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 of dim. m and n . Consider the locus of f associated to prescribed singularity types: • mono-singularity type η : C m , 0 → C n , 0 , • multi-singularity type η ml = ( η 1 , · · · , η s ) : C m , { x 1 , · · · , x s } → C n , 0

1. Thom polynomials: { } x ∈ M | the germ f : M , x → N , f ( x ) is of type η η ( f ) = ⊂ M , { } y ∈ N | ∃ S ⊂ f − 1 ( y ) s.t. f : M , S → N , y is of type η ml η ml ( f ) = ⊂ N become locally closed submanifolds of M and N , respectively, if f is appropriately generic. Thom polynomials for η and for η ml universally express Dual [ η ( f )] ∈ H ∗ ( M ) Dual [ η ml ( f )] ∈ H ∗ ( N ) and – R. Thom (’57) for mono-sing., M. Kazarian (2003) for multi-sing. Notation: [Chern class associated to f ] c i = c i ( f ) : = c i ( f ∗ TN − T M ) and ¯ c i ( f ) : = c i ( T M − f ∗ TN ) , where c i = ¯ c ( E ) = 1 + c 1 ( F ) + c 2 ( F ) + · · · c ( F − E ) = c ( F ) 1 + c 1 ( E ) + c 2 ( E ) + · · · .

1. Thom polynomials: Example For instance, in case of m = n + 1 , there are typical types such as A 1 : ( x , y ) �→ x 2 + y 2 A 2 : ( x , y , u ) �→ ( x 3 + ux + y 2 , u ) , A 3 : ( x , y , u 1 , u 2 , v ) �→ ( x 4 + u 1 x 2 + u 2 x + y 2 , u 1 , u 2 , v ) , A k 1 : singular value whose fiber has k nodes ( A 1 -sing.)

1. Thom polynomials: Example For instance, in case of m = n + 1 , there are typical types such as A 1 : ( x , y ) �→ x 2 + y 2 A 2 : ( x , y , u ) �→ ( x 3 + ux + y 2 , u ) , A 3 : ( x , y , u 1 , u 2 , v ) �→ ( x 4 + u 1 x 2 + u 2 x + y 2 , u 1 , u 2 , v ) , A k 1 : singular value whose fiber has k nodes ( A 1 -sing.) The dual to the closure of the loci are expressed by ∈ H ∗ ( M ) tp ( A 1 ) = c 2 tp ( A 2 ) = 2 c 1 ( c 2 tp ( A 3 ) = 5 c 4 1 − 4 c 2 1 − c 2 , 1 − c 2 ) , 1 c 2 − c 1 c 3 1 ) = ( s 2 − s 01 ) 2 − s 0001 + 8 s 001 s 01 − 7 s 3 n ( A 2 ∈ H ∗ ( N ) . Notation: [Landweber-Novikov classes in cobordism theory] s I = s I ( f ) : = f ∗ ( c I ( f )) = f ∗ ( c i 1 1 ( f ) · · · c i k k ( f )) , I = ( i 1 , · · · , i k )

1. Thom polynomials: Remark 1: In real C ∞ category (or real analytic/algebraic), we work with the Stiefel-Whitney class w i or the Pontrjagin class p j , instead of Chern class c i . Remark 2: the Tp of a real algebraic singularity type η : R m , 0 → R n , 0 is obtained from the Tp of the complexified singularity η C : C m , 0 → C n , 0 by switching c i �→ w i 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 A k . Now work on real C ∞ category. Let M n + ℓ ( ℓ ≥ 0 ) , N n be C ∞ manifolds, and f : M → N a Morin map. Put A k ( f ) = { x ∈ M | the germ f at x is of type A k } their closures form a filtration of closed submanifolds: M = ¯ A 0 ( f ) ⊃ ¯ A 1 ( f ) ⊃ ¯ A 2 ( f ) ⊃ · · · Denote by ι k : ¯ A k ( f ) → M the inclusion. If N = R n , we can take an orthogonal projection of R n to a line so that p ◦ f ◦ ι k : ¯ A k ( 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 → R n , ∑ χ ( ¯ ( modulo 2) χ ( M ) + A k ( f )) = 0 k ≥ 1 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 → R n , ∑ χ ( ¯ ( modulo 2) χ ( M ) + A k ( f )) = 0 k ≥ 1 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 → R n , for any i ≤ n − 1 , ∑ ι k ∗ W i ( ¯ W i ( M ) + A k ( f )) = 0 ∈ H ∗ ( M ; Z 2 ) . k ≥ 1 where W i ( M ) = w m − 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 , a i 1 1 W i . α = where a i ∈ Z , and W i are subvarieties. The function 1 1 W ∈ F ( X ) is defined by 1 1 W ( x ) = 1 if x ∈ W , 0 otherwise. For proper morphisms f : X → Y , we define f ∗ : F ( X ) → F ( Y ) : ∫ ∑ a i χ ( W i ∩ f − 1 ( y )) f ∗ α ( y ) : = α = ( y ∈ Y ) f − 1 ( 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 ∫ ∑ α : = pt ∗ α = a i χ ( W i ) . X For proper f : X → Y , pt ∗ f ∗ = ( pt ◦ f ) ∗ thus, ∫ ∫ Fubini-Thm : α = f ∗ α. X Y

3. Constructible functions and Yomdin-Nakai’s formula Let η : C n + ℓ , 0 → C n , 0 be an ℓ -dimensional isolated complete intersection singularity (ICIS). Then, the Milnor number of η is defined by µ η = ( − 1) ℓ ( χ ( B ǫ (0)) ∩ η − 1 ( c )) − 1) , where B ǫ (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 : M n + ℓ → N n 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 : M n + ℓ → N n , where M is compact, connected, and N is connected, it holds that ( ) 1 M + ( − 1) ℓ µ ( f ) f ∗ 1 = χ ( F )1 1 N where F is a generic fibre of f .

3. Constructible functions and Yomdin-Nakai’s formula In fact, for any critical value y ∈ N , ∫ ( ) ( ) 1 M + ( − 1) ℓ µ ( f ) 1 M + ( − 1) ℓ µ ( f ) f ∗ 1 ( y ) = 1 f − 1 ( y ) = 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 ∫ ∫ ∫ ( ) ( ) 1 M + ( − 1) ℓ µ ( f ) 1 M + ( − 1) ℓ µ ( f ) 1 = f ∗ 1 = χ ( F ) 1 1 N , M N N that is, ∫ µ ( f ) = ( − 1) ℓ ( χ ( F ) χ ( N ) − χ ( M )) . M

3. Constructible functions and Yomdin-Nakai’s formula Applying the Fubini-theorem to the lemma, we have ∫ ∫ ∫ ( ) ( ) 1 M + ( − 1) ℓ µ ( f ) 1 M + ( − 1) ℓ µ ( f ) 1 = f ∗ 1 = χ ( F ) 1 1 N , M N N that is, ∫ µ ( f ) = ( − 1) ℓ ( χ ( F ) χ ( N ) − χ ( M )) . M A Morin map f : M → N is of finite-type. Let ι k : ¯ A k ⊂ M be the inclusion. ∑ µ ( f ) = ι k ∗ 1 1 ¯ A k ( f ) . k ≥ 1 ∑ A k ( f )) = ( − 1) ℓ ( χ ( F ) χ ( N ) − χ ( M )) . χ ( ¯ k ≥ 1 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 • C i ( α ) ∈ H 2 i ( X ) , C ∗ ( α ) = C 0 ( α ) + · · · + C r ( 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 1 X ) = c ( TX ) ⌣ [ X ] . The Chern-Schuwartz-MacPherson class of X is C S M ( X ) : = C ∗ (1 1 X ) ∈ 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 → Z 2 , we can asso- ciate a total homology class W ∗ ( α ) ∈ H ∗ ( X ; Z 2 ) so that • W i ( α ) ∈ H i ( X ; Z 2 ) , W ∗ ( α ) = W 0 ( α ) + · · · + W r ( 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 1 X ) = w ( TX ) ⌣ [ X ] 2 .