# Factorization of birational maps for qe schemes in characteristic 0

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1 Factorization of birational maps for qe schemes in characteristic 0 AMS special session on Algebraic Geometry joint work with M. Temkin (Hebrew University) Dan Abramovich Brown University October 24, 2014 Abramovich (Brown) Factorization of birational maps October 24, / 1

2 Factorization of birational maps: varieties Theorem (W lodarczyk, ℵ-Karu-Matsuki-W lodarczyk (2002)) Let φ : X 1 X 2 be the blowing up of a coherent ideal sheaf I on a variety X 2 over a field of characteristic 0 and let U X 2 be the complement of the support of I. Assume X 1, X 2 are regular. Then φ can be factored, functorially for smooth surjective morphisms on X 2, into a sequence of blowings up and down of smooth centers disjoint from U: X 1 = V 0 ϕ 1 V 1 ϕ 2... ϕ l 1 V l 1 ϕ l V l = X 2. Abramovich (Brown) Factorization of birational maps October 24, / 1

3 Factorization of birational maps: qe schemes Theorem (ℵ-Temkin) Let φ : X 1 X 2 be the blowing up of a coherent ideal sheaf I on a qe scheme X 2 over a field of characteristic 0 and let U X 2 be the complement of the support of I. Assume X 1, X 2 are regular. Then φ can be factored, functorially for regular surjective morphisms on X 2, into a sequence of blowings up and down of regular centers disjoint from U: X 1 = V 0 ϕ 1 V 1 ϕ 2... ϕ l 1 V l 1 ϕ l V l = X 2. Abramovich (Brown) Factorization of birational maps October 24, / 1

4 Regular morphisms and qe schemes Definition A morphism of schemes f : Y X is said to be regular if it is (1) flat and (2) all geometric fibers of f : X Y are regular. Definition A locally noetherian scheme X is a qe scheme if: for any scheme Y of finite type over X, the regular locus Y reg is open; and For any point x X, the completion morphism Spec Ô X,x Spec O X,x is a regular morphism. Qe schemes are the natural world for resolution of singularities. Abramovich (Brown) Factorization of birational maps October 24, / 1

5 Places where qe rings appear A of finite type over a field or Z, and localizations. A = the formal completion of the above. A = O(X ), where X is an affinoid germ 1 of a complex analytic space. A = O(X ), where X is an affinoid Berkovich k-analytic space, A = O(X ), where X is an affinoid rigid space over k. In all these geometries we deduce factorization in characteristic 0 from factorization over Spec A, which requires GAGA. 1 intersection with a small closed polydisc Abramovich (Brown) Factorization of birational maps October 24, / 1

6 Factorization step 1: birational cobordism We follow W lodarczyk s original ideas Given X 1 = Bl I (X 2 ). On B 0 = P 1 X 2 consider I = I + I 0, where I 0 is the defining ideal of {0} X 2. Set B 1 = Bl I B 0. It contains X 1 as the proper transform of {0} X 2, as well as X 2 Apply canonical resolution of singularities to B 1, resulting in a regular scheme B, projective over X 2, with G m action. So far, this works for schemes. Abramovich (Brown) Factorization of birational maps October 24, / 1

7 Factorization step 2: GIT Equivariant embedding B P X2 (E a1 E ak ), with a i < a i+1. B a1 G m = X 1, B ak G m = X 2. Denoting W ai = B ai G m, W ai + = B ai +ɛ G m, W ai = B ai ɛ G m we have and a sequence W ai + = W (ai+1 ) X 1 W a2 W a3 ϕ2 ϕ 2+ ϕ 3 W a2... ϕ (l 1)+ W al X 2 Abramovich (Brown) Factorization of birational maps October 24, / 1

8 Factorization step 3: local description B ss a i W ai is affine. If b Ba ss i is a fixed point, can diagonalize T B ss a i,b. Tangent eigenvectors lift to eigenfunctions. Locally on W ai get equivariant Ba ss i A dim +1. This chart is regular and inert. Abramovich (Brown) Factorization of birational maps October 24, / 1

9 Factorization step 4: Luna s fundamental lemma Definition (Special orbits) An orbit G m x B ss a i Definition (Inert morphisms) is special if it is closed in the fiber of B ss a i W ai. The G m -equivariant Ba ss i A dim +1 is inert if (1) it takes special orbits to special orbits and (2) it preserves inertia groups. Theorem (Luna s fundamental lemma, [Luna,Alper,ℵ-Temkin]) The regular and inert G m -equivariant Ba ss i equivariant, namely A dim +1 is strongly B ss a i = A dim +1 A dim +1 G m W ai. Abramovich (Brown) Factorization of birational maps October 24, / 1

10 Factorization step 4: torification This is compatible with W ai ± W ai. Locally on W i the transformations W ai ± W ai have toric charts. The process of torification allows us to assume they are toroidal transformations. Toroidal factorization is known [W lodarczyk, Morelli, ℵ-Matsuki-Rashid]. Abramovich (Brown) Factorization of birational maps October 24, / 1

11 Factorization in terms of ideals Recall that X 1 = Bl I (X 2 ). Our factorization X 1 = V 0 ϕ 1 V 1 ϕ 2... ϕ l 1 V l 1 ϕ l V l = X 2 gives a sequence of ideals J i such that V k = Bl Ji (X 2 ), and smooth Z i such that ϕ ±1 i is the blowing up of Z i. Abramovich (Brown) Factorization of birational maps October 24, / 1

12 Transporting to other categories 1: covering Assume X 1 X 2 is a blowing up of an ideal in one of our categories: local, formal, complex, Berkovich, rigid. Say we have a finite cover X 2 = U α X 2 by patches such that O(U α ) is a qe ring which determines U α : ideals correspond to closed sub-objects, coherent sheaves are acyclic, correspond to modules, etc. In complex analysis, U α = X 2 D with D a closed polydisc with the restricted sheaf [Frisch, Bambozzi]. Write X 2 = Spec O(U α), write I corresponding to I, and X 1 = Bl I X 2, which is regular. Factorization of X 1 X 2 gives ideals J i and smooth Z i V k := Bl J i (X 2 ). Abramovich (Brown) Factorization of birational maps October 24, / 1

13 Transporting to other categories 2: GAGA Theorem (Serre s Théorème 3) In each of these categories, the pullback functor h : Coh(P r X 2 ) Coh(Pr X 2 ) is an equivalence which induces isomorphisms on cohomology groups and preserves regularity. Abramovich (Brown) Factorization of birational maps October 24, / 1

14 Transporting to other categories 2: patching X 2 cover X 2 h X 2 We had ideals J i O X 2 and smooth Z i V k := Bl J i (X 2 ) factoring X 1 X 2. We get ideals J i O X 2 and smooth Z i V k := Bl J i (X 2 ) factoring X 1 X 2. Functoriality and Temkin s trick ensure that these agree on X 2 X 2 X 2. Descent gives ideals J i O X2 and smooth Z i V k := Bl Ji (X 2 ) factoring X 1 X 2, as needed. Abramovich (Brown) Factorization of birational maps October 24, / 1

15 An analytic GAGA: statement Theorem (Serre s Théorème 3) Let D be a closed polydisk, A = O(D). The pullback functor h : Coh(P r A ) Coh(Pr D ) is an equivalence which induces isomorphisms on cohomology groups. Abramovich (Brown) Factorization of birational maps October 24, / 1

16 An analytic GAGA: lemmas Lemma (Dimension Lemma) We have H i (P r A, F) = Hi (P r D, h F) = 0 for i > r and all F. Proof. Use Çech covers of P r D by closed standard polydisks! Lemma (Structure Sheaf Lemma) We have H i (P r A, O) = Hi (P r D, O) for all i. Proof. Use proper base change for P r D P r CP n π ϖ D CP n. Abramovich (Brown) Factorization of birational maps October 24, / 1

17 An analytic GAGA: Serre s proof - cohomology Lemma (Twisting Sheaf Lemma) We have H i (P r A, O(n)) = Hi (P r D, O(n)) for all i, r, n. Proof. Induction on r and 0 O P r D (n 1) O P r D (n) O P r 1(n) 0 D H i 1 (P r 1, O(n)) A H i (P r A, O(n 1)) H i (P r A, O(n)) H i (P r 1, O(n)) A H i 1 (P r 1 D, O(n)) H i (P r D, O(n 1)) H i (P r D, O(n)) H i (P r 1 D, O(n)). So the result for n is equivalent to the result for n 1. By the Structure Sheaf Lemma it holds for n = 0 so it holds for all n. Abramovich (Brown) Factorization of birational maps October 24, / 1

18 An analytic GAGA: Serre s proof - cohomology Proposition (Serre s Théorème 1) Let F be a coherent sheaf on P r A. The homomorphism h : H i (P r A, F) Hi (P r D, h F) is an isomorphism for all i. Proof. Descending induction on i for all coherent P r A modules, the case i > r given by the Dimension Lemma. Choose a resolution 0 G E F 0 with E a sum of twisting sheaves. Flatness of h implies 0 h G h E h F 0 exact. H i (P r A, G) H i (P r A, E) H i (P r A, F) H i+1 (P r A, G) H i+1 (P r A, E) = = = H i (P r D, G) H i (P r D, E) H i (P r D, F) H i+1 (P r D, G) H i+1 (P r D, E) so the arrow H i (P r A, F) Hi (P r D, h F) surjective, and so also for G, and finish by the 5 lemma. Abramovich (Brown) Factorization of birational maps October 24, / 1

19 An analytic GAGA: Serre s proof - Homomorphisms Proposition (Serre s Théorème 2) For any coherent P r A-modules F, G the natural homomorphism Hom P r A (F, G) Hom P r D (h F, h G) is an isomorphism. In particular the functor h is fully faithful. Proof. By Serre s Théorème 1, suffices to show that h Hom P r A (F, G) Hom P r D (h F, h G) is an isomorphism. ( ) h Hom P r A (F, G) = Hom O x x (F x, G x ) Ox O x = Hom Ox (F x Ox O x, G x Ox O x ) = Hom P r X (h F, h G) x. by flatness. Abramovich (Brown) Factorization of birational maps October 24, / 1

20 An analytic GAGA: Serre s proof - generation of twisted sheaves Proposition (Cartan s Théorème A) For any coherent sheaf F on P r D there is n 0 so that F(n) is globally generated whenever n > n 0. Proof. Induction on r. Suffices to generate stalk at x. Choose H x, and get an exact sequence 0 O( 1) O O H 0. This gives F( 1) ϕ 1 F ϕ 0 F H 0 which breaks into 0 G F( 1) P 0 and 0 P F F H 0, where G and F H are coherent sheaves on H, Abramovich (Brown) Factorization of birational maps October 24, / 1

21 Serre s proof - generation of twisted sheaves (continued) Proof of Cartan s Théorème A, continued. 0 G F( 1) P 0 and 0 P F F H 0, so right terms in and H 1 (P r D, F(n 1)) H1 (P r D, P(n)) H2 (H, G(n)) H 1 (P r D, P(n)) H1 (P r D, F(n)) H1 (H, F H (n)) vanish for large n. So h 1 (P r D, F(n)) is descending, and when it stabilizes H 1 (P r D, P(n)) H1 (P r D, F(n)) is bijective so H 0 (P r X, F(n)) H0 (H, F H (n)) is surjective. Sections in H 0 (H, F H (n)) generate F H (n) by dimension induction, and by Nakayama the result at x H follows. Abramovich (Brown) Factorization of birational maps October 24, / 1

22 An analytic GAGA: Serre s proof - the equivalence Peoof of Serre s Théorème 3. Choose a resolution O( n 1 ) k ψ 1 O( n 0 ) k 0 F 0. By Serre s Théorème 2 the homomorphism ψ is the analytification of an algebraic sheaf homomorphism ψ, so the cokernel F of ψ is also the analytification of the cokernel of ψ. Abramovich (Brown) Factorization of birational maps October 24, / 1

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