Singularities and Characteristic classes for Differentiable Maps - - PowerPoint PPT Presentation

Singularities and Characteristic classes for Differentiable Maps

（可微分写像の特異点と特性類）

Toru Ohmoto, Hokkaido Univ. July 20, Nihon Univ.

Contents

• 1. Thom polynomials
• 2. Fukuda’s formula for Morin maps
• 3. Constructible functions and Yomdin-Nakai’s formula
• 4. Chern-MacPherson transformation
• 5. Ando’s higher Thom polynomials
• 6. Universal Segre-SM class
• 7. Classical Milnor number formulas, revisited
• 8. Conclusion

• 1. Thom polynomials:

Thom polynomial was introduced (in R. Thom’s talk at the Strasbourg seminar, 1957) as the simplest universal cohomological obstruction for the appearence of prescribed singularities of functions/maps. Let f : M → N be a complex holomorphic map between complex manifolds

• f dim. m and n.

Consider the locus of f associated to prescribed singularity types:

• mono-singularity type η : Cm, 0 → Cn, 0,
• multi-singularity type ηml = (η1, · · · , ηs) : Cm, {x1, · · · , xs} → Cn, 0

• 1. Thom polynomials:

η(f) = { x ∈ M | the germ f : M, x → N, f(x) is of type η } ⊂ M, ηml(f) = { y ∈ N | ∃ S ⊂ f −1(y) s.t. f : M, S → N, y is of type ηml } ⊂ N become locally closed submanifolds of M and N, respectively, if f is appropriately generic. Thom polynomials for η and for ηml universally express Dual [η(f)] ∈ H∗(M) and Dual [ηml(f)] ∈ H∗(N) – R. Thom (’57) for mono-sing., M. Kazarian (2003) for multi-sing. Notation: [Chern class associated to f] ci = ci(f) := ci(f ∗TN − T M) and ¯ ci = ¯ ci(f) := ci(T M − f ∗TN), where c(F − E) = c(F) c(E) = 1 + c1(F) + c2(F) + · · · 1 + c1(E) + c2(E) + · · ·.

• 1. Thom polynomials: Example

For instance, in case of m = n + 1, there are typical types such as A1 : (x, y) → x2 + y2 A2 : (x, y, u) → (x3 + ux + y2, u), A3 : (x, y, u1, u2, v) → (x4 + u1x2 + u2x + y2, u1, u2, v), Ak

1 : singular value whose fiber has k nodes (A1-sing.)

• 1. Thom polynomials: Example

For instance, in case of m = n + 1, there are typical types such as A1 : (x, y) → x2 + y2 A2 : (x, y, u) → (x3 + ux + y2, u), A3 : (x, y, u1, u2, v) → (x4 + u1x2 + u2x + y2, u1, u2, v), Ak

1 : singular value whose fiber has k nodes (A1-sing.)

The dual to the closure of the loci are expressed by tp(A1) = c2

1 − c2,

tp(A2) = 2c1(c2

1 − c2),

tp(A3) = 5c4

1 − 4c2 1c2 − c1c3

∈ H∗(M) n(A2

1) = (s2 − s01)2 − s0001 + 8s001s01 − 7s3

∈ H∗(N). Notation: [Landweber-Novikov classes in cobordism theory] sI = sI(f) := f∗(cI(f)) = f∗(ci1

1 ( f) · · · cik k (f)),

I = (i1, · · · , ik)

• 1. Thom polynomials:

Remark 1: In real C∞ category (or real analytic/algebraic), we work with the Stiefel-Whitney class wi or the Pontrjagin class p j, instead of Chern class ci. Remark 2: the Tp of a real algebraic singularity type η : Rm, 0 → Rn, 0 is

• btained from the Tp of the complexified singularity ηC : Cm, 0 → Cn, 0 by

switching ci → wi and coefficients being modulo two (Borel-Haefliger, 1962)

• 2. Fukuda’s formula for Morin maps

We say f : M → N is a Morin map if f is stable and admits only singularities of type Ak. Now work on real C∞ category. Let Mn+ℓ (ℓ ≥ 0) , Nn be C∞ manifolds, and f : M → N a Morin map. Put Ak(f) = { x ∈ M | the germ f at x is of type Ak } their closures form a filtration of closed submanifolds: M = ¯ A0(f) ⊃ ¯ A1(f) ⊃ ¯ A2( f) ⊃ · · · Denote by ιk : ¯ Ak(f) → M the inclusion. If N = Rn, we can take an orthogonal projection of Rn to a line so that p ◦ f ◦ ιk : ¯ Ak(f) → R is a Morse function. Then counting the number of critical points leads us to find a formula:

• 2. Fukuda’s formula for Morin maps
• Thm. (Fukuda) For Morin maps f : M → Rn,

χ(M) + ∑

k≥1

χ( ¯ Ak( f)) = 0 (modulo 2) More generally, for Morin maps f : M → N, the RHS is changed to be χ(F)χ(N), where F is a generic fibre of f (O. Saeki).

• 2. Fukuda’s formula for Morin maps
• Thm. (Fukuda) For Morin maps f : M → Rn,

χ(M) + ∑

k≥1

χ( ¯ Ak( f)) = 0 (modulo 2) More generally, for Morin maps f : M → N, the RHS is changed to be χ(F)χ(N), where F is a generic fibre of f (O. Saeki).

• Thm. (Fukuda) For Morin maps f : M → Rn, for any i ≤ n − 1,

Wi(M) + ∑

k≥1

ιk∗ Wi( ¯ Ak(f)) = 0 ∈ H∗(M; Z2). where Wi(M) = wm−i(T M) ⌣ [M]2, the Whitney homology class for real manifolds.

• 3. Constructible functions and Yomdin-Nakai’s formula

F(X): the abelian group of constructible functions over X. α : X → Z, α = ∑ ai1 1Wi. where ai ∈ Z, and Wi are subvarieties. The function 1 1W ∈ F(X) is defined by 1 1W(x) = 1 if x ∈ W, 0 otherwise. For proper morphisms f : X → Y, we define f∗ : F(X) → F(Y): f∗α(y) := ∫

f −1(y)

α = ∑ ai χ(Wi ∩ f −1(y)) (y ∈ Y) For proper f : X → Y, g : Y → Z, (g ◦ f)∗ = g∗ ◦ f∗

• 3. Constructible functions and Yomdin-Nakai’s formula

So F : Var → Ab becomes a covariant functor from the category of complex algebraic varieties (resp. real algebraic varieties, subanalyitic sets ...) and proper morphisms to the category of abelian groups. In particular for pt : X → pt, X being compact, α ∈ F(X), the integration of α based on Euler characteristic measure is defined to be ∫

X

α := pt∗α = ∑ ai χ(Wi). For proper f : X → Y, pt∗ f∗ = (pt ◦ f)∗ thus,

Fubini-Thm:

∫

X

α = ∫

Y

f∗α.

• 3. Constructible functions and Yomdin-Nakai’s formula

Let η : Cn+ℓ, 0 → Cn, 0 be an ℓ-dimensional isolated complete intersection singularity (ICIS). Then, the Milnor number of η is defined by µη = (−1)ℓ(χ(Bǫ(0)) ∩ η−1(c)) − 1), where Bǫ(0) is a sufficiently small closed ball centered at 0 and c is a regular value sufficiently close to the critical value 0. Holomorphic map f : Mn+ℓ → Nn is finite-type if for every point x ∈ M, the germ f : M, x → N, f(x) is an ICIS. Then, the Milnor number constructible function is defined: µ(f) : M → Z, x → Milnor number of f at x.

• 3. Constructible functions and Yomdin-Nakai’s formula
• Lem. For maps of finite-type f : Mn+ℓ → Nn, where M is compact,

connected, and N is connected, it holds that f∗ ( 1 1M + (−1)ℓµ(f) ) = χ(F)1 1N where F is a generic fibre of f.

• 3. Constructible functions and Yomdin-Nakai’s formula

In fact, for any critical value y ∈ N, f∗ ( 1 1M + (−1)ℓµ( f) ) (y) = ∫

f −1(y)

( 1 1M + (−1)ℓµ(f) ) = χ(red) + χ(orenge) + (χ(green) − 1) = χ(red) + χ(green) = χ(F)

• 3. Constructible functions and Yomdin-Nakai’s formula

Applying the Fubini-theorem to the lemma, we have ∫

M

( 1 1M + (−1)ℓµ(f) ) = ∫

N

f∗ ( 1 1M + (−1)ℓµ(f) ) = χ(F) ∫

N

1 1N, that is, ∫

M

µ(f) = (−1)ℓ (χ(F)χ(N) − χ(M)) .

• 3. Constructible functions and Yomdin-Nakai’s formula

Applying the Fubini-theorem to the lemma, we have ∫

M

( 1 1M + (−1)ℓµ(f) ) = ∫

N

f∗ ( 1 1M + (−1)ℓµ(f) ) = χ(F) ∫

N

1 1N, that is, ∫

M

µ(f) = (−1)ℓ (χ(F)χ(N) − χ(M)) . A Morin map f : M → N is of finite-type. Let ιk : ¯ Ak ⊂ M be the inclusion. µ(f) = ∑

k≥1

ιk∗1 1 ¯

Ak( f).

∑

k≥1

χ( ¯ Ak(f)) = (−1)ℓ (χ(F)χ(N) − χ(M)) . In real case (and modulo 2), this is Fukuda-Saeki’s formula.

• 4. Chern-MacPherson transformation

For any singular variety X, c(TX) no longer exists. But a certain substitute exists uniquely as follows: Axiom of MacPherson’s Chern class transformation: For each constructible function α : X → Z, we can associate a total homology class C∗(α) ∈ H∗(X) so that

• Ci(α) ∈ H2i(X), C∗(α) = C0(α) + · · · + Cr(α), r = dim supp(α),
• C∗(α + β) = C∗(α) + C∗(β),
• For proper morphisms f : X → Y, f∗C∗(α) = C∗(f∗(α)),
• For non-singular X, it holds that

C∗(1 1X) = c(TX) ⌣ [X]. The Chern-Schuwartz-MacPherson class of X is CS M(X) := C∗(1 1X) ∈ H∗(X).

• 4. Chern-MacPherson transformation

For any singular variety X, c(TX) no longer exists. But a certain substitute exists uniquely as follows: Real version (Sullivan, MacPherson etc): For each algebraically constructible function α : X → Z2, we can asso- ciate a total homology class W∗(α) ∈ H∗(X; Z2) so that

• Wi(α) ∈ Hi(X; Z2), W∗(α) = W0(α) + · · · + Wr(α), r = dim supp(α),
• W∗(α + β) = W∗(α) + W∗(β),
• For proper morphisms f : X → Y, f∗W∗(α) = W∗(f∗(α)),
• For non-singular X, it holds that

W∗(1 1X) = w(TX) ⌣ [X]2.

• 4. Chern-MacPherson transformation

Apply C∗ to the equality f∗(1 1M + (−1)ℓµ(f)) = χ(F)1 1N. ⇒ “total class version" of the Euler number formula (Yomdin-Nakai’s formula) f!(c(T M) + (−1)ℓDualC∗(µ(f))) = χ(F)c(TN) where f! : H∗+ℓ(M) → H∗(N) is the Gysin homomorphism. In particular, for Morin map f : M → N, (ιk : ¯ Ak ⊂ M the inclusion ), ∑

k≥0

f!ιk! c(T ¯ Ak(f)) = (−1)ℓ(χ(F)c(TN) − f!c(T M)) ∈ H∗(N; Z)

• 5. Ando’s higher Thom polynomials

For Morin maps (in complex case), the closure of Ak-type singular set ¯ Ak(f), that is Σℓ+1,1k−1,0, is a closed submanifold in M. Thus the pushforwad of Chern class of the submanifold makes sense ιk! c(T ¯ Ak(f)) ∈ H∗(M). Its leading term is nothing but Tp: ι!(1) = ι∗[ ¯ Ak(f)] = tp(Ak)(c(f)).

Yoshifumi Ando called the total class ιk! c(T ¯

Ak(f)) “higher Thom polynomials": However, in general, the closure ¯ Ak(f) admits its singularities, so c(T ¯ Ak(f)) does not make sense. Instead, the CSM class C∗( ¯ Ak(f)) always exists, that would give a mostly natural definition.

• 6. Universal Segre-SM class

For α ∈ F(M), where M is smooth, we define the Segre-SM class of α to be sS M(α, M) := c(T M)−1 ⌣ C∗(α) ∈ H∗(M). In particular, for α = 1 1W, write it by sS M(W, M). If W is smooth, then it is the inverse normal Chern class sS M(W, M) = ι∗c(ν−1

W ) = ι∗c(TW − ι∗T M)

where ι : W ⊂ M is the inclusion.

• 6. Universal Segre-SM class
• Thm. [O] For a K-singularity type η of map-germs, there exists a power

series tpS M(η) in variables c1, c2, · · · s.t. for any appropriately generic maps f : M → N, it holds that sS M(η(f), M) = tpS M(η)(c(f)) ⌣ [M]. Proof: I have established an equivariant theory of the Chern-SM class (i.e., MacPherson’s natural transformation C∗ for classifying stacks). As a corollary, the above theorem is shown in completely the same way as the proof of Tp theorem.

• 6. Universal Segre-SM class
• Thm. [O] For a K-singularity type η of map-germs, there exists a power

series tpS M(η) in variables c1, c2, · · · s.t. for any appropriately generic maps f : M → N, it holds that sS M(η(f), M) = tpS M(η)(c(f)) ⌣ [M].

• Rem. leading term of tpS M(η) is equal to tp(η).
• Ex. For 1-dimensional ICIS Cn+1, 0 → Cn, 0

tpS M(A1) = (c2

1 − c2) − (2c3 1 − 3c1c2 + c3) + (3c4 1 − 6c2 1c2 + 4c1c3 − c4) + h.o.t.

tpS M(A2) = 2c1(c2

1 − c2) − 2c1(4c3 1 − 5c1c2 + c3) + h.o.t.

· · · · · ·

• 6. Universal Segre-SM class

Application of this theorem to f : Mn+ℓ → Nn of finite type. We have already seen f! ( c(T M) + (−1)ℓDualC∗(µ(f)) ) = χ(F)c(TN). (Yomdin-Nakai)

• 6. Universal Segre-SM class

Application of this theorem to f : Mn+ℓ → Nn of finite type. We have already seen f! ( c(T M) + (−1)ℓDualC∗(µ(f)) ) = χ(F)c(TN). (Yomdin-Nakai) By projection formula, f! ( c(T M − f ∗TN) + (−1)ℓc(f ∗TN)−1 · DualC∗(µ(f))) ) = χ(F).1. Now, by the theorem of tpS M, c(T M)−1 · DualC∗(µ(f)) = Dual sS M(µ(f), M) = tpS M(µ)(c(f)). Thus we may rewrite it by f! ( c(T M − f ∗TN) + (−1)ℓc(T M − f ∗TN) · tpS M(µ)(c(f)) ) = χ(F).1.

• 6. Universal Segre-SM class

f! ( ¯ c(f) + (−1)ℓ¯ c( f) · tpS M(µ)(c(f)) ) = χ(F).1. (Yomdin-Nakai) On the other hand, it is obvious to see f!(1 + ¯ c1(f) + · · · + ¯ cℓ(f)) = χ(F).1 ∈ H0(N). Hence ∆ := (¯ cℓ+1( f) + ¯ cℓ+2(f) + · · · ) + (−1)ℓ¯ c(f) · tpS M(µ)(c(f)) ∈ ker f!. Fact 1: by our theorem ∆ is a series in ci(f) of order greater than ℓ; Fact 2: Landweber-Novikov classes sI(f) = f!(cI(f)) where |I| > ℓ are linearly independent. =⇒ ∆ is identically zero.

• 6. Universal Segre-SM class
• Thm. [O] The Segre-Schwartz-MacPherson class sS M of µ is universally

expressed by tpS M(µ) = (−1)ℓ+1(1 + c1 + c2 + · · · )(¯ cℓ+1 + ¯ cℓ+2 + · · · ) where ci = ci(f) = ci(f ∗TN − T M), ¯ ci = ¯ ci(f) = ci(T M − f ∗TN). In particular,

• Cor. [O] For maps f : M∗+ℓ → N∗ of finite type, it holds that

C∗(µ(f)) = c(T M) ⌣ sS M(µ(f), M) = (−1)ℓ+1c(f ∗TN) · (¯ cℓ+1(f) + ¯ cℓ+2(f) + · · · ) ⌣ [M].

• 6. Universal Segre-SM class

Remark: In real category, the theorem remains true after changing ci → wi and C∗ → W∗: tpS M(µ) = (−1)ℓ+1(1 + w1 + w2 + · · · )( ¯ wℓ+1 + ¯ wℓ+2 + · · · ) For real Morin maps f : M → N, W∗(µ(f)) = ∑

k≥1

W∗( ¯ Ak(f)) = w(f ∗TN) · ( ¯ wℓ+1(f) + · · · ) ⌣ [M]2. In particular, for Morin maps f : M → Rn, ( ¯ wℓ+1(f) + · · · ) ⌣ [M]2 = Wn−1(M) + Wn−2(M) + · · · + W0(M), thus Fukuda’s second formula is recovered: Wi(M) + ∑

k≥1

Wi( ¯ Ak(f)) = 0. (i ≤ n − 1)

• 7. Classical Milnor number formula, revisited

Let η : Cm, 0 → C, 0 (ℓ = m − 1) be a quasi-homog. isolated hypersurface singularity, with weight w1, · · · , wm and degree d, i.e., η(αw1 x1, · · · , αwm xm) = αdη(x1, · · · , xm) α ∈ T = C∗. That is, T = C∗ ⊂ A = Aut((Cm, 0) × (C, 0)) is the stabilizer subgroup of η: Cm, 0

η

−→ C, 0

ρ1 = ∏ α⊗wi ↓

↓ α⊗d = ρ2 Cm, 0

η

−→ C, 0

• 7. Classical Milnor number formula, revisited

This naturally induces E1 := γ⊗w1 ⊕ · · · ⊕ γ⊗wm

fη

−→ E2 := γ⊗d ց ւ P∞ where γ is the tautological bundle of the projective space PN with N large

• enough. Let t = c1(γ).

c(E1) = cT(ρ1) = ∏ (1 + wit), c(E2) = cT(ρ2) = 1 + dt in H∗(P∞) = Z[t].

• 7. Classical Milnor number formula, revisited

This naturally induces E1 := γ⊗w1 ⊕ · · · ⊕ γ⊗wm

fη

−→ E2 := γ⊗d ց ւ P∞ By the corollary, the m-th term of C∗(µ(fη)) is µ(f)(0) Dual [η(fη)] = (−1)ℓ+1¯ cℓ+1( fη) = (−1)mcm(E1 − E2) in Hm(E1) = Hm(P∞) = Z[t]. On the other hand, the η-type singular set η(fη) is the zero section of E1, hence Dual [η(fη)] = cm(E1) ∈ Hm(P∞). Thus µ(η)(0) = (−1)m cm(E1 − E2) cm(E1)

• 7. Classical Milnor number formula, revisited

This naturally induces E1 := γ⊗w1 ⊕ · · · ⊕ γ⊗wm

fη

−→ E2 := γ⊗d ց ւ P∞ The well-known Milnor number formula is recovered by

• Cor. For a quasi-homog. hypersurface sing. in Cm,

µ(η)(0) = (−1)m cm(E1 − E2) cm(E1) = top. (−1)m ∏(1 + (wi − d)t)

• top. ∏(1 + wit)

=

m

∏

i=1

d − wi wi

• 7. Classical Milnor number formula, revisited

Similarly we can work for a quasi-homogeneous ICIS η : Cm, 0 → Cn, 0 (ℓ = m − n ≥ 0) with weight wi and degree d j. As a direct consequence from our theorem, Dual C∗(µ(fη)) = (−1)ℓc(E2) ∑

i≥ℓ+1

ci(E1 − E2). The m-th term of this formula gives

Cor (Greuel-Hamm, Guisti, Damon) For ℓ-dim. quasi-homog. ICIS,

µ(η)(0) = (−1)m−n ( cn(E2) cm(E1)cm−n(E1 − E2) − 1 )

• cf. J. Damon’s ‘Cohen-Macaulay-Bezout theorem’ can be interpreted by this

formula.

• 8. Conclusion

Other type formulae due to T. Fukuda:

• K. Aoki, T. Fukuda and T. Nishimura,

counting formula for the number of branches of real curve germs f −1(0) of f : Rn+1, 0 → Rn, 0. (=⇒ Montaldi-van Straten, Damon, etc made generalizations)

• G. Ishikawa and T. Fukuda,

the number of cusps in stable puturbations of C2, 0 → C2, 0 (=⇒ Gaffney-Mond, Saia, Fukui-Weyman etc made generalizations) ... etc These formulae are also related to ‘generalized Tp theory’ when we work

• n quasi-homogeneous singularities.

Thanks for your attention, and ...

Happy birthday to Fukuda-sensei !