Remarks on the Milnor number e Seade 1 Jos´ 1 Instituto de Matem´ aticas, Universidad Nacional Aut´ onoma 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 : ( C n + 1 , 0 ) → ( C , 0 ) with a critical point at 0. Let V = f − 1 ( 0 ) and K = V ∩ S ε the link. Milnor’s classical theorem (1968) says that we have a locally trivial fibration : φ := f → S 1 . | f | : S ε \ K − Alternative description. Given ε > 0 as above, choose 0 < δ << ε and set N ( ε, δ ) = f − 1 ( ∂ D δ ) ∩ B ε . Then: → ∂ D δ ∼ = S 1 f : N ( ε, δ ) − 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: F t := f − 1 ( t ) ∩ B ε has the homotopy type of a bouquet of spheres of middle dimension: � S n F t ≃ µ 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 H n ( F t ) 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: O n + 1 , 0 µ = dim C 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 := ( f 1 , · · · , f k ) : ( C n + k , 0 ) → ( C k , 0 ) Such a germ also has an associated Milnor fibration and a well-defined Milnor number: the rank of the middle-homology of 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 := ( f 1 , · · · , f k ) : ( C n + k , 0 ) → ( C k , 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: 2 g E i − 2 = E i · ( K + E i ) for each irreducible component of the exceptional divisor in � V . Laufer (1977) proved: V ) + K 2 + 12 ρ g ( V ) µ ( V ) + 1 = χ ( � where: χ = usual Euler characteristic ; K 2 = self-intersection number ; and ρ g := dim H 1 ( � 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 V ) + K 2 + 12 ρ g ( V ) , La ( V , p ) = χ ( � 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 V ) + K 2 + 12 ρ g ( V ) , La ( V , p ) = χ ( � 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 µ GS : The 2nd Betti-number of a smoothing (and b 1 vanishes). Theorem (Steenbrink 1981) Furthermore, this invariant satisfies Laufer’s formula: V ) + K 2 + 12 ρ g ( V ) µ GS + 1 = χ ( � That is, for smoothable Gorenstein singularities the Laufer invariant is µ GS + 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 µ GS : The 2nd Betti-number of a smoothing (and b 1 vanishes). Theorem (Steenbrink 1981) Furthermore, this invariant satisfies Laufer’s formula: V ) + K 2 + 12 ρ g ( V ) µ GS + 1 = χ ( � That is, for smoothable Gorenstein singularities the Laufer invariant is µ GS + 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: → H 4 ( X ; R ) ∼ H 2 ( X ; R ) ∪ H 2 ( 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 ω 2 ( M ) = 0. Not all manifolds are. Yet: Every closed oriented 4-manifold is spin c : There is a class in H 2 ( M ; Z ) whose reduction modulo 2 is ω 2 ( M ) = 0. Seade Remarks on the Milnor number
If M is a complex surface it is canonically spin c and its canonical class K M reduced modulo 2 gives ω 2 ( M ) . Definition (Rochlin, 1970s) Let W be an oriented 2-submanifold of a closed oriented M 4 . W is a characteristic submanifold if [ W ] ∈ H 2 ( M ; Z ) reduced modulo 2 is ω 2 ( M ) . Notice K M can always be smoothed C ∞ and the smoothing is a characteristic submanifold. If K M 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 ≡ 8 Arf W mod ( 16 ) where Arf W ∈ { 0 , 1 } is an invariant associated to H 1 ( W ; Z 2 ) . 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 p 1 ( M )[ M ] where p 1 = Pontryagin class. For compact complex surfaces one has p 1 = c 2 1 − 2 c 2 , c 1 ( M ) = − K M and c 2 ( M )[ M ] = χ ( M ) . 12 ( c 2 1 Recall the 2nd Todd polynomial is 1 + c 2 ) . Hence σ ( M ) − K 2 = − 8 Td ( M )[ M ] K is a C ∞ smoothing of the canonical divisor K , Thus, if W = � 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 , O M ) so Rohlin’s theorem can be restated as: χ ( M , O M ) ≡ 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 = K M ⊗ D − 2 Notice that such W = K M − 2 D 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 H 0 ( 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 = − K M one has D = K M : h ( − K ) = dim H 0 ( − K , K M | K ) mod 2 Seade Remarks on the Milnor number
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