Hirzebruch genera and functional equations Victor M. Buchstaber - PowerPoint PPT Presentation

Hirzebruch genera and functional equations Victor M. Buchstaber Steklov Mathematical Institute, Russian Academy of Sciences International Conference Geometry Days in Novosibirsk — 2014 dedicated to 85th anniversary of Yuri Grigorievich Reshetnyak September 24–27, 2014 Novosibirsk, Russia 1/64

We will consider a smooth oriented manifold with a smooth action of a compact torus, such that all fixed points are isolated. Such manifolds naturally appear in different areas of mathematics. They are the key objects of toric geometry, toric topology, and the theory of homogeneous spaces of compact Lie groups. 2/64

The theory of Hirzebruch genera of manifolds is a well-known area of algebraic topology. It has important applications in the theory of differential operators on manifolds, mathematical physics and combinatorics. In the case of manifolds with compact torus action there is an equivariant Hirzebruch genus and arises the famous rigidity problem for this genus. In many cases this problem is equivalent to the problem of fiberwise multiplicativity of Hirzebruch genera. 3/64

The localization formulas for equivariant genus appear. They give the value of this genus in terms of torus representation in the tangent space at fixed points. The rigidity conditions and localization formulas lead to functional equations that characterize the fundamental fiberwise multiplicative genera. In the talk we will describe the general approach to rigid Hirzebruch genera problem and demonstrate the results for the homogeneous manifolds of compact Lie groups. 4/64

The main construction Let us consider a set Λ = { Λ i , i = 1 , . . . , m } of ( k × n ) -matrices Λ i with integer coefficients and a map ε : [1 , m ] → {− 1 , 1 } . Let A be a commutative associative ring over Q . We associate to each series f ( x ) = x + a 1 x 2 + a 2 x 3 + . . . ∈ A [[ x ]] the characteristic function of the pair (Λ , ε ) : m n 1 � � L (Λ , ε ; f )( t ) = ε ( i ) . (1) f ( � Λ j i , t � ) i =1 j =1 Here t = ( t 1 , . . . , t k ) , Λ j i , j = 1 , . . . , n are k -dimensional column i , t � = Λ j , 1 vectors of Λ i and � Λ j i t 1 + . . . + Λ j , k i t k . 5/64

Admissible pairs Set x f ( x ) = Q ( x ) , Q (0) = 1 . We have:   n n n 1 � � � � � � Λ j L (Λ , ε ; f )( t ) = ε ( i ) Q i , t � . (2)   � Λ j i , t � i =1 j =1 j =1 The pair (Λ , ε ) is called admissible if L (Λ , ε ; f )( t ) ∈ A [[ t ]] for any ring A and any series f ( x ) = x + a 1 x 2 + a 2 x 3 + . . . ∈ A [[ x ]] . 6/64

The universal series It is sufficient to check that the pair (Λ , ε ) is admissible for the universal series a q x q +1 ∈ A [[ x ]] , � f u ( x ) = x + q � 1 where A = � A − 2 n = Q [ a 1 , . . . , a q , . . . ] , deg a q = − 2 q . n � 0 Set deg t l = 2 for l = 1 . . . , k . If the pair (Λ , ε ) is admissible, then � P ω t ω , L (Λ , ε ; f u )( t ) = (3) ω where each ω = ( i 1 , . . . , i k ) is a set of non-negative integers, t ω = t i 1 1 . . . t i k k , | ω | = i 1 + . . . + i k and P ω ∈ A − 2( n + | ω | ) . Note L (Λ , ε ; f u )(0) = P ( a 1 , . . . , a n ) , where P ( · ) = P ∅ ( · ) , deg P ∅ = − 2 n . 7/64

Rigid pairs The pair (Λ , ε ) is called rigid for a family of series F if L (Λ , ε ; f )( t ) ≡ L (Λ , ε ; f )(0) = P ( a 1 , . . . , a n ) ∈ A for any series f ∈ F . Problem Find the solution of rigidity functional equation L (Λ , ε ; f )( t ) ≡ C where C is constant in t , that is, for a given pair (Λ , ε ), find the family of series F and calculate the polynomial C = P ( a 1 , . . . , a n ). 8/64

Manifolds with torus action Theorem For any smooth oriented manifold M 2 n with a smooth action of the compact torus T k such that all the fixed points are isolated there is the correspondence L : ( M 2 n , T k ) → (Λ , ε ) . Proof. Let x 1 , . . . , x m be the set of all fixed points. Then in the tangent space τ i ≃ R 2 n of the point x i a representation of the torus T k is defined. Given a basis in T k one can choose a set of weights i = { Λ j , 1 Λ j i , . . . , Λ j , k i } , j = 1 , . . . , n . 9/64

Manifolds with torus action One can define the map ε : [1 , m ] → {− 1 , 1 } , where ε ( i ) = 1 , if the orientation in τ i , induced by the orientation of the manifold M 2 n , coincides with the orientation in τ i , defined by the set of weights Λ j i , and ε ( i ) = − 1 otherwise. Therefore we have the correspondence L . 10/64

Normal complex T k -manifolds Let ( M 2 n , T k ) be a smooth manifold M 2 n with an action of a torus T k . There is a linear representation of the torus T k in R 2 N ≃ C N and an equivariant embedding M 2 n ⊂ C N . Let ν N ( M 2 n ) be the normal bundle of this embedding. The pair ( M 2 n , T k ) is called normal complex T k -manifold if there exists N such that ν N ( M 2 n ) is a complex T k -bundle. If ( M 2 n , T k ) is a normal complex T k -manifold, then M 2 n is a stably-complex T k -manifold and therefore it is orientable. 11/64

Hirzebruch genus (complex case) Let a q x q +1 ∈ A [[ x ]] , � f ( x ) = x + as before. q � 1 The series n t i � f ( t i ) i =1 can be presented in the form L f ( σ 1 , ..., σ n ) , where σ k is the k -th elementary symmetric polynomial of t 1 , ..., t n . L f ( σ 1 , ..., σ n ) = 1 − a 1 σ 1 +( a 2 1 − a 2 ) σ 2 1 +(2 a 2 − a 2 We have 1 ) σ 2 + . . . 12/64

The Hirzebruch genus L f of a stably complex manifold M 2 n with tangent Chern classes c i = c i ( τ ( M 2 n )) and fundamental cycle � M 2 n � is defined by the formula L f ( M 2 n ) = ( L f ( c 1 , ..., c n ) , � M 2 n � ) ∈ A − 2 n . The universal series f u ( x ) determines the isomorphism L f u : Ω U ⊗ Q → Q [ a 1 , . . . , a q , . . . ] , where Ω U is the ring of cobordisms of stably-complex manifolds and a q , q = 1 , 2 , . . . are the coefficients of f . Any series f ( x ) ∈ A [[ x ]] gives a ring homomorphism L f : Ω U → A . 13/64

Equivariant genus Let ( M 2 n , T k ) be a normal complex T k -manifold M 2 n with an action of a torus T k . Then for any series f ( x ) there is the equivariant genus L f ( M 2 n , T k )( t ) = L f ([ M 2 n ]) + � Q ω t ω , | ω | > 0 where Q ω = L f ( B 2( n + | ω | ) ) . ω Here [ M 2 n ] ∈ Ω − 2 n is the complex cobordism class of M 2 n U and B 2( n + | ω | ) ∈ Ω − 2( n + | ω | ) ⊗ Q for all ω . ω U 14/64

The construction of admissible pairs From localization theorem for equivariant genus (V. Buchstaber, T. Panov, N. Ray IMRN, 2010), we obtain Corollary Let ( M 2 n , T k ) be a normal complex T k -manifold with isolated fixed points. Then the correspondence L : ( M 2 n , T k ) → (Λ , ε ) gives the admissible pair (Λ , ε ) and L f ( M 2 n , T k )( t ) = L ( L ( M 2 n , T k ) , f )( t ) . In particular, for every L ( M 2 n , T k ) the equation holds: m n 1 � � ε ( i ) ≡ 0 . � Λ j i , t � i =1 j =1 15/64

Complex and almost complex manifolds A pair ( M 2 n , T k ) is called a complex T k -manifold, if M 2 n is a complex manifold with a holomorphic action of a torus T k . A pair ( M 2 n , T k ) is called an almost complex T k -manifold, if on the tangent bundle τ ( M 2 n ) there exists a structure of a complex T k -bundle. The structure of a complex or almost complex T k -manifold ( M 2 n , T k ) defines the structure of a normal complex manifold ( M 2 n , T k ) and therefore an admissible pair (Λ , ε ) . For each such pair ε ( i ) = 1 , i = 1 , . . . , m . 16/64

Complex projective spaces C P n = { ( z 1 : . . . : z n +1 ); ( z 1 , . . . , z n +1 ) ∈ C n +1 } has the canonical structure of T n +1 -complex manifold k , . . . , δ n +1 with the fixed points e k = ( δ 1 ) , k = 1 , . . . , n + 1 , k δ i k = 0 if i � = k and δ k k = 1 . The weights at e k are the n -dimensional vectors such that � Λ k j , t � = t j − t k , j � = k , and the signs are ε ( e k ) ≡ 1 . For any series f ( x ) ∈ A [[ x ]] such that f (0) = 0 , f ′ (0) = 1 we get n +1 1 � � f ( t j − t i ) ∈ A [[ t 1 , . . . , t n +1 ]] . i =1 j � = i 17/64

Complex projective line C P 1 = { ( z 1 : z 2 ); ( z 1 , z 2 ) ∈ C 2 } . The action of T 2 on C P 1 : ( z 1 : z 2 ) → ( t 1 z 1 : t 2 z 2 ) has two fixed points (1 : 0) and (0 : 1) . Rigidity functional equation: 1 1 f ( t 2 − t 1 )+ f ( t 1 − t 2 ) ≡ C , where f ( x ) = x + . . . , C = − 2 a 1 . The general analytic solution of this equation is x f ( x ) = q ( x 2 ) − a 1 x , where q (0) = 1 . 18/64

Hirzebruch L -genus — the signature of the manifold Rigidity functional equation for C P 2 is 1 1 1 f ( t 1 − t 2 ) f ( t 1 − t 3 )+ f ( t 2 − t 1 ) f ( t 2 − t 3 )+ f ( t 3 − t 1 ) f ( t 3 − t 2 ) ≡ C . From this equation we get 1 − 2 a 1 a 2 + a 3 ) 2 = 0 . C = 3(2 a 2 (2 a 2 1 − a 2 )( a 3 1 − a 2 ) , If f ( x ) is a solution of this equation and f ( − x ) = − f ( x ) , then f ( x ) + f ( y ) f ( x + y ) = 1 + Cf ( x ) f ( y ) , √ 1 that is f ( x ) = C th ( Cx ) . √ This series determines the most famous Hirzebruch genus, namely, the signature . 19/64