Ordinary Flops

Ordinary Flops

CHIN-LUNG WANG (NCU and NCTS)

(Joint work with H.-W. Lin) September 6, 2004

CONTENTS

1. Ordinary flips/flops and local models 2. Chow motives and Poincar´e pairing

3. Ordinary/quantum product for simple flops 4. Deformations and degenerations

5. Slicing — Mukai flops

1.1 Ordinary (r, r⁰) Flips

X smooth projective,

ψ : X → ¯X log-extremal small contraction, R = R⁺[C], the log-extremal ray,

Z ⊂ X and S ⊂ ¯X: ψ exceptional sets, ψ = ψ|¯ _Z : Z → S, Z_s := ¯ψ⁻¹(s).

ψ is a (r, r⁰) flipping contraction if

(i) ¯ψ : Z = P_S(F ) → S for some rank r + 1 vector bundle F over a smooth base S,

(ii) N_Z/X|_Zs =∼ O_P^r(−1)^⊕(r0+1).

Fact: Let ¯ψ : Z = P_S(F ) → S and V → Z a vector bundle such that V |_Zs is trivial ∀s ∈ S.

Then V ∼= ¯ψ^∗F⁰ for some vector bundle F⁰.

Apply to V = O_P

S(F )(1) ⊗ N_Z/X, we get N_Z/X =∼ O_P

S(F )(−1) ⊗ ¯ψ^∗F⁰. Since O_P

Z(L⊗F )(−1) = ¯φ^∗L ⊗ O_P

Z(F )(−1) for L ∈ Pic(Z), on the blow-up φ : Y = Bl_ZX → X,

E = P_Z(N_Z/X) ∼= P_Z( ¯ψ^∗F⁰) = ¯ψ^∗P_S(F⁰) = P_S(F )×_SP_S(F⁰), N_E/Y = O_PZ(NZ/X)(−1) = ¯φ^∗O_{PS(F )}(−1) ⊗ ¯φ^0∗O_PS(F⁰₎(−1).

Basic diagram: g = ψ ◦ φ : Y → ¯X, ¯g = g|_E,

E = P_S(F ) ×_S P_S(F⁰) ⊂ Y

φ¯

ssffffffffffffffffffffff

φ¯^{0WWWWWWWW ++W}

WW WW WW WW WW W

¯ g ⊂g

²²

Z = P_S(F ) ⊂ X

ψ¯

++XX XX XX XX XX XX XX XX XX XX XX XX

X Z⁰ = P_S(F⁰)

ψ¯⁰

ssggggggggggggggggggggggg

S ⊂ ¯X

The pair (F, F⁰) is unique up to a twisting:

(F, F⁰) ∼ (F ⊗ L, F⁰ ⊗ L^∗) for all L ∈ Pic(S).

Theorem 1 Ordinary (r, r⁰)-flip f : X 99K X⁰ exists. Moreover, Y = ¯Γ_f = X ×_X¯ X⁰ ⊂ X × X⁰.

Proof. From 0 → T_C → T_X|_C → N_C/X → 0 and N_C/X =∼ O_C(1)^⊕(r−1) ⊕ O_C(−1)^⊕(r0+1),

K_X.C = 2g(C) − 2 − ((r − 1) − (r⁰ + 1)) = r⁰ − r.

Pick a line C_Y ∈ ¯φ⁻¹(pt), φ(C_Y ) = C. Then

K_Y.C_Y = (φ^∗K_X + r⁰E).C_Y = (r⁰ − r) − r⁰ = −r < 0.

Let H be very ample on X and L a supporting divisor of C. Let c = H.C. For large k,

kφ^∗L − (φ^∗H + cE)

is big and nef, and vanishes precisely on [C_Y ].

Thus C_Y is a K_Y -negative extremal ray and

0 0

1.2 Analytic Local Models

F → S, F⁰ → S: holomorphic vector bundles, ψ : Z =¯ P_S(F ) → S, ¯ψ⁰ : Z⁰ = P_S(F⁰) → S,

E = Z ×_S Z⁰ with projections ¯φ and ¯φ⁰.

Y = total space of N := ¯φ^∗O_Z(−1)⊗¯φ^0∗O_Z0(−1), E = zero section, N_E/Y = N .

We have analytic contraction diagram

E

φ¯

}}zzzzzzzz EEEE

φ¯⁰

""E EE

  j _//

Y

φ

||yyyyyyyy φ⁰

""F FF FF FF F

Z

ψC¯CCCCC!!C C

  i _//

X

ψDDDDDDDZ_!!D ⁰

zzzz

ψ¯⁰

}}zzzz

  i⁰ _//

X⁰

ψ⁰

||yyyyyyyyy

S ^ _j0 //X

X and X⁰ are smooth, S = Sing( ¯X).

X = total space of N_Z/X = O_{PS(F )}(−1) ⊗ ¯ψ^∗F⁰, X⁰ = total space of N_Z⁰_/X⁰ = O_PS(F⁰₎(−1) ⊗ ¯ψ^0∗F .

Again, (F, F⁰) and (F₁, F₁⁰) define isomorphic analytic local model if and only if (F₁, F₁⁰) = (F ⊗ L, F⁰ ⊗ L^∗) for some L ∈ Pic(S).

An ordinary (r, r)-flip is called an ordinary P^r flop or simply a P^r flop.

2.1 Canonical Correspondences

M: category of motives. Objects = smooth varieties, morphisms = correspondences

Hom_M( ˆX₁, ˆX₂) = A^∗(X₁ × X₂)

under composition law: for U ∈ A^∗(X₁ × X₂), V ∈ A^∗(X₂ × X₃), p_ij : X₁ × X₂ × X₃ → X_i × X_j,

V ◦ U = p_13∗(p^∗₁₂U.p^∗₂₃V ),

[U ] : A^∗(X₁) → A^∗(X₂); a 7→ p_2∗(U.p^∗₁a).

Induced map on T -valued points Hom( ˆT , ˆX_i):

U_T : A^∗(T × X₁) −→ A^{U ◦ ∗}(T × X₂).

Identity Principle: Let U, V ∈ Hom( ˆX, ˆX⁰).

Then U = V if and only if U_T = V_T for all T . (U_X ◦ ∆_X = V_X ◦ ∆_X0 implies U = V .)

Theorem 2 For ordinary flops f : X 99K X⁰, the graph closure F := ¯Γ_f induces ˆX ∼= ˆX⁰ via F^∗ ◦ F = ∆_X and F ◦ F^∗ = ∆_X0.

Proof. For any T , id_T × f : T × X 99K T × X⁰ is also an ordinary flop. By the identity principle we only need to show F^∗F = id on A^∗(X).

FW = p⁰_∗(¯Γ_f.p^∗W ) = φ⁰_∗φ^∗W.

φ^∗W = ˜W + j_∗^³c(E).¯φ^∗s(W ∩ Z, W )^´

dim W, where 0 → N_E/Y → φ^∗N_Z/X → E → 0 and s(W ∩ Z, W ) is the relative Segre class.

Observation: the error term is lying over W ∩Z.

P^r × P^{r //}_E^{2r+s  //}

²²

Y ^2r+s+1

P^{r //}Z^r+s

²²S^s

Let W ∈ A_k(X). By Chow’s moving lemma we may assume that W intersects Z transversally.

dim W ∩ Z := ` = k + (r + s) − (2r + s + 1) = k − r − 1. Since dim φ<sup>−1</sup>(W ∩ Z) = ` + r < k, we get φ^∗W = ˜W and φ⁻¹(W ) ∩ E = φ⁻¹(W ∩ Z).

Hence FW = W⁰, the proper transform of W in X⁰. (W⁰ may not be transversal to Z⁰.)

Let B be an irreducible component of W ∩ Z and ¯B = ¯ψ(B) ⊂ S with dimension `<sub>B</sub> ≤ `.

Notice that W⁰∩Z⁰ has irreducible components { ¯ψ⁰⁻¹( ¯B)}_B. Let φ^0∗W⁰ = ˜W + ^P_B E_B.

E_B ⊂ ¯φ⁰⁻¹ψ¯⁻¹( ¯B), a P^r × P^r bundle over ¯B.

For the generic point s ∈ ¯B, we thus have dim E_B,s ≥ k − `<sub>B</sub> = r + 1 + (` − `_B) > r = ¹₂2r.

In particular, E_B,s contains positive dimensional fibers of φ and φ⁰ and φ_∗(E_B) = 0. So F^∗FW = W . The proof is completed. ¤

2.2 The Poincar´e Pairing

Corollary 3 Let f : X 99K X⁰ be an ordinary flop. If dim α + dim β = dim X, then

(Fα.Fβ) = (α.β).

That is, F is orthogonal with respect to (−.−).

Proof. α.β = φ^∗α.φ^∗β = (φ^0∗Fα + ξ).φ^∗β = (φ^0∗Fα).φ^∗β = Fα.(φ_∗⁰ φ^∗β) = Fα.Fβ. ¤ Remark: F⁻¹ = F^∗ both in the sense of corre-

3.1 Triple Product for Simple Flops

f : X 99K X⁰ a simple P^r flop, S = pt, h = hyperplane class of Z = P^r,

h⁰ = hyperplane class of Z⁰,

x = [h × P^r], y = [P^r × h⁰] in E = P^r × P^r.

φ^∗[h^s] = x^sy^r − x^s+1y^r−1 + · · · + (−1)^r−sx^ry^s, F[h^s] = (−1)^r−s[h^0s],

φ^0∗α⁰ = φ^∗α+(α.h^r−i) xⁱ + (−1)ⁱ⁻¹yⁱ

x + y , α ∈ Aⁱ(X).

Theorem 4 For simple P^r-flops, α ∈ Aⁱ(X), β ∈ A^j(X), γ ∈ A^k(X) with i ≤ j ≤ k ≤ r, i + j + k = dim X = 2r + 1,

Fα.Fβ.Fγ = α.β.γ + (−1)^r(α.h^r−i)(β.h^r−j)(γ.h^r−k).

Example: r = 2, dim X = 5, (i, j, k) = (1, 2, 2):

T α.T β.T γ = α⁰.β⁰.γ⁰ = φ^0∗α⁰.φ^0∗γ⁰.φ^∗γ⁰

= (φ^∗α + (α.h)E)(φ^∗β + (β.Z)(x − y))(φ^∗γ + (γ.Z)(x − y))

= α.β.γ + (β.Z)(γ.Z)φ^∗α.(x − y)²

+ (α.h)(γ.Z)φ^∗β.E.(x − y) + (α.h)(β.Z)φ^∗γ.E.(x − y) + (α.h)(β.Z)(γ.Z)E.(x − y)²

= α.β.γ + (α.h)(β.Z)(γ.Z).

3.2 Quantum Corrections (Outline)

The three point functions hα, β, γi = ^X

d∈A₁(X) hα, β, γi_0,3,d

= α.β.γ + ^X

k∈N hα, β, γi_0,3,k[C] q^k[C]

+ ^X

d6=k[C] hα, β, γi_0,3,d q^d

and (−, −) determine the quantum product.

The difference of α.β.γ is already determined.

Deformations to the normal cone: X = X ×P¹, Φ : M → X be the blowing-up along Z × {∞}.

M_t = X for all t 6= ∞ and M∼ _∞ = Y ∪ ˜E where E =˜ P_S(N_Z/X ⊕ O). Y ∩ ˜E = E = P_S(N_Z/X) is the infinity part of ˜E. Similarly Φ⁰ : M⁰ → X⁰ = X⁰×P¹ and M_∞⁰ = Y ⁰∪ ˜E⁰. Y = Y ⁰ and E = E⁰.

When S = pt, ˜E ∼= ˜E⁰. J. Li’s degeneration formula (A. Li and Y. Ruan) implies the equiv- alence of hα, β, γi_0,3,d with d 6= k[C].

For simple P¹-flops, the second term gives

X

k(α.k[C])(β.k[C])(γ.k[C])^DI_0,0,k[C]^Eq^k[C].

= (α.C)(β.C)(γ.C) q^[C]

1 − q^[C]

by the multiple cover formula (Voisin).

For simple P²-flops of type (1, 2, 2), hα, β, γi_0,3,k[C] = k(α.C)(β.Z)(γ.Z)

×

Z

M_0,2(P²,k)c_3(k−1)(R¹π_∗e^∗₃O(−1)^⊕3), with e₃ : M_0,3(P², k) → X and π : M_0,3(P², k) → M_0,2(P², k). (Work in progress.)

3.3 Some Explicit Formulae

For P^r-flop with non-trivial base S, α ∈ A^∗(Z) has the form α = ^Pξⁱψ¯^∗a_i; ξ = c₁(O_{P(F )}(−1)), a_i ∈ A^∗(S). E = ¯φ^∗O_{P(F )}(−1) ⊗ ¯φ^0∗Q_F0.

Fα = ^XF(ξⁱ). ¯ψ^0∗a_i = ^X φ¯⁰_∗(c_r(E).¯φ^∗ξⁱ). ¯ψ^0∗a_i.

Fξ¹ = (−1)^r−1(ξ⁰ − ¯ψ^∗[c₁(F ) + c₁(F⁰)]).

Fξ² = (−1)^r−2(ξ⁰² − ¯ψ^∗[(c₁ + c⁰₁).ξ⁰ + (c²₁ + c₁c⁰₁ − c₂ + c⁰₂)]).

Fξ³ = (−1)^r−3(ξ⁰³ − ¯ψ^0∗[(c₁ + c⁰₁)ξ⁰² + (c²₁ + c₁c⁰₁ − c₂ + c⁰₂)ξ⁰ + (c³₁ − 2c₁c₂ − c₂c⁰₁ + c²₁c⁰₁ + c₁c⁰₂ + c₃)]).

4.1 Deformations

Theorem 5 Ordinary flips deform in families:

let f : X 99K X⁰ be an (r, r⁰) flip with base S and X → ∆ be a smooth family with X₀ = X. Then there is a smooth family X⁰ → ∆ and a ∆- birational map F : X 99K X⁰ such that F₀ = f . Moreover, F is also an (r, r⁰) flip, with base S → ∆ an one parameter deformations of S.

Key: the ray [C] is stable in deformations.

Idea. Hilb_C/X is a G(2, r + 1) bundle over S.

N_C/X =∼ O(1)^⊕(r−1) ⊕ O(−1)^⊕(r0+1) ⊕ O^s+1. H¹(C,O(k)) = 0 for all k ≥ −1 implies that Hilb_C/X is smooth at [C] for all C ⊂ Z and the natural map π : Hilb_C/X → ∆ is a smooth fibration with special fiber Hilb_C/X. By the stability of Grassmannian bundles we obtain Z → S → ∆. The supporting line bundles L for C on X is the unique extension of the sup- porting line bundle L for C on X. ¤

4.2 Degenerations

Fact. Every three dimensional smooth flop is the limit of composite of P¹ flops.

Question: What is the closure of composite of general ordinary flops?

5.1 Generalized Mukai Flops

ψ : (X, Z) → ( ¯X, S) with N_Z/X = T_Z/S^∗ ⊗ ¯ψ^∗L, L ∈ Pic(S). Will construct the local model as a section of ordinary flops with F⁰ = F^∗ ⊗ L.

E = P_S(F ) ×_S P_S(F⁰) ⊂ Y

Φ

ssgggggggggggggggggggggg Φ⁰

++XX XX XX XX XX XX XX XX XX XX XX

g

²²

Z = P_S(F ) ⊂ X

ΨXXXXXXXXXXXXX_{XX ++X} XX

XX XX XX

X Z⁰ = P_S(F⁰) ⊂ X⁰

Ψ⁰

ssffffffffffffffffffffffffff

S ⊂ ¯X

Suppose ∃ bi-linear map F ×_S F⁰ → η_S to a line bundle η_S over S. O_{P(F )}(−1) → ¯ψ^∗F pulls back to ¯φ^∗O_P (−1) → ¯g^∗F , hence a linear map

φ¯^∗O_Z(−1) ⊗_E φ¯^0∗O_Z0(−1) → ¯g^∗(F ⊗_S F⁰) → ¯g^∗η_S. Y := inverse image of the zero section of ¯g^∗η_S in Y. X = Φ(Y ) ⊃ Z, X⁰ = Φ⁰(Y ) ⊃ Z⁰, ¯X = g(Y ) ⊃ S with restriction maps φ, φ⁰, ψ, ψ⁰. By tensoring the Euler sequence

0 → O_Z(−1) → ¯ψ^∗F → Q → 0

with S^∗ = O_Z(1) and notice that S^∗⊗Q ∼= T_Z/S, we get by dualization

0 → T_Z/S^∗ → O_Z(−1) ⊗ ¯ψ^∗F^∗ → O_Z → 0.

The inclusion maps Z ,→ X ,→ X leads to 0 → N_Z/X → N_Z/X → N_X/X|_Z → 0.

N_X/X|_Z = O(X)|_Z = ¯ψ^∗O( ¯X)|_S. Denote O( ¯X)|_S by L. Recall N_Z/X =∼ O_P

S(F )(−1) ⊗ ¯ψ^∗F⁰. By tensoring with ¯ψ^∗L^∗, we get

0 → N_Z/X ⊗ ¯ψ^∗L^∗ → O_{PS(F )}(−1) ⊗ ¯ψ^∗(F⁰ ⊗ L^∗) → O_Z → 0.

So F⁰ = F^∗⊗L if and only if N_Z/X = T∼ _Z/S^∗ ⊗ ¯ψ^∗L.

5.2 Mukai Flops as Limits of Isomorphisms

For Mukai flops, L ∼= O_S, F⁰ = F^∗ with duality pairing F ×_S F^∗ → O_S. Consider π : Y → C via

Y → ¯g^∗O_S = O_E =^∼ E × C−→^π2 C.

We get a fibration with Y_t := π⁻¹(t), being smooth for t 6= 0 and Y₀ = Y ∪ E. E = Y ∩ E restricts to the degree (1, 1) hypersurface over each fiber along E → S. Let X_t, X⁰_t and ¯X_t be the proper transforms of Y_t in X, X⁰ and ¯X.

For t 6= 0, all maps in the diagram

Yt

}}{{{{{{{{

C!!C CC CC CC

Xt

BÃÃB BB BB

BB X⁰_t

~~||||||||

X¯_t

are all isomorphisms. For t = 0 this is the Mukai flop. Thus Mukai flops are limits of iso- morphisms. They preserve all interesting in- variants like diffeomorphism type, Hodge type (Chow motive via [Y ]+[E]) and quantum rings etc. In fact all quantum corrections are zero.