Torus actions in the normalization problem Jasmin Raissy Dipartimento di Matematica "L. Tonelli" Università di Pisa School-Conference in Complex Analysis and Geometry CIRM, July 13–17, 2009 Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 1 / 21
Normalization Problem Setting Let f : ( C n , p ) → ( C n , p ) be germ of biholomorphism fixing p . Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 2 / 21
Normalization Problem Setting Let f : ( C n , O ) → ( C n , O ) be germ of biholomorphism fixing O . Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 2 / 21
Normalization Problem Setting Let f : ( C n , O ) → ( C n , O ) be germ of biholomorphism fixing O . Locally, using multi-index notation, � f Q z Q , f ( z ) = Λ z + Q ∈ N n | Q |≥ 2 where n � z Q := z q 1 1 · · · z q n n , f Q ∈ C n , | Q | := q j , j = 1 Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 2 / 21
Normalization Problem Setting Let f : ( C n , O ) → ( C n , O ) be germ of biholomorphism fixing O . Locally, using multi-index notation, � f Q z Q , f ( z ) = Λ z + Q ∈ N n | Q |≥ 2 where n � z Q := z q 1 1 · · · z q n n , f Q ∈ C n , | Q | := q j , j = 1 with Λ in Jordan normal form, i.e., Λ = Diag ( λ 1 , . . . , λ n ) + N N = nilpotent matrix and λ 1 , . . . , λ n ∈ C ∗ not necessarily distinct. Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 2 / 21
Normalization Problem Setting Let f : ( C n , O ) → ( C n , O ) be germ of biholomorphism fixing O . Locally, using multi-index notation, � f Q z Q , f ( z ) = Λ z + Q ∈ N n | Q |≥ 2 where n � z Q := z q 1 1 · · · z q n n , f Q ∈ C n , | Q | := q j , j = 1 with Λ in Jordan normal form, i.e., Λ = Diag ( λ 1 , . . . , λ n ) + N N = nilpotent matrix and λ 1 , . . . , λ n ∈ C ∗ not necessarily distinct. Want to know whether ∃ ϕ : ( C n , O ) → ( C n , O ) , local holomorphic change of coordinates, d ϕ O = Id , s.t. ϕ − 1 ◦ f ◦ ϕ has a simple form. Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 2 / 21
Normalization Problem Linearization Linearization problem simple = linear Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 3 / 21
Normalization Problem Linearization Linearization problem simple = linear Idea: first to search for a formal solution of (1) f ◦ ϕ = ϕ ◦ Λ and then to study the convergence of ϕ . Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 3 / 21
Normalization Problem Linearization Linearization problem simple = linear Idea: first to search for a formal solution of (1) f ◦ ϕ = ϕ ◦ Λ and then to study the convergence of ϕ . We have to recursively solve, for each coordinate j , ϕ j ( z ) = z j + � | Q |≥ 2 ϕ Q , j z Q . λ Q := λ q 1 1 · · · λ q n n ( λ Q − λ j ) ϕ Q , j = Polynomial ( f P , j , ϕ R , k , with P ≤ Q , R < Q ) (2) Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 3 / 21
Normalization Problem Linearization Linearization problem simple = linear Idea: first to search for a formal solution of (1) f ◦ ϕ = ϕ ◦ Λ and then to study the convergence of ϕ . We have to recursively solve, for each coordinate j , ϕ j ( z ) = z j + � | Q |≥ 2 ϕ Q , j z Q . λ Q := λ q 1 1 · · · λ q n n ( λ Q − λ j ) ϕ Q , j = Polynomial ( f P , j , ϕ R , k , with P ≤ Q , R < Q ) (2) lexicographic order on N n Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 3 / 21
Normalization Problem Resonances Definition A resonant multi-index for λ ∈ ( C ∗ ) n , rel. to j ∈ { 1 , . . . , n } is Q ∈ N n , with | Q | ≥ 2, s.t. λ Q = λ j . (3) Res j ( λ ) := { Q ∈ N n | | Q | ≥ 2 , λ Q = λ j } . Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 4 / 21
Normalization Problem Resonances Definition A resonant multi-index for λ ∈ ( C ∗ ) n , rel. to j ∈ { 1 , . . . , n } is Q ∈ N n , with | Q | ≥ 2, s.t. λ Q = λ j . (3) Res j ( λ ) := { Q ∈ N n | | Q | ≥ 2 , λ Q = λ j } . Resonances = obstruction to formal linearization. Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 4 / 21
Normalization Problem Poincaré-Dulac normal forms Theorem (Poincaré-Dulac, 1904) ϕ O = Id , s.t. ∀ f as above ∃ � ϕ formal change of coord., d � ϕ − 1 ◦ f ◦ � ] n where g ( O ) = O, dg O = df O and g ϕ = g ∈ C [ [ z 1 , . . . , z n ] � has only resonant monomials, � g Q , j z Q . g j ( z ) = λ j z j + ε j z j + 1 + | Q |≥ 2 λ Q = λ j Moreover, the resonant terms of � ϕ can be arbitrarily chosen, and that choice determines uniquely g res and the remaining terms of � ϕ . A germ of the form Λ + g res , with g res containing only resonant monomials is said in Poincaré-Dulac normal form. Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 5 / 21
Normalization Problem Normalization Problem Given f , ∃ ? ϕ : ( C n , O ) → ( C n , O ) , holomorphic change of coordinates, d ϕ O = Id , s.t. ϕ − 1 ◦ f ◦ ϕ is in Poincaré-Dulac normal form? Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 6 / 21
Normalization Problem Normalization Problem Given f , ∃ ? ϕ : ( C n , O ) → ( C n , O ) , holomorphic change of coordinates, d ϕ O = Id , s.t. ϕ − 1 ◦ f ◦ ϕ is in Poincaré-Dulac normal form? Problem Not uniqueness of the formal change of coordinates � ϕ given by Poincaré-Dulac theorem, and not having explicit expression for g res , make very difficult to give estimates for the convergence of � ϕ . Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 6 / 21
Torus Actions Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 7 / 21
Torus Actions The same problem can be stated for germs of holomorphic vector field near a singular point. Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 7 / 21
Torus Actions The same problem can be stated for germs of holomorphic vector field near a singular point. In 2002, N.T. Zung found that to find a Poincaré-Dulac holomorphic normalization for a germ of holomorphic vector field is the same as to find (and linearize) a suitable torus action which preserves the vector field. Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 7 / 21
Torus Actions Germs commuting with a torus action Theorem (–, 2009) f commutes with a holom. effective T r -action on ( C n , O ) , 1 ≤ r ≤ n, with weight matrix Θ ∈ M n × r ( Z ) � ∃ ϕ local holom. change of coord. s.t. ϕ − 1 ◦ f ◦ ϕ contains only Θ -resonant monomials. Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 8 / 21
Torus Actions Germs commuting with a torus action Theorem (–, 2009) f commutes with a holom. effective T r -action on ( C n , O ) , 1 ≤ r ≤ n, with weight matrix Θ ∈ M n × r ( Z ) � ∃ ϕ local holom. change of coord. s.t. ϕ − 1 ◦ f ◦ ϕ contains only Θ -resonant monomials. f commutes with A : T r × ( C n , O ) → ( C n , O ) , A ( x , O ) = O , means f ( A ( x , z )) = A ( x , f ( z )) . Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 8 / 21
Torus Actions Germs commuting with a torus action Theorem (–, 2009) f commutes with a holom. effective T r -action on ( C n , O ) , 1 ≤ r ≤ n, with weight matrix Θ ∈ M n × r ( Z ) � ∃ ϕ local holom. change of coord. s.t. ϕ − 1 ◦ f ◦ ϕ contains only Θ -resonant monomials. f commutes with A : T r × ( C n , O ) → ( C n , O ) , A ( x , O ) = O , means f ( A ( x , z )) = A ( x , f ( z )) . A lin is semi-simple and Sp ( A lin ( x , · )) = { exp ( 2 π i � r k = 1 x k θ k j ) } j = 1 ,..., n where Θ = ( θ k j ) ∈ M n × r ( Z ) is the weight matrix of A . Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 8 / 21
Torus Actions Germs commuting with a torus action Theorem (–, 2009) f commutes with a holom. effective T r -action on ( C n , O ) , 1 ≤ r ≤ n, with weight matrix Θ ∈ M n × r ( Z ) � ∃ ϕ local holom. change of coord. s.t. ϕ − 1 ◦ f ◦ ϕ contains only Θ -resonant monomials. f commutes with A : T r × ( C n , O ) → ( C n , O ) , A ( x , O ) = O , means f ( A ( x , z )) = A ( x , f ( z )) . A lin is semi-simple and Sp ( A lin ( x , · )) = { exp ( 2 π i � r k = 1 x k θ k j ) } j = 1 ,..., n where Θ = ( θ k j ) ∈ M n × r ( Z ) is the weight matrix of A . z Q e j , with | Q | ≥ 1, is Θ -resonant if n � � Q , θ k � := q h θ k h = θ k ∀ k = 1 , . . . , r . j h = 1 Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 8 / 21
Torus Actions Definition An additive resonant multi-index for θ ∈ C n , rel. to j ∈ { 1 , . . . , n } is Q ∈ N n , with | Q | ≥ 2, s.t. (4) � Q , θ � = θ j . j ( θ ) := { Q ∈ N n | | Q | ≥ 2 , � Q , θ � = θ j } . Res + Jasmin Raissy (Università di Pisa) Torus actions and normalization CIRM July 2009 9 / 21
Recommend
More recommend
