Factorization of birational maps on steroids
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1 Factorization of birational maps on steroids IAS, April 14, 2015 Dan Abramovich Brown University April 14, 2015 This is work with Michael Temkin (Jerusalem) Abramovich (Brown) Factorization of birational maps April 14, / 23
2 Statement Theorem (ℵ-Temkin, after W lodarczyk 03, ℵ-Karu-Matsuki-W l 02) Let φ : X Y be a projective birational morphism of regular, noetherian qe schemes. Assume either char = 0 or strong resolution holds. Then φ factors as X = V 0 ϕ 1 V 1 ϕ 2... ϕ l 1 V l 1 ϕ l V l = Y, with V i regular, projective over Y, and ϕ i or ϕ 1 i is the blowing up of a regular Z i ( X i or X i+1 ). The factorization is functorial for regular surjective Y 1 Y, namely for X 1 = X Y Y 1 get the factorization with (V i ) 1 = V i Y Y 1 etc. Question What s a regular morphism? what s a qe scheme? Abramovich (Brown) Factorization of birational maps April 14, / 23
3 Regular morphisms and qe schemes Definition f : Y X is regular if flat and all geometric fibers of f : X Y are regular. Definition X is a qe scheme if: locally noetherian, for any Y /X of finite type, Y reg Y is open; and For any x X, Spec Ô X,x X is a regular morphism. I ll give examples if you ask. Lesson: Commutative rings are as bad as you feared. Abramovich (Brown) Factorization of birational maps April 14, / 23
4 Nature Why? Qe schemes are the natural world for resolution of singularities. (Temkin) they show up in nature. IAS nature = C analytic. Can we factor? consider Y = C n, and for X blow up (1, 0) once (2, 0) twice... (n, 0) n times at infinitely near points. There is no way to factor this in finitely many steps. Problem The local rings are noetherian, but not the Stein patches. Abramovich (Brown) Factorization of birational maps April 14, / 23
5 Affinoids and germs Consider closed polydisc D C r ( Stein compact ) and sheaf O D := O C r D (overconvergent functions). Theorem (Frisch 67, Matsumura) A D := Γ(D, O D ) is an excellent regular noetherian ring. Abramovich (Brown) Factorization of birational maps April 14, / 23
6 Correspondences There is an algebraization correspondence: closed complex subspaces of D correspond to closed subschemes of D alg := Spec A D etc. No weird boundary phenomena. There is an analytification functor from schemes of finite type over D alg complex spaces over D. It preserves regularity. Use these as patches to build complex geometry of analytic germs. There is a similar picture with affinoids in rigid analytic or Berkovich spaces, affine formal schemes, etc. Abramovich (Brown) Factorization of birational maps April 14, / 23
7 Analytic factorization Theorem (ℵ-Temkin, generalizing the compact complex manifold case (ℵKMW)) Let Y be a compact nonsingular analytic germ. Any X Y projective bimeromorphic can be factored into blowings up and down as before. This requires GAGA. Theorem (GAGA, Serre s Théorème 3) Analytification induces a cohomology-preserving equivalence Lemma (correspondence) Coh(P n D alg ) Coh(P n D ). For an affinoid Y, analytification induces bijections {Blowings up X /Y alg } {Blowings up X an /Y }, {Factorizations X Y alg } {Factorizations X an Y }. Abramovich (Brown) Factorization of birational maps April 14, / 23
8 Analytic factorization given correspondence Lemma Write Y = Y i with Y i affinoids so Y alg i Write X i := Bl I Y i = X Y Y i, so X alg i Get blowup X alg i Y alg i. qe schemes. = Bl I alg Y alg I Apply algebraic factorization X alg i Y alg i. Analytification gives corresponding X i Y i. The theorem follows from the claim below: Claim (Analytic Patching) regular. Let Y Y 1 Y 2 be affinoid, and X = Bl I Y. Then the restrictions of X 1 Y 1 and X 2 Y 2 to X Y coincide. Abramovich (Brown) Factorization of birational maps April 14, / 23
9 Claim and Lemma By the Correspondence Lemma, Analytic Patching follows from Lemma (Algebraic Patching) Let X alg = Bl I alg Y alg X alg 2 Y alg 2 to X alg Proof.. Then the restrictions of X alg 1 Y alg Y alg coincide. 1 and Let Z = Y alg 1 Y alg 2 and W = Z Y alg. The embeddings Y alg Y alg i and the identity Z Z give two maps h i : W Z. These are regular (Temkin!) and surjective. Write X Z = Bl I alg Z = X alg 1 X alg 2. Note that h1 X Z = h2 X Z, since they are the blowings up of the same ideal sheaf. Functoriality for regular surjective morphisms gives the Lemma. Abramovich (Brown) Factorization of birational maps April 14, / 23
10 About GAGA It is magnificent. You can too: Lemma (Dimension Lemma) We have H i (P n D alg, F) = H i (P n D, F an ) = 0 for i > n and all F. Lemma (Structure Sheaf Lemma) We have H i (P n, O) = H i (P n D alg D, O) for all i. Abramovich (Brown) Factorization of birational maps April 14, / 23
11 Proof of lemmas Proof of Dimension Lemma. Use Čech covers of P n D = n i=0 Dn [1 + ɛ] D by closed standard polydisks. Proof of Structure Sheaf Lemma. By proper base change for P n D P n CP r get π D ϖ CP r R i π O P n D = R i ϖ O P n CP r. More on GAGA in the appendix. Abramovich (Brown) Factorization of birational maps April 14, / 23
12 Factorization step 1: birational cobordism We follow W lodarczyk s original ideas. Much works for schemes. Claim There is (functorially) a regular projective (B Y, O B (1)), with G m action, such that: B ss a min G m = X, B ss a max G m = Y. Proof. If X = Bl I Y, then take the deformation of the normal cone of Z(I ) and resolve singularities to get B. Abramovich (Brown) Factorization of birational maps April 14, / 23
13 Factorization step 2: VGIT Claim The quotient B ss a B ss a G m is affine, a min a a max, Claim There is a functorial factorization of φ into a sequence of B ss a i G m B ss a i G m B ss a i + G m. For this, study relatively affine actions of diagonalizable groups on locally noetherian schemes. The maps result from B ss a i B ss a i B ss a i +. Abramovich (Brown) Factorization of birational maps April 14, / 23
14 Factorization step 3: torific blowups Claim There is (functorially) an invariant ideal J i on B ss a i so that B tor a i := Bl Ji B ss a i, with its exceptional divisor, is toroidal and the G m action is a toroidal action. Over a field k, a pair (B, E) is toroidal if locally it has a regular morphism to a toric variety (X σ, D) with its toric divisor D = X σ T. In general there is a criterion by Kato. The action is toroidal if the map is equivariant for a subgroup of T. The proof requires studying logarithmically regular schemes. The ideal is the torific ideal of ℵ-de Jong and ℵKMW. Abramovich (Brown) Factorization of birational maps April 14, / 23
15 Factorization step 4: Luna s fundamental lemma Definition (Special orbits) An orbit G m x X is special if it is closed in the fiber of X X G m. Definition (Inert morphisms) A G m -equivariant X Y is inert if (1) it takes special orbits to special orbits and (2) it preserves inertia groups. Theorem (Luna s fundamental lemma, [Luna 73,Bardsley-Richardson 85, Alper 10,ℵ-Temkin]) A regular and inert G m -equivariant X Y is strongly regular, namely (1) X G m Y G m is regular and (2) X = Y Y Gm X G m. Abramovich (Brown) Factorization of birational maps April 14, / 23
16 Factorization step 5: torification is torific Claim The following diagram is toroidal: B tor a i B tor a i B tor a i + B tor a i G m Ba tor i G m Ba tor i + G m This uses Luna and the properties of torific ideals. Abramovich (Brown) Factorization of birational maps April 14, / 23
17 Factorization step 6: resolving and patching An argument using canonical resolution of ℵKMW allows one to replace Ba tor i G m by regular toroidal schemes so that Ba tor i 1 + G m Ba tor i G m is a sequence of blowings down and up of nonsingular centers. Finally we have Claim (Morelli 96, W l97, ℵ-Matsuki-Rashid, ℵKMW, ℵ-Temkin) There is a toroidal factorization of B tor a i G m B tor a i + G m, functorial with respect to regular surjective morphisms. This requires generalized cone complexes of ℵ-Caporaso-Payne. Abramovich (Brown) Factorization of birational maps April 14, / 23
18 GAGA appendix: 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 April 14, / 23
19 GAGA appendix: 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 April 14, / 23
20 GAGA appendix: 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 April 14, / 23
21 GAGA appendix: 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 April 14, / 23
22 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 April 14, / 23
23 GAGA appendix: 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 April 14, / 23
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