on the lê-geuel formula for the milnor numberrottner/gmg70/download/seade.pdf · thenmilnor’s...

On the Le-Geuel Formula for the Milnornumber

Jose Seade1

1Instituto de Matematicas,Universidad Nacional Autonoma de Mexico.

Lambrecht, Germany22nd June, 2015

To Gert-Martin

always an example for his quality and elegance inmathematics, his diversity, kindness, generosity, and more !!

Seade On the Le-Geuel Formula for the Milnor number

THE GENERAL SETTING• Springs from classical ideas (Lefschetz, Thom and others:)

• Study the topology of complex singularities by an inductiveprocess, taking slices by general hyperplanes. Then use theseto build-up the topology (of the Milnor fibre, of the space, ....)

• A natural continuation is to consider slices by a function withan isolated singularity, instead of by a linear function.

• Then we are in the setting we’ll study today:

The classical Le-Greuel formula• This somehow goes the other way round: given an ICIS germdefined by (f1, ..., fk ,g), study its topology by comparing it withthe ICIS defined by (f1, ..., fk ).


• I talk first about complex singularities.

• Then I’ll move to real singularities.


• Starting point is the classical: Milnor’s fibration theorem andits generalizations by Hamm, Le and others.

• Consider a holomorphic function with a critical point at 0.

f : (Cn+1,0)→ (C,0)

Set V = f−1(0) and K = V ∩ Sε the link.

• Given ε > 0 as above, choose 0 < δ << ε ; set D∗δ := Dδ \ {0}and consider the “Milnor tube”:

N(ε, δ) = f−1(D∗δ) ∩ Bε

Then Milnor’s theorem can be stated as saying that we have alocally trivial fibration:

f : N(ε, δ) −→ D∗δ


Figure : A Milnor tube N(ε, δ) = f−1(D∗δ) ∩ Bε


Moreover, we know that if f has an isolated critical point, then:

Ft :=(

f−1(0) ∩ Bε)'∨µ(f )

Sn

a bouquet of spheres, the number of spheres in this bouquet isthe Milnor number µ(f ). We get:

χ(Ff ) = 1 + (−1)nµ(f )

and µ(f ) can be computed as the local IndPH(∇f ). Hence

µ(f ) = dimCOn+1,0(

∂f∂z1, · · · , ∂f

∂zn+1

) .


Soon after Milnor, H. Hamm extended this theorem for ICIS.So ICIS also have a well-defined Milnor number := rank of themiddle homology of Milnor fibre.

Since for hypersurfaces the Milnor number is:


∂f∂z1, · · · , ∂f

∂zn+1

) .It is natural to search for an algebraic expression for thisinvariant in the case of ICIS germs

In very early 1970’s Le Dung Trang and Gert-Martin Greuelproved, independently, what we know today as the Le-Greuelformula for the Milnor number of an ICIS germ:


Consider f = (f1, · · · , fk ) and g, holomorphic map germs(Cn+k ,0)→ (C,0) such that f and (f ,g) define isolatedcomplete intersection germs.Then their Milnor numbers are related by:

µ(f ) + µ(f ,g) = dimCOn+k ,0

(f , Jack+1(f ,g)),

where Jack+1(f ,g) denotes ideal generated by determinants ofall (k + 1) minors of the Jacobian matrix.


AIM: Generalize this theorem to:

i) Holomorphic map-germs on singular varieties; and

ii) For real singularities.

• Related work by J. J. Nuno-Ballesteros, Bruna Okamoto, andJ. N. Tomazella.

• and also by N. Dutertre and N. de Goes Grulha.

In the two cases the idea of the proof is inspired by my paperwith Brasselet and Le about the local Euler obstruction.

Key point was constructing appropriate vector fields on thesingular variety, and lift these to the Nash blow up.

Here the constructions are similar, but we replace the Nashblow up by the Milnor fibre.

We discuss first the complex case.


THE COMPLEX CASE

This is joint work with R. Callejas-Bedregal, M. Morgadoand M. Saia. To appear in Tohoku Math. J.

Recall:

Theorem (Milnor- Le fibration theorem ≈ 1975)

Let (X ,0) be a germ of complex analytic variety andf : (X ,0)→ (C,0) holomorphic. Set V = f−1(0), K = V ∩ Sεthe link and N(ε, δ) = f−1(∂Dδ) ∩ Bε. Then:

f : N(ε, δ) −→ ∂Dδ ∼= S1

is a locally trivial fibration.

Notice that if the germ (X ,0) is not an isolated singularity, thenthe Milnor fibre Ff is itself singular.


Want a “Le-Greuel type” formula for functions on X :


(f , Jack+1(f ,g)),

Yet, in general there is not a Milnor number in this setting,hence a Le-Greuel formula makes no sense as such.

This can be re-formulated as:

χ(Ff ) = χ(Ff ,g) + indPH(g,0,∇(g|Ff )

)where the latter term is the Poincare-Hopf index.

Notice that the term on the right, indPH(g,0,∇(g|Ff )

), is the

so-called GSV-index of the vector field ∇(g|V\{0})

This is the expression that generalizes.


Recall usual definition: Given an ICIS germ at 0 defined byf : (Cm,0)→ (Ck ,0), m > k , and the restriction to V (f ) of acontinuous vector field v in Cm, which has an isolatedsingularity at 0 and is tangent to V (f ), then:

DefinitionThe GSV-index of v on V (f ) is the Poincare-Hopf index of acontinuous extension of v to a Milnor fibre of f .

If f is regular at 0, this is the usual local index of P-H.


In other words, can think we have a family of vector fields vttangent to the Milnor fibers and varying continuously with theparameter t .

The singularities of vt ’s collapse into 0 as the Milnor fibersdegenerate to V (f ). GSV-index = sum of the PH-indices in theMilnor fibre.

We need to extend this notion to the more general setting weenvisage here, where the Milnor fibre is itself singular, sothe PH-index makes no sense.


We use the radial index, introduced by H. King and D. Trotmanin 1992, just published in 2014. (Later Ebeling-Gusein Zadeand Aguilar-Seade-Veejovsky.) This is defined always for everycontinuous vector field on arbitrary real or complex analyticsingularity germs:

Indrad(v ; V ) = 1 + dif(v , vrad) .


Now X is a complex analytic singular variety of dimension n + kin an open neighborhood of 0 ∈ Cm,

f : (X ,0)→ (Ck ,0) is a holomorphic function which isgenerically a submersion with respect to some complexanalytic Whitney stratification {Sα} of X .

We assume further that V := V (f ) has dimension more than 0and f has the Thom af -property with respect to the abovestratification.

Consider too a holomorphic map-germ g : (X ,0)→ (C,0) withan isolated critical point at 0, both on X and also on V (f ) (forthe given Whitney stratification).


Recall that if V (f ) is an ICIS in Cm then the GSV-index of v isthe PH-index of a continuous extension of v to a Milnor fibre.

Now V (f ) is an ICIS in X and may have arbitrary singular set.

DefinitionThe GSV-index of g on V relative to the function f is, bydefinition, the total Schwartz index of the conjugate gradientvector field ∇Ff (g) on the Milnor fiber Ff :

indGSV(g,0; f ) := indSch(∇Ff (g); Ff ) .

Easy to see this coincides with previous definition when X isitself an ICIS.


Me may now state:

Theorem (Generalized Le-Greuel formula)

Let g : (X ,0)→ (C,0) have isolated critical point at 0 in thestratified sense, both in X and in V (f ); let Ff ,g be Milnor fibre of(f ,g). Equip Ff with obvious stratification. Then:

χ(Ff ) = χ(Ff ,g) + indGSV(g,0; f ) ,

The proof is mostly topological, done by constructingappropriate vector fields and counting indices.

Notice that in the classical formula the right hand sidespecializes to an invariant determined by the local algebra ofV (f ). This does not happen in general.


REAL ANALYTIC MAP-GERMS

Joint work with: J.-L- Cisneros and Nivaldo de GoesGrulha. To appear in Int. J. Maths.

Recall Milnor proved in his book that given an analyticmap-germ

f : (Rn,0)→ (Rp,0) , n > p ≥ 1

with an isolated critical point at 0, one has a locally trivialfibration in a tube:

f : Nf (ε, δ) := Bε ∩ f−1(∂Dp) −→ ∂Dp ∼= Sp−1

Alas this is very restrictive.

Later [Pichon-Seade] and others: isolated critical value ... moregeneral, still very stringent.


We consider the following fairly general setting: an analyticmap-germ with arbitrary critical value at 0,

f : (Rn,0)→ (Rp,0) , n > p ≥ 2 ,

with the Thom property (can be relaxed). Assume further thatdim V (f ) > 0 and f is locally surjective at 0:

Set: Bε = closed ε-ball around 0 in Rn. Consider restriction f |Bε

(denoted just f ).

Set Cf = critical points of f in Bε.

Denote ∆f = f (Cf )= discriminant of f .


Dpδ be an open Milnor ball in Rp around 0, radius 0 < δ � ε.

Nf (ε, δ) = Bε ∩ f−1(Dpδ \∆f ) solid Milnor tube

Nf (ε, δ) = Bε ∩ f−1(∂Dpδ \∆f ) Milnor tube

Notice blue part is now inverse image of the whole discriminant


Following result is essentially due to Pham(“Singularities ofintegrals”, Springer textbook, 2011):

Theorem (Milnor-Le type fibrations)

Let f : (Rn,0)→ (Rp,0) be as before. Then the restrictions

f : Nf (ε, δ)→ Dpδ \∆f ,

and

f : Nf (ε, δ)→ ∂Dpδ \∆f ,

are locally trivial fibrations.

Notice the base can have several connected components fibers with possibly different topology


Example:

f is homogeneous degree d ≥ 2.

Homogeneity implies that if y ∈ Rk is a critical value, then thewhole line of points of the form {t y} with t ∈ R are criticalvalues.

⇒ discriminant is a union of straight lines.

Case p = 2, degree = 2 get several sectors in R2. This hasbeen studied by Santiago Lopez de Medrano (ProceedingsHironaka’s fest in Valladolid). He determines topology of thefibers in each sector.


As before: Ff is a Milnor fibre of Rn f→ Rp, analytic with criticalpoint at 0, with the Thom af -property and dim V (f ) > 0.

Consider map (Rn,0)g→ (Rk ,0), k ≥ 1, with isolated critical

point in Rn with respect to a Whitney stratification adapted toV (f ).

TheoremThe map-germs f and (f ,g) have associated locally trivialfibrations of the Milnor-Le type (in tubes) and

χ(Ff ) = χ(Ff ,g) + IndPH∇g|Ff ,

where g : Rn → R is given by g(x) = ‖g(x)− t0‖2 with t0 ∈ Rk

such that Ff ,g = g|−1Ff

(t0).


Example: Let (f ,g) : R4 → R3 where:

f (t , x , y , z) = (t , x2 + y2 − z2) and g(t , x , y , z) = t + z

V (f ) is the cone x2 + y2 − z2 = 0 in the hyperplane t = 0.

The critical set Cf is the t-axis and f |Cfis 1-to-1.

R2 \ discriminant has two connected components.

Natural stratification of R4: the origin; the t-axis minus theorigin; the cone V (f ) minus the origin; and the complement ofV (f ) ∪ Cf in R4.

Stratification of R2: the origin; the discriminant ∆ minus theorigin; and one 2-dimensional stratum given by the complementof ∆ in R2.

f satisfies the Thom af for stratification.


The Milnor fibres of f are of two types (take a constant):

F(a,b) = f−1(a,b) =

{a one-sheeted hyperboloid if b > 0,a two-sheeted hyperboloid if b < 0.

g has an isolated critical point at the origin of R4 with respect tothe stratification.

When b > 0 , g restricted to F (a,b) has no critical points.When b < 0 the critical points of g are the vertices of thetwo-sheeted-hyperboloid.

Easy computation shows χ(Fh) = 0 in all cases, and theLe-Greuel formula yields:

χ(F(a,b)) =

{0 + 0 = 0 if b > 0 (F(a,b) is one-sheeted hyperboloid),0 + 2 = 2 if b < 0 (F(a,b) is two-sheeted hyperboloid).


Algebraic formulaRecall that in holomorphic case, if v = (v1, ..., vn) is a hol.vector field with iso. sing. at 0 ∈ Cn then:

IndPH(v ,0) = dimCOn,0

(v1, ..., vn)

This leads to algebraic interpretation of the Milnor number forhypersurfaces:


∂f∂z1, · · · , ∂f

∂zn+1

) .and eventually leads to the Le Greuel formula:


(f , Jack+1(f ,g)),

In our discussion, Milnor numbers are being replaced by theEuler characteristics of the fibers. What about the expressionon the right? Recall this is an index of vector fields.


Real analytic case is more delicate.

A first difference is that the index can be negative. So no hopethis can be computed as the dimension of some vector space.

What follows is largely inspired by work of X. Gomez-Mont, L.Giraldo and P. Mardesic, which is itself inspired by thecelebrated algebraic formula for the local index byEisenbud-Levine and Khimshiasvili.

This expresses the local index as the signature of a certainbilinear form defined on local algebra of v . Let us explain:

Let ARn,0= local ring of germs of real analytic real-valuedfunctions.v = (v1, · · · , vn) at 0 ∈ Rn, germ of analytic vector field withalgebraically isolated sing. at 0Bv be the local algebra of v ,

Bv = ARn,0/(v1, · · · , vn),


Denote by Jv ∈ Bv the function x 7→ determinant of theJacobian matrix of v .For every linear form φ : Bv → R one gets a bilinear form:〈 , 〉φ : Bv × Bv → R by:

〈f ,g〉φ = φ(f g).

Theorem (Eisenbud-Levine, Khimshiasvili)

One can always choose the linear form φ so that φ(Jv ) > 0, andin this case one has:

IndPH(v ,0) = Signature(〈 , 〉φ),

independently of the choice of φ.


More generally, one can consider analytic vector fields on realanalytic hypersurface germs in Rn+1: work by X. Gomez-Mont,L. Giraldo and P. Mardesic

1 Get algebraic formulae for the total index of vector field ona Milnor fibre.

2 If n is even, formula depends on right Milnor fibers and leftfibers.

Using their work we get an algebraic Le-Greuel formula for realsingularities which are complete intersections defined by twoequations.


Theorem

Let f ,g be real analytic map-germs (Rn+1,0)→ (R,0) with analgebraically isolated critical point at 0, which define a completeintersection germ in Rn+1. Suppose (for simplicity) that n > 1 isodd. Set V (f ) := f−1(0), let g be the restriction of g to V (f ) andlet v := ∇g be the gradient vector field of g.Then one has:

χ(Ff ) = χ(Ff ,g) + Sgn(V (f ),0)(v)− SgnA(hv ),

where Ff and Ff ,g are as before. The termSgn(V (f ),0)(v)− SgnA(hv ) is determined as above by the localalgebra of f and g (signatures of certain quadratic formsassociated to local algebras which are concentrated on thelocal germs defined by f , g.)

If their work is extended to ICIS in general, then we can extendformula above to ICIS in general (not done yet)


I believe these are some steps in a direction in which there isyet a lot to be done.

Danke schon

und ...


Gluckwunsche Gert-Martin!!


on the lê-geuel formula for the milnor numberrottner/gmg70/download/seade.pdf · thenmilnor’s...

Documents