vanishing cycles of polynomial functions

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