Remarks on the Milnor number

Transcription

1 José 1 1 Instituto de Matemáticas, Universidad Nacional Autónoma de México. Liverpool, U. K. March, 2016 In honour of Victor!!

2 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 f : S ε \ K S 1. Alternative description. Given ε > 0 as above, choose 0 < δ << ε and set N(ε, δ) = f 1 ( D δ ) B ε. Then: f : N(ε, δ) D δ = S 1 is a locally trivial fibration, equivalent to previous one.

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4 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: F t S n 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 )

5 Milnor also proved that µ actually is the Poincaré-Hopf local index of the gradient vector field f. Hence it is an intersection number: µ = dim C O n+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.

6 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.

7 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 : Ṽ 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.

8 Let (V, p) be a normal surface singularity germ, Ṽ a good resolution; K its canonical class, well defined by the adjunction formula: 2g Ei 2 = E i (K + E i ) for each irreducible component of the exceptional divisor in Ṽ. Laufer (1977) proved: µ(v ) + 1 = χ(ṽ ) + K ρ g (V ) where: χ = usual Euler characteristic ; K 2 = self-intersection number; and ρ g := dim H 1 (Ṽ, 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.

9 Definition For every normal numerically Gorenstein surface singularity germ (V, p) we may call the integer La(V, p) = χ(ṽ ) + K ρ g (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.

10 Definition For every normal numerically Gorenstein surface singularity germ (V, p) we may call the integer La(V, p) = χ(ṽ ) + K ρ g (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.

11 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.

12 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.

13 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: µ GS + 1 = χ(ṽ ) + K ρ g (V ) 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.

14 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: µ GS + 1 = χ(ṽ ) + K ρ g (V ) 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.

15 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 2 (X; R) H 2 (X; R) H 4 (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.

16 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)

17 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 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:

18 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 2c 2, c 1 (M) = K M and c 2 (M)[M] = χ(m). Recall the 2nd Todd polynomial is 1 12 (c2 1 + c 2). 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),

19 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: We get: Theorem χ(m, O M ) Arf K mod (2). 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.

20 Definition (Esnault--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 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 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

21 Definition (Esnault--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 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 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

22 Definition (Esnault--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 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 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

23 We have: Theorem (Esnault--Viehweg) The parity of the analytic Euler characteristic coincides with the mod (2) index h( K ): h( K ) = χ(m, O M ) mod (2). More generally, let W = K M 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)i h i (M, D) is the analytic Euler characteristic of M with coefficients in D.

24 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.

25 Theorem Let M and W = K M 2D be as above. If W 0, then: χ(m, D) = h 0 (W ; D W ) R, with R an even integer associated to the divisor W : R = h 1 (M; D) 2h 2 (M; D) + dim Ker( ˆβ), where ˆβ is a skew symmetric bilinear form on H 1 (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

26 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--Viehweg) The parity of the geometric genus coincides with the mod (2) index h( K ). That is: dim H 1 (Ṽ, O Ṽ ) dim H0 ( K, KṼ K ) mod 2 Furthermore, if the resolution Ṽ is minimal, then: dim H 1 (Ṽ, O Ṽ ) = dim H0 ( K, K K ) and for all vertical divisors D 0 and W = 2D K we have: dim H 1 (Ṽ, O Ṽ ) = dim H0 (W, D W ) (W 2 K 2 ).

27 4 Laufer s formula revisited We know that if (V, P) is a normal Gorenstein smoothable singularity, then: µ(v ) + 1 = χ(ṽ ) + K ρ g (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:

28 Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let L V 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 T L V which is compatible with the complex structure on V. We call P the canonical framing of L V 2 The element in the framed cobordism group Ω fr 3 = Z 24 represented by the pair (L V, P) depends only on the analytic structure of the germ (V, p).

29 Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let L V 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 T L V which is compatible with the complex structure on V. We call P the canonical framing of L V 2 The element in the framed cobordism group Ω fr 3 = Z 24 represented by the pair (L V, P) depends only on the analytic structure of the germ (V, p).

30 Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let L V 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 T L V which is compatible with the complex structure on V. We call P the canonical framing of L V 2 The element in the framed cobordism group Ω fr 3 = Z 24 represented by the pair (L V, P) depends only on the analytic structure of the germ (V, p).

31 Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let L V 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 T L V which is compatible with the complex structure on V. We call P the canonical framing of L V 2 The element in the framed cobordism group Ω fr 3 = Z 24 represented by the pair (L V, P) depends only on the analytic structure of the germ (V, p).

32 Theorem Let (V, p) be a normal Gorenstein surface singularity germ. Let L V 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 T L V which is compatible with the complex structure on V. We call P the canonical framing of L V 2 The element in the framed cobordism group Ω fr 3 = Z 24 represented by the pair (L V, P) depends only on the analytic structure of the germ (V, p).

33 In order to determine the element in Z 24 that (L V, 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: e R : Ω fr 3 Z 24 Adams definition in 1966 is via homotopy theory. Conner and Floyd gave an interpretation using spin cobordism. refined it later, using complex cobordism. One gets

34 Let (V, p) be normal Gorenstein surface singularity, and (L V, P) its link equipped with its canonical framing, Theorem If X is a compact 4-manifold with boundary L V, whose interior has a complex structure compatible with P. Then: e R ([L V, P]) = K 2 X + χ(x) + 12 Arf(K X ) mod (24), where K X H 2 (X) is dual of Chern class of canonical bundle of X relative to P and Arf(K X ) {0, 1} is the Arf invariant of a certain quadratic form associated to K X. Furthermore: If X = Ṽ is a good resolution of (V, p), then K X is the canonical class (determined by the adjunction formula), independently of the choice of P. If (V, p) is smoothable and X = F V is a smoothing, then K X = 0 and Arf(K X ) = 0.

35 Corollary If (V, p) is Gorenstein and smoothable, then its Laufer invariant modulo 24 is: La(V, p) e R ([L V, P]) K 2 + χ(ṽ ) + 12Arf(K ) mod(24) 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.

36 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.

37 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.

38 Indices of vector fields on singular varieties Several possible extensions of local Poincaré-Hopf. One is radial index (Schwartz for radial vector fields. Then King-Trotman, Ebeling and Gusein-Zade, Aguilar--Verjovsky).

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

40 Need to extend notion of GSV index to vector fields on singularities which are not ICIS: homological index: Recall definition. (Gómez-Mont 1990s.) Consider normal isolated singularity (V, p) (any dimension n ) in some C m, and A germ of holomorphic vector field ω in C n tangent to V with isolated singularity at p, For each j 0, let Ω j (V, p) be space of j-forms on germ (V, P).

41 One has a complex (Ω V,p, ω): 0 Ω n (V, p) ω Ω n 1 (V, p) ω ω Ω 0 (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 Ind hom (ω; (V, p)) is the Euler characteristic of this complex: Ind hom (ω; (V, p)) := n h i (Ω V,p, ω). i=0

42 If (V, p) is an ICIS, then [v. Bothmer, Ebeling, Gómez-Mont 2008] Ind hom (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 Ind hom (ω; (V, p))? Does it yield to the LHS in Laufer s formula?

43 Congratulations Victor!!!!

44 THANKS A LOT!!