Remarks on the Milnor number e Seade 1 Jos 1 Instituto de Matem - - PowerPoint PPT Presentation

Remarks on the Milnor number

Jos´ e Seade1

1Instituto de Matem´

aticas, Universidad Nacional Aut´

• noma de M´

exico.

Liverpool, U. K. March, 2016 In honour of Victor!!

Seade Remarks on the Milnor number

§1 The Milnor number

Consider a holomorphic map-germ f : (Cn+1, 0) → (C, 0) with a critical point at 0. Let V = f −1(0) and K = V ∩ Sε the

• link. Milnor’s classical theorem (1968) says that we have a

locally trivial fibration: φ := f |f| : Sε \ K − → S1. Alternative description. Given ε > 0 as above, choose 0 < δ << ε and set N(ε, δ) = f −1(∂Dδ) ∩ Bε. Then: f : N(ε, δ) − → ∂Dδ ∼ = S1 is a locally trivial fibration, equivalent to previous one.

Seade Remarks on the Milnor number

Seade Remarks on the Milnor number

When f has an isolated critical point, Milnor proved that the fiber: Ft := f −1(t) ∩ Bε has the homotopy type of a bouquet of spheres of middle dimension: Ft ≃

• µ

Sn The number of spheres in this wedge is, by definition, the Milnor number of f; usually denoted µ(f) (or simply µ) By definition one has: µ = Rank Hn(Ft)

Seade Remarks on the Milnor number

Milnor also proved that µ actually is the Poincar´ e-Hopf local index of the gradient vector field ∇f. Hence it is an intersection number: µ = dimC On+1,0 Jac(f) where Jac(f) is the Jacobian ideal of f (generated by its partial derivatives). This number is also known as the Milnor number of the hypersurface germ (V, 0) where V = f −1(0). This is an important invariant that has played a key-role in singularity theory.

Seade Remarks on the Milnor number

These results were soon generalized by H. Hamm to ICIS: f := (f1, · · · , fk) : (Cn+k, 0) → (Ck, 0) Such a germ also has an associated Milnor fibration and a well-defined Milnor number: the rank of the middle-homology

• f the Milnor fibre (= the number of corresponding spheres).

So that ICIS germs also have a well-defined Milnor number.

Seade Remarks on the Milnor number

§2 Laufer’s Formula for the Milnor number

Two natural approaches to studying ICIS V := f −1(0), f := (f1, · · · , fk) : (Cn+k, 0) → (Ck, 0) i) Looking at local non-critical levels f −1(t) and the way how these degenerate to V; ii) Looking at resolutions of the singularity π : V → V. Laufer (1977) built a bridge between these two viewpoints. This was for n = 2. Later generalized by Looijenga to higher

• dimensions. We focus on case n = 2.

Seade Remarks on the Milnor number

Let (V, p) be a normal surface singularity germ,

• V a good resolution; K its canonical class, well defined by the

adjunction formula: 2gEi − 2 = Ei · (K + Ei) for each irreducible component of the exceptional divisor in

• V. Laufer (1977)

proved: µ(V) + 1 = χ( V) + K 2 + 12ρg(V) where: χ = usual Euler characteristic ; K 2 = self-intersection number; and ρg := dim H1( V, O) = geometric genus. Left hand side has no a priori meaning if the singularity is not an ICIS. Right hand side is always a well-defined integer for all normal, numerically Gorenstein, surface singularities, independent of all choices.

Seade Remarks on the Milnor number

Definition For every normal numerically Gorenstein surface singularity germ (V, p) we may call the integer La(V, p) = χ( V) + K 2 + 12ρg(V) , the Laufer invariant of (V, p). Question What is La(V, p) when the germ is not an ICIS? I will consider two cases: a) the singularity germ is smoothable; b) The germ is non-smoothable.

Seade Remarks on the Milnor number

Definition For every normal numerically Gorenstein surface singularity germ (V, p) we may call the integer La(V, p) = χ( V) + K 2 + 12ρg(V) , the Laufer invariant of (V, p). Question What is La(V, p) when the germ is not an ICIS? I will consider two cases: a) the singularity germ is smoothable; b) The germ is non-smoothable.

Seade Remarks on the Milnor number

From now on (V, p) is a normal surface singularity. Recall: 1) The germ of V at p is Gorenstein if its canonical bundle K = ∧2(T ∗(V \ {p}) is holomorphically trivial. (In this setting, this is equivalent to usual definition of a Gorenstein singularity) The germ of V at p is numerically Gorenstein if the bundle K is topologically trivial. 2) The germ of V at p is smoothable if there exists a 3-dimensional complex analytic space W and a flat map F : W → C such that F−1(0) is isomorphic to the germ (V, p) and F−1(t) is smooth for t = 0.

Seade Remarks on the Milnor number

Some remarks before continuing: 1) All hypersurface germs are Gorenstein and smoothable; the Milnor fibration is the smoothing (unique up to equivalence). 2) The same statement holds for ICIS germs. 3) There exist normal surface Gorenstein singularities which are non-smoothable. 4) There exist normal surface singularities which have many non-equivalent smoothings.

Seade Remarks on the Milnor number

One has Theorem (Greuel-Steenbrink 1981) Every smoothable Gorenstein normal surface singularity (V, p) has a well-defined Milnor number µGS: The 2nd Betti-number of a smoothing (and b1 vanishes). Theorem (Steenbrink 1981) Furthermore, this invariant satisfies Laufer’s formula: µGS + 1 = χ( V) + K 2 + 12ρg(V) That is, for smoothable Gorenstein singularities the Laufer invariant is µGS + 1. What if there is no smoothing? We come back to this later. First a digression.

Seade Remarks on the Milnor number

One has Theorem (Greuel-Steenbrink 1981) Every smoothable Gorenstein normal surface singularity (V, p) has a well-defined Milnor number µGS: The 2nd Betti-number of a smoothing (and b1 vanishes). Theorem (Steenbrink 1981) Furthermore, this invariant satisfies Laufer’s formula: µGS + 1 = χ( V) + K 2 + 12ρg(V) That is, for smoothable Gorenstein singularities the Laufer invariant is µGS + 1. What if there is no smoothing? We come back to this later. First a digression.

Seade Remarks on the Milnor number

§3 Rochlin’s signature theorem and the geometric genus

Recall that if X is a closed oriented 4-manifold, cup product determines a non-degenerate bilinear form: H2(X; R) ∪ H2(X; R) − → H4(X; R) ∼ = R Its signature is the signature of X, σ(X) ∈ Z. Classical Rochlin’s theorem (1951) says that if X is spin, then its signature is a multiple of 16: σ(M) ≡ 0 mod (16) What if M is not-necessarily spin? Recall spin means Stiefel-Whitney class ω2(M) = 0. Not all manifolds are. Yet: Every closed oriented 4-manifold is spinc: There is a class in H2(M; Z) whose reduction modulo 2 is ω2(M) = 0.

Seade Remarks on the Milnor number

If M is a complex surface it is canonically spinc and its canonical class KM reduced modulo 2 gives ω2(M). Definition (Rochlin, 1970s) Let W be an oriented 2-submanifold of a closed oriented M4. W is a characteristic submanifold if [W] ∈ H2(M; Z) reduced modulo 2 is ω2(M) . Notice KM can always be smoothed C∞ and the smoothing is a characteristic submanifold. If KM is even, ∅ is characteristic (and M is spin)

Seade Remarks on the Milnor number

Theorem (Rochlin, Kervaire, Milnor, Casson, Kirby-Freedman 1970s) Let W be a characteristic sub manifold of M, then σ(M) − W 2 ≡ 8Arf W mod (16) where Arf W ∈ {0, 1} is an invariant associated to H1(W; Z2). If W is characteristic in M, then W is equipped with a spin structure and: Arf W = 0 ⇔ W is a spin boundary This theorem has a nice re-interpretation for complex surfaces:

Seade Remarks on the Milnor number

Remark first Thom-Hirzebruch signature theorem: σ(M) = 1 3 p1(M)[M] where p1 = Pontryagin class. For compact complex surfaces

• ne has

p1 = c2

1 − 2c2

, c1(M) = −KM and c2(M)[M] = χ(M). Recall the 2nd Todd polynomial is

1 12(c2 1 + c2). Hence

σ(M) − K 2 = −8Td(M)[M] Thus, if W = K is a C∞ smoothing of the canonical divisor K, then Rohlin’s theorem can be restated as: Td(M)[M] ≡ Arf K (24),

Seade Remarks on the Milnor number

Furthermore, by Hirzebruch-Riemann-Roch’s theorem the Todd genus equals the analytic Euler characteristic: Td(M)[M] = χ(M, OM) so Rohlin’s theorem can be restated as: χ(M, OM) ≡ Arf K mod (2) . We get: Theorem For complex surfaces, Rochlin’s theorem is equivalent to saying that the analytic Euler characteristic is an integer and its parity is determined by the invariant Arf K. Want similar expression in algebraic geometry, not with a topological smoothing of K.

Seade Remarks on the Milnor number

Definition (Esnault-Seade-Viehweg 1992) A characteristic divisor W of M is a divisor of a bundle L of the form L = KM ⊗ D−2 Notice that such W = KM − 2D represents a homology class whose reduction modulo 2 coincides with that of K Definition Let W be a characteristic divisor of M. Define its mod (2)-index by: h(W) = dim H0(W, D|W) mod 2 Theorem (Atiyah, Libgober) If W is non-singular, this is the invariant in Rochlin’s theorem In particular, for anti-canonical class −K = −KM one has D = KM: h(−K) = dim H0(−K, KM|K) mod 2

Seade Remarks on the Milnor number

Definition (Esnault-Seade-Viehweg 1992) A characteristic divisor W of M is a divisor of a bundle L of the form L = KM ⊗ D−2 Notice that such W = KM − 2D represents a homology class whose reduction modulo 2 coincides with that of K Definition Let W be a characteristic divisor of M. Define its mod (2)-index by: h(W) = dim H0(W, D|W) mod 2 Theorem (Atiyah, Libgober) If W is non-singular, this is the invariant in Rochlin’s theorem In particular, for anti-canonical class −K = −KM one has D = KM: h(−K) = dim H0(−K, KM|K) mod 2

Seade Remarks on the Milnor number

Definition (Esnault-Seade-Viehweg 1992) A characteristic divisor W of M is a divisor of a bundle L of the form L = KM ⊗ D−2 Notice that such W = KM − 2D represents a homology class whose reduction modulo 2 coincides with that of K Definition Let W be a characteristic divisor of M. Define its mod (2)-index by: h(W) = dim H0(W, D|W) mod 2 Theorem (Atiyah, Libgober) If W is non-singular, this is the invariant in Rochlin’s theorem In particular, for anti-canonical class −K = −KM one has D = KM: h(−K) = dim H0(−K, KM|K) mod 2

Seade Remarks on the Milnor number

We have: Theorem (Esnault-Seade-Viehweg) The parity of the analytic Euler characteristic coincides with the mod (2) index h(−K): h(−K) = χ(M, OM) mod (2) . More generally, let W = KM − 2D be a characteristic divisor, where D is a divisor of some holomorphic bundle D. Then: h(W) = χ(M, D) mod 2 , where χ(M, D) = 2

i=0(−1)ihi(M, D) is the analytic Euler

characteristic of M with coefficients in D.

Seade Remarks on the Milnor number

Question: Is this congruence mod (2) reduction of some equality? This would provide an integral lifting of Rochlin’s theorem in the case of complex manifolds. In the case of complex manifolds, this is equivalent to asking: Can Rochlin’s theorem be improved to something like: χ(M, D) = G(W) for some integral invariant of a characteristic divisor W? The answer is positive.

Seade Remarks on the Milnor number

Theorem Let M and W = KM − 2D be as above. If W = 0, then: χ(M, D) = h0(W; D|W) − R , with R an even integer associated to the divisor W: R = h1(M; D) − 2h2(M; D) + dim Ker(ˆ β) , where ˆ β is a skew symmetric bilinear form on H1(M; D). Problem now is understanding R, perhaps relating it with more recent invariants of low dimensional manifolds. Of course this can be regarded from the viewpoint of the Atiyah-Singer index theorem

Seade Remarks on the Milnor number

Now back to singularities: Consider a normal surface singularity germ (V, p) that we can assume algebraic. Take a compactification of it and resolve all singularities. With some extra work we get: Theorem (Esnault-Seade-Viehweg) The parity of the geometric genus coincides with the mod (2) index h(−K). That is: dim H1( V, O

V) ≡ dim H0(−K, K V|K)

mod 2 Furthermore, if the resolution V is minimal, then: dim H1( V, O

V) = dim H0(−K, K|K)

and for all vertical divisors D ≥ 0 and W = 2D − K we have: dim H1( V, O

V) = dim H0(W, D|W) + 1

8(W 2 − K 2) .

Seade Remarks on the Milnor number

§4 Laufer’s formula revisited

We know that if (V, P) is a normal Gorenstein smoothable singularity, then: µ(V) + 1 = χ( V) + K 2 + 12ρg(V) As noted before, RHS well defined even for non-smoothable. Who ought to be in LHS when there is not a smoothing? A weak answer can be given via cobordism, using previous discussion. Need some facts on the geometry and topology of Gorenstein singularities:

Seade Remarks on the Milnor number

Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let LV be its link and set V ∗ = V \ {p}. Then:

1

A choice of a never-vanishing holomorphic 2-form ω around p determines:

A reduction to SU(2) ∼ = Sp(1) of the structure group of tangent bundle TV ∗ A canonical trivialization P of the tangent bundle TLV which is compatible with the complex structure on V. We call P the canonical framing of LV

2

The element in the framed cobordism group Ωfr

3 ∼

= Z24 represented by the pair (LV, P) depends only on the analytic structure of the germ (V, p).

Seade Remarks on the Milnor number

Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let LV be its link and set V ∗ = V \ {p}. Then:

1

A choice of a never-vanishing holomorphic 2-form ω around p determines:

A reduction to SU(2) ∼ = Sp(1) of the structure group of tangent bundle TV ∗ A canonical trivialization P of the tangent bundle TLV which is compatible with the complex structure on V. We call P the canonical framing of LV

2

The element in the framed cobordism group Ωfr

3 ∼

= Z24 represented by the pair (LV, P) depends only on the analytic structure of the germ (V, p).

Seade Remarks on the Milnor number

Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let LV be its link and set V ∗ = V \ {p}. Then:

1

A choice of a never-vanishing holomorphic 2-form ω around p determines:

A reduction to SU(2) ∼ = Sp(1) of the structure group of tangent bundle TV ∗ A canonical trivialization P of the tangent bundle TLV which is compatible with the complex structure on V. We call P the canonical framing of LV

2

The element in the framed cobordism group Ωfr

3 ∼

= Z24 represented by the pair (LV, P) depends only on the analytic structure of the germ (V, p).

Seade Remarks on the Milnor number

Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let LV be its link and set V ∗ = V \ {p}. Then:

1

A choice of a never-vanishing holomorphic 2-form ω around p determines:

A reduction to SU(2) ∼ = Sp(1) of the structure group of tangent bundle TV ∗ A canonical trivialization P of the tangent bundle TLV which is compatible with the complex structure on V. We call P the canonical framing of LV

2

The element in the framed cobordism group Ωfr

3 ∼

= Z24 represented by the pair (LV, P) depends only on the analytic structure of the germ (V, p).

Seade Remarks on the Milnor number

Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let LV be its link and set V ∗ = V \ {p}. Then:

1

A choice of a never-vanishing holomorphic 2-form ω around p determines:

A reduction to SU(2) ∼ = Sp(1) of the structure group of tangent bundle TV ∗ A canonical trivialization P of the tangent bundle TLV which is compatible with the complex structure on V. We call P the canonical framing of LV

2

The element in the framed cobordism group Ωfr

3 ∼

= Z24 represented by the pair (LV, P) depends only on the analytic structure of the germ (V, p).

Seade Remarks on the Milnor number

In order to determine the element in Z24 that (LV, P) represents we may use a classical invariant coming from algebraic topology: The Adams e-invariant: an integer well defined modulo 24, provides a group isomorphism: eR : Ωfr

3 −

→ Z24

• Adams’ definition in 1966 is via homotopy theory.
• Conner and Floyd gave an interpretation using spin

cobordism.

• Seade refined it later, using complex cobordism. One gets

Seade Remarks on the Milnor number

Let (V, p) be normal Gorenstein surface singularity, and (LV, P) its link equipped with its canonical framing, Theorem If X is a compact 4-manifold with boundary LV, whose interior has a complex structure compatible with P. Then: eR([LV, P]) = K 2

X + χ(X) + 12 Arf(KX)

mod (24) , where KX ∈ H2(X) is dual of Chern class of canonical bundle of X relative to P and Arf(KX) ∈ {0, 1} is the Arf invariant of a certain quadratic form associated to KX. Furthermore:

• If X =

V is a good resolution of (V, p), then KX is the canonical class (determined by the adjunction formula), independently of the choice of P.

• If (V, p) is smoothable and X = FV is a smoothing, then

KX = 0 and Arf(KX) = 0.

Seade Remarks on the Milnor number

Corollary If (V, p) is Gorenstein and smoothable, then its Laufer invariant modulo 24 is: La(V, p) ≡ eR([LV, P]) mod(24) ≡ K 2 + χ( V) + 12Arf(K) mod(24) Improving this result from the viewpoint we follow here is therefore equivalent in a way to finding an appropriate 3-manifolds invariant that provides an integral lifting of the e-invariant of the link: a reminiscent of the Casson invariant.

Seade Remarks on the Milnor number

§5 On the Milnor number

The question of what is the Laufer invariant for non-smoothable Gorenstein singularities is much related to the question: Who ought to be the Milnor number when there is no smoothing or when there are several smoothings with non-equivalent topology? Many tentative definitions. A very interesting one by Buchweitz-Greuel for curves. For smoothable curves this invariant essentially is the Euler characteristic of the smoothing.

Seade Remarks on the Milnor number

At least two other related viewpoints: i) Via indices of vector fields and 1-forms. ii) Via Chern classes for singular. Let us say a few words about these.

Seade Remarks on the Milnor number

Indices of vector fields on singular varieties

• Several possible extensions of local Poincar´

e-Hopf.

• One is radial index (Schwartz for radial vector fields. Then

King-Trotman, Ebeling and Gusein-Zade, Aguilar-Seade-Verjovsky).

Seade Remarks on the Milnor number

Another notion in case of vector fields on ICIS: GSV index, One has: µ = −1n(IndGSV − Indradial), independent of vector field. This can be taken as idea to extend notion of Milnor number.

Seade Remarks on the Milnor number

Need to extend notion of GSV index to vector fields on singularities which are not ICIS: homological index: Recall definition. (G´

• mez-Mont 1990s.)

Consider normal isolated singularity (V, p) (any dimension n ) in some Cm, and A germ of holomorphic vector field ω in Cn tangent to V with isolated singularity at p, For each j ≥ 0, let Ωj(V, p) be space of j-forms on germ (V, P).

Seade Remarks on the Milnor number

One has a complex (Ω•

V,p, ⊢ ω):

0 − → Ωn(V, p)

⊢ω

− → Ωn−1(V, p)

⊢ω

− → · · ·

⊢ω

− → Ω0(V, p) − → 0 where ⊢ ω is the contraction of forms by the vector field. One has the homology of this complex in usual way. Definition Homological index Indhom(ω; (V, p)) is the Euler characteristic

• f this complex:

Indhom(ω; (V, p)) :=

n

• i=0

hi(Ω•

V,p, ∧ω).

Seade Remarks on the Milnor number

If (V, p) is an ICIS, then [v. Bothmer, Ebeling, G´

• mez-Mont

2008]

• Indhom(ω; (V, p)) coincides with GSV-index
• Hence the Milnor number equals the difference [homological

index - radial index], independently of choice of vector field. If germ (V, p) is not an ICIS:

• what is Indhom(ω; (V, p))?
• Does it yield to the LHS in Laufer’s formula?

Seade Remarks on the Milnor number

Congratulations Victor !!!!

Seade Remarks on the Milnor number

THANKS A LOT !!

Seade Remarks on the Milnor number