Motivic Milnor fiber at infinity Pierrette Cassou-Nogu` es, Michel - PowerPoint PPT Presentation

Motivic Milnor fiber at infinity Pierrette Cassou-Nogu` es, Michel Raibaut IMB (Universit´ e Bordeaux ), LAMA, (Universit´ e Savoie Mont-Blanc) Bielefeld, May 2017

Definitions 1) Equivariant Grothendieck group All algebraic varieties are over C . Let X be an algebraic variety and G an al- gebraic group acting on X . We say that the action is good if all G -orbit is contained in an affine open subset of X . In the following we shall take G = G m the multiplicative group and assume that the actions are good. Let S be an algebraic variety.

We denote by Var G m S × G m the category whose objects are X → S × G m , ( p S , p G m ) , σ • where σ is a good action of G m on X • ∃ n, ∀ x ∈ X, λ ∈ G m , p G m ( σ ( λ, x )) = λ n p G m ( x ) • ∀ x ∈ X, λ ∈ G m , p S ( σ ( λ, x )) = p S ( x )

We consider the Grothendieck ring K 0 (Var G m S × G m ). It is generated by classes [ X → S × G m , σ ], with [ X → S × G m , σ ] = [ Y → S × G m , σ ]+[ X \ Y → S × G m , σ ] for all Y closed in X and invariant by G m . The ring operation is given by [ X × S × G m X ′ → S × G m , σ X × σ X ′ ] = [ X → S × G m , σ X ][ X ′ → S × G m , σ X ′ ] with two technical conditions:

[ X × A n → S × G m , σ ] = [ X × A n → S × G m , σ ′ ] where σ and σ ′ lift the same action on X on X × A n [ X → S × G m , σ ] = [ X → S × G m , σ k ] where for all k > 0, σ k ( λ, x ) = σ ( λ k , x )

We denote by L = [ S × G m × A 1 → S × G m , p S × G m , τ ] where p S × G m is the projection on S × G m and τ ( λ, ( s, µ, x )) = ( s, λµ, x ). S × G m )[ L − 1 ] M G m S × G m = K 0 (Var G m

Let l : S → S ′ be a morphism. There exists a morphism l ! (direct image) from M G m S × G m to M G m S ′ × G m such that l ! ([ X → S × G m , ( p S , p G m ) , σ ]) = [ X → S ′ × G m , ( l ◦ p S , p G m ) , σ ] . Let T ⊂ S and i : T → S the injection, there exists a morphism (restriction) from M G m S × G m to M G m T × G m such that i − 1 ([ X → S × G m , ( p S , p G m ) , σ ]) = [ p − 1 S ( T ) → T × G m , ( p S , p G m ) , σ ] .

2) Arc spaces. Let X be a C -variety. For any natural num- ber n , we denote by L n ( X ) the space of n - jets of X . This set is an algebraic variety whose K -rational points, for any field exten- sion K/ C are the K [ t ] /t n +1 -rational points of X . There are canonical morphisms L n +1 ( X ) → L n ( X ).

The arc space of X , denoted by L ( X ), is the projective limit of this system. This set is a C -scheme and we denote by π n : L ( X ) → L n ( X ) the canonical morphisms. For a non zero element φ in C [[ t ]] or C [ t ] /t n +1 , we denote by ord φ the valuation of φ and by ac ( φ ) its first non zero coefficient. The group G m acts on L ( X ) by σ ( λ, φ )( t ) = φ ( λt ).

a) The motivic Zeta function. Let X be a smooth variety and f : X �→ A 1 a morphism. Let X 0 ( f ) = { x ∈ X | f ( x ) = 0 } and for n ≥ 1 X n ( f ) = { φ ∈ L ( X ) | ord f ( φ ) = n } We can consider [ X n ( f ) → X 0 ( f ) × G m , p, σ ] ∈ M G m X 0 ( f ) × G m where σ the standard action on L ( X ) and p = ( p X 0 ( f ) , p G m )( φ ) = ( φ (0) , ac ( f ( φ ))).

We denote by X m n ( f ), the image by π m of X n ( f ). We have for m ≥ n n ( f )] L − md = [ X n [ X m n ( f )] L − nd And call this the motivic measure of X n ( f ). (Kontsevitch) We define (Denef Loeser) mesX n ( f ) T n ∈ M G m � Z f ( T ) = X 0 ( f ) × G m [[ T ]] . n ≥ 1

b) The modified Zeta function. Let Z be a smooth variety, and U a dense open subset of Z , let F be its complement and let f : Z → A 1 be a morphism. Let n and δ be two positive integers, we consider the arc space Z δ n ( f ) := { ϕ ∈ L ( Z ) | ord f ( ϕ ) = n, ord ϕ ∗ I F ≤ nδ } endowed with the arrow “origin, angular com- ponent” and the standard action of G m on arcs. Then, we consider the modified mo- tivic zeta function (Guibert, Loeser, Merle) n ( f )) T n ∈ M G m Z δ mes( Z δ � f,U ( T ) := Z 0 ( f ) × G m [[ T ]] . n ≥ 1

It is proven that there exists an integer δ 0 such that for all integer δ ≥ δ 0 , the series Z δ f,U ( T ) is rational and its limit when T goes to infinity is independent of δ . We will de- note by S f,U the limit − lim T →∞ Z δ f,U ( T ).

� � c) The motivic Milnor fiber at infinity. Let f be a polynomial in C [ x, y ]. A com- pactification of f is a data ( X, i, ˆ f ) with X an algebraic variety, ˆ f a proper map and i an open dominant immersion, such that the following diagram is commutative A 2 i X , ˆ f � f A 1 � P 1 j where j is the following open dominant im-

mersion A 1 P 1 : → j a �→ [1 : a ] With these notations, we denote by 1 / ˆ f the f − 1 (0). extension of 1 /f on X \ ˆ Let ( X, i, ˆ f ) be a compactification. Let con- For any integer n ∈ N ∗ , we sider δ > 0. consider ord ϕ ∗ I X \ i ( A 2 ) ≤ nδ   X δ   n, A 2 (1 / ˆ  ϕ ∈ L ( X ) f ) = ord 1 f ( ϕ ( t )) = n ˆ 

As n ≥ 1, for any arc ϕ in X δ n , ϕ (0) belongs f − 1 ( ∞ ). So, we have a canonical map to ˆ X δ f − 1 ( ∞ ) × G m n, A 2 (1 / ˆ ˆ f ) → � � ϕ (0) , ac 1 �→ ϕ f ( ϕ ( t )) . ˆ and a G m -action given by ( λ, ϕ )( t ) equal to ϕ ( λt ). In particular, the motivic measure of f ) belongs to M G m X δ n, A 2 (1 / ˆ f − 1 ( ∞ ) × G m . ˆ f )) T n ∈ M G m Z δ mes( X δ � n, A 2 (1 / ˆ f , A 2 ( T ) = f − 1 ( ∞ ) × G m [[ T ]] 1 ˆ ˆ n ≥ 1

Theorem 1 Let ( X, i, ˆ f ) be a compactifica- tion of f and δ > 0 . The zeta function Z δ ( T ) is rational for δ large enough and 1 f , A 2 ˆ C has a limit when T goes to infinity. We de- note ( A 2 T →∞ Z δ ( T ) ∈ M G m S 1 C ) = − lim f − 1 ( ∞ ) × G m . 1 f , A 2 ˆ ˆ ˆ C f S f, ∞ ( A 2 ( A 2 C ) ∈ M G m C ) = ˆ f ! S 1 {∞}× G m ˆ f does not depend on the chosen compactifi- cation and is called motivic Milnor fiber at infinity of f .

(Raibaut, Matsui, Takeuchi) Our aim is to compute S f, ∞ ( A 2 C ) Michel Raibaut has computed S f, ∞ ( A 2 C ) in the case where f is non degenerate for its Newton polygon. We want to generalize these results for all polynomials in two vari- ables.

Newton algorithm 1) Newton polygon at infinity. Let f ( x, y ) = � c a,b x a y b + c 0 , 0 with c 0 , 0 generic. Let Supp f = { ( a, b ) ∈ N 2 | c a,b � = 0 } Let N ∞ ( f ) be the set of compact faces of the convex hull of Supp f and N 0 ∞ ( f ) the set of faces of N ∞ ( f ) which do not contain the origin.

Write pα + qβ = N , ( | p | , | q | ) = 1, the equation of a face γ of dimension 1 of N 0 ∞ ( f ). c a,b x a y b = x a γ y b γ ( x q − µy p ) ν µ � � f γ ( x, y ) = µ ∈ R γ,f ( a,b ) ∈ γ We say that f is non degenerate if ν µ = 1 for all γ ∈ N 0 ∞ ( f ), all µ ∈ R γ,f .

To ( p, q, µ ) we associate a Newton map at infinity : f σ p,q,µ ( v, w ) = f ( µ q ′ v − p , v − q ( w + µ p ′ )) ∈ C [ v − 1 , v, w ] where p ′ p − q ′ q = 1 when p > 0, f σ p,q,µ ( v, w ) = f ( v − p ( w + µ p ′ ) , µ q ′ v − q ) ∈ C [ v − 1 , v, w ] when p ≤ 0.

2) Newton algorithm for C [ v − 1 , v, w ]. Let f ( v, w ) = � d a,b v a w b + d 0 , 0 ∈ C [ v − 1 , v, w ] with d 0 , 0 generic. We consider N 0 ( f ), the set of compact faces of the convex hull of { ( a, b ) + R 2 ≥ 0 , ( a, b ) ∈ Supp f } Let pα + qβ = N , ( p, q ) = 1, the equation of a face γ of dimension 1 of N 0 ( f ). d a,b x a y b + d 0 , 0 � f γ ( x, y ) = ( a,b ) ∈ γ ( x q − µy p ) ν µ = x a γ y b γ � µ ∈ R γ,f

We say that f is non degenerate if ν µ = 1 for all µ and all γ ∈ N 0 ( f ) , all µ ∈ R γ,f . To ( p, q, µ ) we associate a Newton map at the origin : f σ p,q,µ ( v, w ) = f ( µ q ′ v p , v q ( w + µ p ′ )) ∈ C [ v − 1 , v, w ] where p ′ p − q ′ q = 1.

Theorem 2 Let f ( v, w ) = � d a,b v a w b + d 0 , 0 ∈ C [ v − 1 , v, w ] with d 0 , 0 generic, then after a finite number of steps, either the Newton polygon at the origin has one face of di- mension 0 , ( − M, 0) , M ≥ 0 , or one face of dimension 1 containing the origin.

Computation of S f, ∞ ( A 2 C ) 1) Compactification In the following, we consider the compacti- fication ( X, i, ˆ f ) of f with X the set of ([ x 0 : x 1 ] , [ y 0 : y 1 ] , [ z 0 : z 1 ]) ∈ ( P 1 ) 3 such that � � x 1 , y 1 0 y d y 0 y d y z 0 x d x = z 1 x d x 0 f 0 x 0 y 0

i and j are the following open dominant im- mersions A 2 i : → X ( x, y ) �→ ([1 : x ] , [1 : y ] , [1 : f ( x, y )]) A 1 P 1 → j : �→ a [1 : a ] ˆ and f is the following projection which is proper P 1 ˆ f : X → ([ x 0 : x 1 ] , [ y 0 : y 1 ] , [ z 0 : z 1 ]) �→ [ z 0 : z 1 ] .

2) First step From a result of Guibert-Loeser-Merle we can write ( A 2 ( G 2 ( { 0 } × G m )+ S 1 C ) = S 1 m ) + S 1 ˆ ˆ ˆ f f f ( G m × { 0 } ) + S 1 ( { (0 , 0) } ) . S 1 ˆ ˆ f f We apply ˆ f ! and we have S f, ∞ ( A 2 C ) = S f, ∞ ( G 2 m )+ S f (0 ,y ) , ∞ + S f ( x, 0) , ∞ + S f (0 , 0) , ∞