Nearby cycles over general bases Weizhe Zheng Morningside Center of Mathematics, Chinese Academy of Sciences Hong Kong Geometry Colloquium November 25, 2017 Weizhe Zheng Nearby cycles over general bases November 2017 1 / 35
The Milnor fibration Plan of the talk The Milnor fibration 1 Nearby cycles over one-dimensional bases 2 Definition and functoriality The quasi-semistable case Constructibility and duality Nearby cycles over general bases 3 Motivation Definition and properties K¨ unneth formula and applications Duality Weizhe Zheng Nearby cycles over general bases November 2017 2 / 35
The Milnor fibration The Milnor fibration Let f : ( C n +1 , 0) → ( C , 0) be a germ of holomorphic function having an isolated critical point at 0. Theorem (Milnor 1967) For ǫ > 0 small, and 0 < η ≪ ǫ , the restriction of f to B ǫ ∩ f − 1 ( D η ) → D η , where B ǫ ⊂ C n +1 is the ball of radius ǫ centered at 0 and D η ⊂ C is the disk of radius η centered at 0 , induces a fibration over D η − { 0 } . Weizhe Zheng Nearby cycles over general bases November 2017 3 / 35
The Milnor fibration The Milnor fibration Let f : ( C n +1 , 0) → ( C , 0) be a germ of holomorphic function having an isolated critical point at 0. Theorem (Milnor 1967) For ǫ > 0 small, and 0 < η ≪ ǫ , the restriction of f to B ǫ ∩ f − 1 ( D η ) → D η , where B ǫ ⊂ C n +1 is the ball of radius ǫ centered at 0 and D η ⊂ C is the disk of radius η centered at 0 , induces a fibration over D η − { 0 } . The fiber M t = f − 1 ( t ) ∩ B ǫ is homotopy equivalent to a bouquet of µ n-spheres S n ∨ · · · ∨ S n , where µ is the Milnor number: µ = dim C { z 0 , . . . , z n } / ( ∂ f /∂ z 0 , . . . , ∂ f /∂ z n ) . Weizhe Zheng Nearby cycles over general bases November 2017 3 / 35
The Milnor fibration The monodromy action We have � Z µ i = n Φ i := Coker( H i (pt) → H i ( M t )) = 0 i � = n . Letting t turn around 0 gives the monodromy operator T ∈ Aut(Φ i ). Weizhe Zheng Nearby cycles over general bases November 2017 4 / 35
The Milnor fibration The monodromy action We have � Z µ i = n Φ i := Coker( H i (pt) → H i ( M t )) = 0 i � = n . Letting t turn around 0 gives the monodromy operator T ∈ Aut(Φ i ). Conjecture (Milnor) T is quasi-unipotent: the eigenvalues of T are roots of unity. Grothendieck proved this using his theory of nearby and vanishing cycles. Weizhe Zheng Nearby cycles over general bases November 2017 4 / 35
Nearby cycles over one-dimensional bases Plan of the talk The Milnor fibration 1 Nearby cycles over one-dimensional bases 2 Definition and functoriality The quasi-semistable case Constructibility and duality Nearby cycles over general bases 3 Motivation Definition and properties K¨ unneth formula and applications Duality Weizhe Zheng Nearby cycles over general bases November 2017 5 / 35
Nearby cycles over one-dimensional bases Definition and functoriality Grothendieck’s nearby and vanishing cycles Grothendieck first mentioned vanishing cycles in a letter to Serre in 1964. Given a family X → S over a one-dimensional base, Grothendieck (1967) constructed in SGA 7 the complex of vanishing cycles, a complex of sheaves measuring: on the one hand, the singularity of the family; and, on the other, the difference between H ∗ ( X s ) and H ∗ ( X t ). He also constructed a closely related complex of sheaves, called the complex of nearby cycles. Settings: ´ etale or complex analytic. We will concentrate on the ´ etale setting. Weizhe Zheng Nearby cycles over general bases November 2017 6 / 35
Nearby cycles over one-dimensional bases Definition and functoriality A dictionary Let S be the spectrum of a Henselian discrete valuation ring. For simplicity assume S strictly local (in other words, the closed point s ∈ S is separably closed). D η : disk S 0 ∈ D η : the center s ∈ S : the closed point D η − { 0 } : punctured disk η ∈ S : the generic point t ∈ D η − { 0 } η : a separable closure of η ¯ π 1 ( D η − { 0 } , t ) ≃ Z : the fund. group I = Gal(¯ η/η ): the inertia group local systems on D η − { 0 } sheaves on η ´ et We have a short exact sequence 1 → P → I → � ℓ � = p Z ℓ (1) → 1. The wild inertia group P is a pro- p -group, where p is the char. of s . Weizhe Zheng Nearby cycles over general bases November 2017 7 / 35
� � � � � � Nearby cycles over one-dimensional bases Definition and functoriality Nearby cycle functor R Ψ Let X → S be a morphism of schemes. Consider Cartesian squares: j i X s X X ¯ η � S s ¯ η. Let Λ = Z / m Z , m invertible on S (or Z ℓ , Q ℓ , etc., ℓ invertible on S ). We etale topoi. D ( X ) := D (Shv( X ´ work with sheaves of Λ-modules in ´ et , Λ)). For K ∈ D + ( X η ), η ) ∈ D + ( X s ) . R Ψ K := i ∗ Rj ∗ ( K | X ¯ Equipped with an action of the inertia group I . Weizhe Zheng Nearby cycles over general bases November 2017 8 / 35
Nearby cycles over one-dimensional bases Definition and functoriality Vanishing cycle functor Φ For K ∈ D + ( X ), distinguished triangle on X s : K | X s → R Ψ( K | X η ) → Φ( K ) → K | X s [1] . Weizhe Zheng Nearby cycles over general bases November 2017 9 / 35
Nearby cycles over one-dimensional bases Definition and functoriality Vanishing cycle functor Φ For K ∈ D + ( X ), distinguished triangle on X s : K | X s → R Ψ( K | X η ) → Φ( K ) → K | X s [1] . For a geometric point x → X s , distinguished triangle � ( R Ψ K ) x � (Φ K ) x � K x [1] . K x � R Γ( X ( x )¯ R Γ( X ( x ) , K ) η , K ) B ǫ : Milnor ball X ( x ) : strict localization M t : Milnor fiber X ( x )¯ η Weizhe Zheng Nearby cycles over general bases November 2017 9 / 35
Nearby cycles over one-dimensional bases Definition and functoriality Functoriality Let h : X → Y be a morphism of schemes over S . For h smooth, the canonical map h ∗ s R Ψ Y → R Ψ X h ∗ η is an isomorphism. In particular, (Φ X Λ) x = 0 at smooth points x of X / S . Weizhe Zheng Nearby cycles over general bases November 2017 10 / 35
Nearby cycles over one-dimensional bases Definition and functoriality Functoriality Let h : X → Y be a morphism of schemes over S . For h smooth, the canonical map h ∗ s R Ψ Y → R Ψ X h ∗ η is an isomorphism. In particular, (Φ X Λ) x = 0 at smooth points x of X / S . For h proper, the canonical map Rh s ∗ R Ψ X → R Ψ Y Rh η ∗ is an isomorphism. In particular, for X / S proper, long exact sequence: sp � H i ( X s , R Ψ K ) � H i ( X s , Φ K ) � H i +1 ( X s , K ) . H i ( X s , K ) H i ( X ¯ η , K ) Weizhe Zheng Nearby cycles over general bases November 2017 10 / 35
Nearby cycles over one-dimensional bases The quasi-semistable case The quasi-semistable case Assume X regular, flat and of finite type over S , X η smooth and ( X s ) red is a divisor with normal crossings. Theorem (Grothendieck, modulo absolute purity) ( R q ΨΛ) P x ≃ Λ[ I t / nI t ]( − q ) ⊗ Z ∧ q C , where x → X s is a geometric point, C = Ker(( n 1 , . . . , n r ): Z r → Z ) . Here n 1 , . . . , n r are the multiplicities of the branches of X s passing through x, and n = gcd( n 1 , . . . , n r ) . Absolute purity was known then for S / Q , and in general by Gabber 1994. Weizhe Zheng Nearby cycles over general bases November 2017 11 / 35
Nearby cycles over one-dimensional bases The quasi-semistable case The quasi-semistable case Assume X regular, flat and of finite type over S , X η smooth and ( X s ) red is a divisor with normal crossings. Theorem (Grothendieck, modulo absolute purity) ( R q ΨΛ) P x ≃ Λ[ I t / nI t ]( − q ) ⊗ Z ∧ q C , where x → X s is a geometric point, C = Ker(( n 1 , . . . , n r ): Z r → Z ) . Here n 1 , . . . , n r are the multiplicities of the branches of X s passing through x, and n = gcd( n 1 , . . . , n r ) . Absolute purity was known then for S / Q , and in general by Gabber 1994. Topological model for the tame Milnor fiber X ( x ) η t : p -prime homotopy fiber of the homomorphism ( S 1 ) r → S 1 � x n i ( x 1 , . . . , x r ) �→ i . i Weizhe Zheng Nearby cycles over general bases November 2017 11 / 35
Nearby cycles over one-dimensional bases The quasi-semistable case Milnor’s conjecture Corollary In the quasi-semistable case, an open subgroup J of I acts trivially on ( R q ΨΛ) P . Weizhe Zheng Nearby cycles over general bases November 2017 12 / 35
Nearby cycles over one-dimensional bases The quasi-semistable case Milnor’s conjecture Corollary In the quasi-semistable case, an open subgroup J of I acts trivially on ( R q ΨΛ) P . An analytic version of this + Hironaka’s resolution of singularities ⇒ Corollary (Milnor’s conjecture) Let f : ( C n +1 , 0) → ( C , 0) be a germ of holomorphic functions having an isolated critical point at 0 . Then T acts quasi-unipotently on Φ i . Weizhe Zheng Nearby cycles over general bases November 2017 12 / 35
Recommend
More recommend
