ON THE COHOMOLOGICAL CREPANT ... - w w w .math.uzh.ch

ON THE COHOMOLOGICAL CREPANT RESOLUTION CONJECTURE FOR WEIGHTED PROJECTIVE SPACES ` ´ SAMUEL BOISSIERE, ETIENNE MANN, AND FABIO PERRONI Abstract. We prove a modified version of the Cohomological Crepant Resolution Conjecture for the weighted projective spaces P(1, 3, 4, 4) and P(1, . . . , 1, n).

1. Introduction The Cohomological Crepant Resolution Conjecture, as proposed by Y. Ruan [Rua06], predicts the existence of an isomorphism between the orbifold cohomology ring of a Gorenstein orbifold Y and the quantum corrected cohomology ring of any crepant resolution ρ : Z → |Y | of the coarse moduli space |Y | of Y , when it exists. The quantum corrected cohomology ring of Z is a deformation of the cohomology ring whose definition involves Gromov-Witten invariants associated to certain exceptional sets. The conjecture belongs to the so called generalized McKay correspondence which broadly speaking can be viewed as a duality between the algebra of finite groups and the geometry of crepant resolutions. The following examples was used to verify it: • the Hilbert scheme of r points on a projective surface S (see [LQ02] for r = 2, [ELQ03] and [LL] for S = P2 r = 3, and [FG03], [QW02], [Uri05] for r general and S with numerically trivial canonical class); • the Hilbert scheme of r points on a quasi-projective surface S carrying a holomorphic symplectic form (see [LQW04], [LS01] and [Vas01]); • the quotient V /G, where V is a complex symplectic vector space and G is a finite subgroup of Sp(V ) (see [GK04]). Note that in all these cases, except [LQ02] and [ELQ03], the quantum corrected cohomology ring coincides with the cohomology ring. In 2005, it was observed independently by J. Bryan, T. Graber and R. Pandharipande [BGP05] on the example [C2 /Z3 ] and by the third named author [Per] on a class of orbifolds with transversal A2 singularities that the conjecture as stated in [Rua06] is too restrictive. Namely, one has to include more general specializations of the quantum parameters. The initial motivation for this paper was to verify the modified Ruan’s Conjecture for Gorenstein weighted projective spaces of dimension 3. In this case crepant resolutions always exist. The main result of the paper concerns the weighted projective space P(1, 3, 4, 4). The singular locus of P(1, 3, 4, 4) is the disjoint union of an isolate singularity of type 31 (1, 1, 1) (we use Reid’s notation [Rei87]) and a transversal A3 -singularity. Therefore the quantum corrected Chow ring of the crepant resolution has 4 quantum parameters, A∗ρ (Z)(q1 , q2 , q3 , q4 ), 3 of them come from the resolution of the Date: April 4, 2007. F.P. was partially supported by SNF, No 200020-107464/1. 1

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` ´ SAMUEL BOISSIERE, ETIENNE MANN, AND FABIO PERRONI

transversal singularity, q1 , q2 , q3 , and the remaining one comes from the isolated singularity. We prove the following result. Theorem 1.1. For (q1 , q2 , q3 , q4 ) equal to (i, i, i, 0) (resp. (−1, −1, −1, 0)), then we have a ring isomorphism ∼ A∗ (P(1, 3, 4, 4)). A∗ρ (q1 , ..., q4 ) = orb Note that the ring isomorphic is explicitly given by (4.3) and (4.4). This result is interesting for two reasons: it confirms for the transversal A3 -case Conjecture 1.9 in [Per] regarding the values of the qi ’s ; secondly the change of variables is inspired from those of [NW03] (see also [BGP05]). In the paper [CCIT06] the authors prove that there is an isomorphism between the small quantum cohomology of P(1, 1, 1, 3) and that of the crepant resolution F3 . It seems that this result together with Theorem 1.1 implies that the map (4.3) (resp. (4.4)) is a ring isomorphism even for q4 = 1, thus proving Conjecture 3.5 below. The paper is organized as follows. In Section 2 we recall some general facts about weighted projective spaces. In Section 3, we review Ruan’s conjecture for weighted projective spaces and write a receipt which we follow for the study of the conjecture. We prove Theorem 1.1 in Section 4. The Gromov-Witten invariants needed for the verification of the conjecture for P(1, 3, 4, 4) are computed in Section 5. In the last Section we prove that, for P(1, . . . , 1, n), the the orbifold Chow ring is isomorphic to the Chow ring of its crepant resolution. 1.a. Acknowledgments. Part of the work was done during the authors’ stay at SISSA, Trieste, to which the authors are grateful for hospitality and support. Particular thanks go to B. Fantechi for very useful discussions and for her interest in the work. The third named author thanks Y. Ruan who suggested the paper [NW03]. 2. Weighted projective spaces In this section we recall some basic facts about weighted projective spaces. Let n ≥ 1 be an integer and w = (w0 , . . . , wn ) a sequence of integers greater or equal than one. Consider the action of the multiplicative group C? on Cn+1 − {0} given by: λ · (x0 , . . . , xn ) := (λw0 x0 , . . . , λwn xn ). The weighted projective space P(w) is defined as the quotient stack [Cn+1 −{0}/C?]. It is a smooth Deligne-Mumford stack whose coarse moduli space, denoted |P(w)|, is a projective variety of dimension n. According to Cox and P [BCS05], P(w) is a toric stack defined by the following stacky fan N := Zn+1 / wi ei , β : Zn+1 → N is the standard projection. For any c in N , we denote c its image in NQ = N ⊗Z Q. We define the simplicial rational fan Σ in NQ as the fan whose cones are generated by any proper subset of the vectors β(e0 ), . . . , β(en ). For any subset I := {i1 , . . . , ik } ⊂ {0, . . . , n}, set wI = (wi1 , . . . , wik ). There is a natural closed embedding ιI : P(wI ) ,→ P(w). The weighted projective space P(w) comes with a natural invertible sheaf, denoted by OP(w) (1), defined as follows: For any scheme X and any stack morphism X → P(w) given by a principal C? -bundle P → X and a C? -equivariant morphism

ON THE CCRC FOR WEIGHTED PROJECTIVE SPACES

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P → Cn+1 − {0}, one defines OP(w) (1)X as the sheaf of sections of the associated line bundle of P . This sheaf is well-behaved with respect to the embedding ιI : P(wI ) ,→ P(w) in the sense that ι∗I OP(w) (1) = OP(wI ) (1). Recall that a Deligne-Mumford stack is reduced if it contains an open dense subscheme, and is Gorenstein if all its ages are integers. Proposition 2.1. (1) The Deligne-Mumford stack P(w) is reduced if and only if gcd(w0 , . . . , wn ) = 1. (2) The Pn Deligne-Mumford stack P(w) is Gorenstein if and only if wi divides j=0 wj for any i.

Proof. (1) In general, the toric stack associated to its stacky fan (N, β, Σ) is reduced if and only if N is a free abelian group. This implies the first part. (2) For any i ∈ {0, . . . , n}, set: Ui := {(x0 , . . . , xn ) ∈ Cn+1 − {0} | xi = 1}.

The trivial C? -bundle Ui × C? and the C? -equivariant morphism: ϕ : Ui × C? → Cn+1 − {0} (u, λ) 7→ λ · u

define an étale morphism Ui → P(w) such that ti Ui → P(w) is a covering. The group Uwi of wi -th roots of the unity acts linearly on Ui with Pndeterminant at a fixed point equal to λ|w| , where we denote |w| := j=0 wj . Therefore λ|w| = 1 if and only if for any i, wi divides |w|. Hence, the stack P(w) has an open cover by the quotient stacks [Ui /Uwi ]. Then P(w) is Gorenstein if and only if the action of Uwi on Ui is in SL(n, C) for all i. This action is in SL(n, C) if and only if wi divides |w|. In dimension 1, the only weighted projective space which is reduced and Gorenstein is P(1, 1) ∼ = P1 . In dimension 2 and 3, the complete list of reduced and Gorenstein weighted projective spaces are given by the following weights :

(2.2)

Dimension 2 (1, 1, 1) (1, 1, 2) (1, 2, 3)

(1, 1, 1, 1) (1, 1, 1, 3) (1, 1, 2, 2) (1, 3, 4, 4)

Dimension 3 (1, 2, 2, 5) (2, 3, 3, 4) (2, 3, 10, 15) (1, 1, 4, 6) (1, 2, 6, 9) (1, 6, 14, 21) (1, 2, 3, 6) (1, 4, 5, 10) (1, 1, 2, 4) (1, 3, 8, 12)

In dimension n, the problem of determining all reduced and Gorenstein P(w) is equivalent to the problem of Egyptian fractions, i.e. the number of solutions of 1 = x10 + · · · + x1n with 1 ≤ x0 ≤ . . . ≤ xn (see [Slo]). Hence, there is a finite number. 3. The Cohomological Crepant Resolution Conjecture In this Section we review the statement of Ruan’s Cohomological Crepant Resolution Conjecture for weighted projective spaces and refer to [Rua06] for the general case.

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Let P(w) be a reduced Gorenstein weighted projective space. Assume we can construct a crepant resolution ρ : Z → |P(w)| which is a projective toric variety obtained by a subdivision Σ0 of the fan Σ that defines |P(w)|. To formulate the conjecture we have to understand the effective 1-cycles in Z which are contracted by ρ. Let N + (Z) ⊂ A1 (Z; Z) be the monoid of effective 1-cycles in Z. It is generated by rational curves which correspond to the (n − 1)-dimensional cones of Σ0 [Rei83]. Set Mρ (Z) := Kerρ∗ ∩ N+ (Z). Notice that Mρ (Z) is a polyhedral cone and we denote with β1 , ..., βm the generators of its rays. For the rest of the paper we assume that β1 , ..., βm are linearly independent1 over Q. Any Γ ∈ Mρ (Z) can be written in a unique way as: Γ=

m X

d ` β` ,

`=1

with d` non-negative integers. We now recall the definition of the quantum corrected Chow ring of Z. We assign a formal variable q` for each β` , hence Γ ∈ Mρ (Z) corresponds to the monomial dm q1d1 · · · qm . The quantum corrected 3-point function is by definition: X d1 dm (3.1) hα1 , α2 , α3 iqc (q1 , ..., qm ) := ΨZ Γ (α1 , α2 , α3 )q1 · · · qm , d1 ,...,dm >0

?

where α1 , α2 , α3 ∈ A (Z) and invariant of Z [Rua06].

ΨZ Γ (α1 , α2 , α3 )

is the genus zero Gromov-Witten

Assumption 1. We assume that (3.1) defines an analytic function of the variables q1 , ..., qm on some region of the complex space Cm . It will be denoted by hα1 , α2 , α3 iqc . In the following, when we evaluate hα1 , α2 , α3 iqc on a point (q1 , ..., qm ), we will implicitly assume that it is defined on such a point. Following [Per], we define a family of rings depending on the parameters q1 , ..., qm . Definition 3.2. The quantum corrected triple intersection hα1 , α2 , α3 iqc (q1 , ..., qm ) is defined by: hα1 , α2 , α3 iρ (q1 , ..., qm ) := hα1 , α2 , α3 i + hα1 , α2 , α3 iqc (q1 , ..., qm ), R where hα1 , α2 , α3 i := Z α1 ∪ α2 ∪ α3 . The quantum corrected cup product α1 ∗ρ α2 is defined by requiring that:

hα1 ∗ρ α2 , αi = hα1 , α2 , αiρ (q1 , ..., qm ) R where hα1 , α2 i := Z α1 ∪ α2 .

∀α ∈ A? (Z),

The following result holds.

Proposition 3.3. For any (q1 , ..., qm ) belonging to the domain of the quantum corrected 3-point function, the quantum corrected cup product ∗ρ satisfies the following properties: Associativity: it is associative on A? (Z), moreover it has a unit which coincides with the unit of the usual cup product of Z. Commutativity: α1 ∗ρ α2 = α2 ∗ρ α1 for any α1 , α2 ∈ A? (Z). 1This condition is satisfied for the weighted projective spaces that we are considering.


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Homogeneity: for any α1 , α2 ∈ A? (Z), deg (α1 ∗ρ α2 ) = deg α1 + deg α2 . We recall the following definition from [Per]. Definition 3.4. The quantum corrected Chow ring of Z is the family of ring structures on the vector space A? (Z) given by ∗ρ . It will be denoted by A?ρ (Z)(q1 , ..., qm ). We finally come to the conjecture, whose study is one of the motivations of this paper. Conjecture 3.5 (Ruan; Bryan and Graber). Let Y be a smooth Gorenstein Deligne-Mumford stack with coarse moduli space denoted by |Y |. Let ρ : Z → |Y | be a crepant resolution of |Y |. Assume that Mρ (Z) is polyhedral and simplicial with generators β1 , . . . , βm . Then there exist roots of the unity c1 , ..., cm and a ring isomorphism : A?ρ (Z)(c1 , ..., cm ) ∼ = A?orb (Y ). This conjecture was stated by Ruan in 2001 with the constants ci all equal to −1, [Rua06], it is called the Cohomological Crepant Resolution Conjecture. A more general form of the conjecture, relating the genus zero Gromov-Witten theory of Y with that of Z has been proposed by Bryan and Graber [BG06], it is called the Crepant Resolution Conjecture. Conjecture 3.5 is a combination of these two conjectures. We present below the recipe that we will apply for P(1, 3, 4, 4) and P(1, . . . , 1, n). We write it for general reduced and Gorenstein weighted projective spaces so that the reader can have a global view of our strategy. Start with a reduced and Gorenstein weighted projective space. (1) Construct the fan of P(w) as explained in Section 2. This gives (n + 1) rays b0 , b1 , . . . , bn . Then add some rays e1 , . . . , ed and define a new fan Σ0 subdividing Σ, in order to obtain a smooth crepant resolution ρ : Z → |P(w)|. In dimension 2, this is the classical Hirzebruch-Jung algorithm. In dimension 3, this is always possible (see [CR02]). The crepancy will be automatic since we shall always add rays generated by points on the junior simplexes. (2) Denote h := ρ∗ c1 (OP(w) (1)). From [Man05] one gets that c1 (T|P(w)|) = |w|η11 and since ρ is crepant: X X ρ∗ c1 (T|P(w)|) = c1 (TZ ) = bi + ej (see [Ful93]). This implies that h =

1 |w|

P

i

i bi

j

+

P

j

ej . Then compute a presentation

of A? (Z) as a quotient of C[h, e1 , . . . , ed ]. (3) If Mρ (Z) is polyhedral and simplicial, then find the generators as follows. Since OP(w) (1) is ample, one has: Mρ (Z) = {σ ∈ Σ0 (n − 1) − Σ(n − 1) | σ · h = 0} .

where Σ(n − 1) denotes the set of (n − 1)-dimensional cones in the fan Σ. (4) Compute the Gromov-Witten invariants in a basis of A? (Z). Deduce a presentation of the quantum corrected ring A?ρ (Z)(q1 , . . . , qm ) as a quotient ring of C((q1 , . . . , qm ))[h, e1 , . . . , ed ]. (5) Compute a presentation of A?orb (P(w)).

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(6) Find the change of variables and roots of the unity c1 , ..., cm that induce an isomorphism between A?orb (P(w)) and A?ρ (Z)(c1 , . . . , cm ).

4. Proof of the Modified Ruan’s Conjecture for P(1, 3, 4, 4) We follow the recipe presented above. The coarse moduli space |P(1, 3, 4, 4)| has a transversal A3 -singularity 14 (1, 3, 4) on the line x0 = x1 = 0 and a singularity 1 3 (1, 1, 1) at the point [0 : 1 : 0 : 0]. (1) The stacky fan of P(1, 3, 4, 4) is given in Z3 by the vectors b0 = (−3, −4, −4), b1 = (1, 0, 0), b2 = (0, 1, 0) and b3 = (0, 0, 1). To resolve the transversal A3 singularity, we add the rays generated by: 3 b1 + 4 1 e2 = (−1, −2, −2) = b1 + 2 1 e3 = (−2, −3, −3) = b1 + 4 e1 = (0, −1, −1) =

To resolve the singularity generated by

1 3 (1, 1, 1)

1 b0 4 1 b0 2 3 b0 . 4

at the point [0 : 1 : 0 : 0], we add the ray

e4 = (−1, −1, −1) =

1 1 1 b0 + b2 + b3 . 3 3 3

The polytope of the crepant resolution Z is given by Figure 1. b1

e1 e2 PSfrag replacements

e3

b0 e4 b3

b2

Figure 1. Polytope of P(1, 3, 4, 4) and a crepant resolution


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(2) We have A? (Z) ∼ = C[h, e1 , e2 , e3 , e4 ]/I where I is the ideal generated by 3he4 , e1 e3 , e1 e4 , e2 e4 , e3 e4 , e21

− 10he1 − 4he2 − 2he3 + 24h2

e1 e2 + 3he1 + 2he2 + he3 − 12h2

e22 − 6he1 − 12he2 − 2he3 + 24h2 e2 e3 + 3he1 + 6he2 + he3 − 12h2

e23 − 6he1 − 12he2 − 14he3 + 24h2 1 16h2 e1 , 16h2 e2 , 16h2 e3 , 16h3 − e34 . 27 (3) The cone Mρ (Z) is generated by the classes β1 := PD(4he1 ), β2 := PD(4he2 ), β3 := PD(4he3 ), β4 := PD(− 13 e24 ). (4) To compute the quantum corrections, we first notice that a curve of homology class d4 β4 is disjoint from any curve of class d1 β1 + d2 β2 + d3 β3 , hence the 3- point function has the following form

=

hα1 α2 α3 iqc (q1 , q2 , q3 , q4 ) 3 Z X Y d1 ,d2 ,d3 >0

+

X

d4 >0

3 Z Y

i=1

i=1

P3

i=1

αi d4 β 4

!

di β i

αi

!

vir d1 d2 d3 q1 q2 q3 deg M0,0 (Z, d1 β1 + d2 β2 + d3 β3 )

deg[M0,0 (Z, d4 β4 )]vir q4d4 .

It follows that ei ∗ρ e4 e4 ∗ρ e4

= ei ∪ e4 if i 6= 4, = (q4 )e4 ∪ e4

and

for some function (q4 ) such that (0) = 1. As in the isomorphism of rings that we will define later, we will put q4 = 0 ; then we only consider classes Γ := d1 β1 + d2 β2 + d3 β3 for di ∈ N. We set βµν := βµ + · · · + βν for µ, ν ∈ {1, 2, 3} and µ ≤ ν. Using Theorem 5.1 we get: ( 0 if Γ 6= dβµν vir deg[M0,0 (Z, Γ)] = 3 1/d if ∃µ, ν ∈ {1, 2, 3} such that Γ = dβµν with d ∈ N. The remaining part of the multiplicative table of A∗ρ (Z)(q1 , q2 , q3 , 0) is as follows: q1 q1 q2 q1 q2 q3 e1 ∗ρ e1 = −24h2 + 10 + 16 he1 +4 +4 1 − q1 1 − q 1 q2 1 − q 1 q2 q3 q2 q1 q2 q2 q3 q1 q2 q3 + 4+4 +4 +4 +4 he2 1 − q2 1 − q 1 q2 1 − q 2 q3 1 − q 1 q2 q3 q1 q2 q3 q2 q3 +4 he3 + 2+4 1 − q 2 q3 1 − q 1 q2 q3

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q1 q1 q2 e1 ∗ρ e2 = −12h + −3 − 8 +4 he1 1 − q1 1 − q 1 q2 q1 q2 q2 q3 q2 he2 +4 −4 + −2 − 8 1 − q2 1 − q 1 q2 1 − q 2 q3 q2 q3 + −1 − 4 he3 1 − q 2 q3 2

e1 ∗ρ e3 =

q1 q2 q1 q2 q3 −4 he1 +4 1 − q 1 q2 1 − q 1 q2 q3 q2 q1 q2 q2 q3 q1 q2 q3 + 4 he2 −4 −4 +4 1 − q2 1 − q 1 q2 1 − q 2 q3 1 − q 1 q2 q3 q1 q2 q3 q2 q3 he3 +4 + −4 1 − q 2 q3 1 − q 1 q2 q3

q1 q1 q2 he1 e2 ∗ρ e2 = −24h + 6 + 4 +4 1 − q1 1 − q 1 q2 q2 q1 q2 q2 q3 + 12 + 16 he2 +4 +4 1 − q2 1 − q 1 q2 1 − q 2 q3 q2 q3 q3 he3 +4 + 2+4 1 − q3 1 − q 2 q3 2

q1 q2 he1 e2 ∗ρ e3 = 12h2 + −3 − 4 1 − q 1 q2 q2 q1 q2 q2 q3 + −6 − 8 −4 +4 he2 1 − q2 1 − q 1 q2 1 − q 2 q3 q3 q2 q3 + −1 − 8 +4 he3 1 − q3 1 − q 2 q3 q1 q2 q1 q2 q3 e3 ∗ρ e3 = −24h2 + 6 + 4 +4 he1 1 − q 1 q2 1 − q 1 q2 q3 q1 q2 q2 q3 q1 q2 q3 q2 +4 +4 +4 he2 + 12 + 4 1 − q2 1 − q 1 q2 1 − q 2 q3 1 − q 1 q2 q3 q3 q2 q3 q1 q2 q3 he3 + 14 + 16 +4 +4 1 − q3 1 − q 2 q3 1 − q 1 q2 q3 (5)For g ∈ {1/3, 1/4, 1/2, 3/4}, denote by 1g the class in A0 (P(w)(exp(2igπ)) ) where P(w)(exp(2igπ)) is the twisted sector corresponding to exp(2igπ). From [BCS05] or [Man05] or [CCLT06], we deduce that the orbifold Chow ring A?orb (P(1, 3, 4, 4)) is generated by H := c1 (OP(w) (1)), E1 := 11/4 , E2 := 11/2 , E3 := 13/4 and E4 := 11/3 , with relations: HE4 , E1 E1 − 3HE2 , E1 E2 − 3HE3 , E1 E3 − 3H 2 ,

E2 E2 − 3H 2 , E2 E3 − HE1 , E3 E3 − HE2 , 16H 3 − E43 .


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(6) The primitive fourth roots of the unity are i and −i. We will construct isomorphisms (4.1) (4.2)

A?ρ (Z)(i, i, i, 0) ∼ = A?orb (P(1, 3, 4, 4)) A? (Z)(−i, −i, −i, 0) ∼ = A? (P(1, 3, 4, 4)). ρ

orb

We first assign to the quantum parameters the values q1 = q2 = q3 = i and q4 = 0, so the quantum corrected cohomology ring A?ρ (Z) is generated by h, e1 , e2 , e3 , e4 with relations: e1 ∗ρ e1 = −24h2 + (−2 + 6i)he1 − 4he2 + (−2 − 2i)he3 e1 ∗ρ e2 = 12h2 + (−1 − 4i)he1 + (2 − 4i)he2 + he3 e1 ∗ρ e3 = −2ihe1 − 2ihe3

e2 ∗ρ e2 = −24h2 + (2 + 2i)he1 + 8ihe2 + (−2 + 2i)he3 e2 ∗ρ e3 = 12h2 − he1 + (−2 − 4i)he2 + (1 − 4i)he3

e3 ∗ρ e3 = −24h2 + (2 − 2i)he1 + 4he2 + (2 + 6i)he3 e4 ∗ρ e4 = e 4 ∪ e 4 .

∼

We construct an isomorphism for A?ρ (Z)(i, i, i, 0) − → A?orb (P(1, 3, 4, 4)) by   h 7−→ H   √ √    E3 e1 7−→ − 2√· E1 − 2i · E2 + 2 · √ (4.3) e2 7−→ −i · 2 · E1 + 2i · E2 − i · 2 · E3  √ √   e3 7−→ 2 · E1 − 2i · E2 − 2 · E3    e 7−→ 3 · exp 2πi · E 4 4 3

For q1 = q2 = q3 = −i and q4 = 0, an isomorphism for (4.2) is given by   h 7−→ H   √ √    · E3 e1 7−→ − √2 · E1 + 2i · E2 + 2√ (4.4) e2 7−→ i · 2 · E1 − 2i · E2 + i · 2 · E3 √ √    e3 7−→ 2 · E1 + 2i · E2 − 2 · E3    e 7−→ 3 · exp 2πi · E 4 4 3

Note that the previous isomorphisms are closely related to these given by W. Nahm and K. Wendland [NW03] or by J. Bryan, T. Graber and R. Pandharipande [BGP05]. 5. Gromov-Witten invariants of the resolution of |P(1, 3, 4, 4)| In this Section, we compute the genus 0 Gromov-Witten invariants of the crepant resolution of |P(1, 3, 4, 4)| with homology class Γ = d1 β1 + d2 β2 + d3 β3 . Our result confirms Conjecture 5.1 [Per]. We follow notation from Section 4. Theorem 5.1. Let ρ : Z → |P(1, 3, 4, 4)| be the crepant resolution of |P(1, 3, 4, 4)| defined in Section 4 and Γ = d1 β1 + d2 β2 + d3 β3 . Then ( Pν 1/d3 if Γ = d i=µ βi , µ ≤ ν ∈ {1, 2, 3} vir deg[M0,0 (Z, Γ)] = 0 otherwise.

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To prove this Theorem, we use the deformation invariance property of the Gromov-Witten invariants. Let U be an open neighborhood of the singular locus |P(4, 4)| of |P(1, 3, 4, 4)|. We construct an explicit deformation of U and a simultaneous resolution. 5.a. The neighborhood. The transversal A3 -singularity is identified with |P(4, 4)| by th closed embedding [x2 : x3 ] 7→ [0 : 0 : x2 : x3 ]. For i ∈ {0, 1, 2, 3}, denote Ui := {[x0 : x1 : x2 : x3 ] ∈ |P(1, 3, 4, 4)| such that xi 6= 0},

then U =:= U2 ∪ U3 ⊂ |P(1, 3, 4, 4)| is an open neighborhood of |P(4, 4)|. Consider the bundle morphism ϕ : O(1) ⊕ O(3) ⊕ O(1) −→ O(4)

(ξ, η, ζ) 7−→ ξ ⊗ η − ζ ⊗4

where we identify O(a) ⊗ O(b) with O(a + b) by the canonical isomorphism. DeO(4) note s0 the zero section of the bundle O(4) → P1 . Then the inverse image O(4) −1 ϕ (Im(s0 )) is a 3-dimensional subvariety of O(1) ⊕ O(3) ⊕ O(1). O(4)

Lemma 5.2. The scheme U is isomorphic to ϕ−1 (Im(s0

)).

Proof. An easy computation shows that U2 ' Spec C[s, u, v, w]/(uv − w 4 ) and U3 ' Spec C[t, x, y, z]/(xy − z 4 ). The affine open subschemes U2 and U3 glue together on U2 ∩ U3 by the following ring isomorphism C[t, 1t , x, y, z] C[s, 1s , u, v, w] −→ (uv − w4 ) (xy − z 4 ) s 7−→ t−1

u 7−→ t−1 x v 7−→ t−3 y

w 7−→ t−1 z. On the other hand, consider a trivialization of the bundle O(1) ⊕ O(3) ⊕ O(1) on the open V0 = {[z0 , z1 ] | z0 6= 0} ⊂ P1 . On such a trivialization, the morphism ϕ is ϕ |V0 : V0 × C3 −→ V0 × C

(s, v1 , v2 , v3 ) 7−→ (s, v1 v2 − v34 ) O(4)

Hence, we have that ϕ−1 (Im(s0 )) |V0 is Spec C[s, v1 , v2 , v3 ]/(v1 v2 − v34 ). We do O(4) the same on V1 = {[z0 , z1 ] | z1 6= 0} ⊂ P1 . We deduce that U and ϕ−1 (Im(s0 )) are union of the same two affine schemes with the same gluing. This proves that they are isomorphic. 5.b. The deformation. We extend the result of E. Brieskorn [Bri66] on simultaneous resolution of deformations of surfaces An -singularities in order to construct a deformation of U and a simultaneous resolution. Consider the bundle morphism ϕ1 : O(1)⊕4 −→ O(1)

(δ1 , . . . , δ4 ) 7−→ δ1 + · · · + δ4

ON THE CCRC FOR WEIGHTED PROJECTIVE SPACES O(1)

O(1)

and let s0 be the zero section of O(1). Set F := ϕ−1 1 (Im(s0 rank 3 in O(1)⊕4 . Consider now the bundle morphism

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)), a subbundle of

ϕ2 : O(1) ⊕ O(3) ⊕ O(1) ⊕ F −→ O(4)

(ξ, η, ζ, δ1 , . . . , δ4 ) 7−→ ξ ⊗ η − ⊗4i=1 (ζ + δi ),

O(4)

O(4)

and let s0 be the zero section of O(4). The inverse image UF := ϕ−1 )) 2 (Im(s0 is a 6-dimensional subvariety of O(1) ⊕ O(3) ⊕ O(1) ⊕ F. We have the following Cartesian diagram O(4)

O(1) ⊕ O(3) ⊕ O(1) ⊃ ϕ−1 (Im(s0

UF ⊂ O(1) ⊕ O(3) ⊕ O(1) ⊕ F

)) ' U O(1)⊕4

s0

P1

F ⊂ O(1)⊕4

Put ∆ := C. For any section σ ∈ H 0 (P1 , F), we get a deformation of U over P1 as follows U∆ UF U P1 × {0}

Σ

P1 × ∆

F

where Σ : P1 × ∆ → F sends (p, t) to t · σ(p) and U∆ is defined by the requirement that the diagram is Cartesian. We now construct a simultaneous resolution of UF → F. Consider the rational map µ : UF 99K P(O(1) ⊕ O(1)) × P(O(1) ⊕ O(2)) × P(O(1) ⊕ O(3))

(ξ, η, ζ, δ1 , ..., δ4 ) 7−→ (ξ, ζ + δ1 ) × (ξ, (ζ + δ1 ) ⊗ (ζ + δ2 )) × (ξ, ⊗3i=1 (ζ + δi )), denote Graph(µ) the graph of µ. Then take the closure of Graph(µ) in UF ×3i=1 P(O(1) ⊕ O(i)), Graph(µ) ⊂ UF ×3i=1 P(O(1) ⊕ O(i)). Finally, we get the commutative diagram (5.3)

UF

Graph(µ)

F

id

F

where the upper row is the composition of the inclusion with the projection. Claim: The diagram (5.3) is a simultaneous resolution of UF → F. Proof. The property of being a simultaneous resolution is local in F. The diagram (6.3) is fibered over P1 . If we restrict it to an open subset of P1 where O(1) is trivial, then the assertion is exactly the result of E. Brieskorn [Bri66]. 5.c. Computation of the invariants. Let δ ∈ H 0 (P1 , O(1)) be a nonzero section, (2` + 1)πi δ` := exp · δ, 4

` ∈ {1, ..., 4}

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and σ := (δ1 , ..., δ4 ) ∈ H 0 (P1 , F). As before, we consider the map Σ : P1 × ∆ → (p, t) 7→

F t · σ(p).

The pull-back of the diagram (5.3) with Σ gives the following diagram Graph(µ)∆ −−−−→   y

(5.4)

P1 × ∆

id

U∆   y

−−−−→ P1 × ∆

which is a simultaneous resolution of U∆ over P1 × ∆. Denote by Graph(µ)t the fiber of Graph(µ)∆ over t ∈ ∆. We have the following commutative diagram ρ−1 (U ) = Graph(µ)0 ρ0

Graph(µ)∆ ρ∆

Graph(µ)t ρt

U

U∆

Ut

P1 × {0}

P1 × ∆

P1 × {t}

Lemma 5.5. Let δ be a global section of O(1) → P1 that vanishes only at one point. Then, for t 6= 0, the variety Graph(µ)t has only one connected nodal complete curve of genus 0 whose dual graph is of type A3 and which is contracted by ρt (see diagram above). This Lemma implies that, for t 6= 0, Graph(µ)t satisfies the hypothesis of Proposition 2.10 of [BKL01]. We deduce the following formula: ( 1/d3 if Γ = d(βµ + ... + βν ), for µ ≤ ν; (5.6) deg[M0,0 (Graph(µ)t , Γ)]vir = 0 otherwise. Proof of Lemma 5.5. Without lost of generality, we can assume that δ vanishes only at the point [1 : 0]. Let V0 := {[x0 : x1 ] ∈ P1 | x0 6= 0}. As our bundles are trivial over V0 , the 3-fold U is given by V0 × V(xy − z 4 ) inside V0 × C3 . The choice of the δ` ’s implies that the 3-fold Ut is given by V0 × V(xy −

4 Y

i=1

(z + δi tδ)) = V0 × V(xy − z 4 − (tδ)4 ) ⊂ V0 × C3 .

For t ∈ ∆, let πt : Ut → P1 × {t} be the projection. By means of πt , Ut is a family of surfaces over P1 . As t 6= 0 and δ([1 : 0]) = 0, the only singular surface of the family is the surface defined by πt−1 ([1 : 0] × {t}) which is a surface A3 singularity. As ρt : Graph(µ)t → Ut is a simultaneous resolution over P1 ×{t}, the fiber Graph(µ)([1:0],t) is a smooth surface with only one complete connected curve of genus 0 whose dual graph is of type A3 and which is contracted by ρt . For any [x0 : x1 ] 6= [1 : 0], the fiber Graph(µ)([x0 :x1 ],t) is isomorphic to the smooth fiber πt−1 ([x0 : x1 ] × {t}).

Hence, the exceptional locus of the resolution πt : Graph(µ)t → Ut has only one connected nodal complete curve of genus 0 whose dual graph is of type A3 and which is contracted by ρt .


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The following lemma and Formula (5.6) implies Proposition 5.1. Lemma 5.7. For any t ∈ ∆, the following equality holds

deg[M0,0 (Graph(µ)t , Γ)]vir = deg[M0,0 (Z, Γ)]vir .

Proof. Since Γ is the homology class of a contracted curve, we have an isomorphism of moduli stacks (see [Per, Lemma 7.1]) M0,0 (Z, Γ) ' M0,0 (ρ−1 (U ), Γ), in particular the right hand side moduli stack is proper with projective coarse moduli space. The tangent-obstruction theory for M0,0 (Z, Γ) depends only on the restriction of the cotangent sheaf ΩZ to the exceptional divisor. Hence the virtual fundamental classes [M0,0 (ρ−1 (U ), Γ)]vir and [M0,0 (Z, Γ)]vir have the same degree. Then it is enough to prove that for any t ∈ ∆, we have (5.8)

deg[M0,0 (Graph(µ)t , Γ)]vir = deg[M0,0 (ρ−1 (U), Γ)]vir .

We use a deformation invariance argument. Gromov-Witten invariants of projective varieties are invariant under deformation of the target variety. In our case, ρ−1 (U ) and Graph(µ)t are not projective. In the following, we will explain why we have the same result as in the projective case. By construction, the morphism p∆ : Graph(µ)∆ → ∆ is a simultaneous resolution of U∆ → ∆ and it contracts the curve of homology class Γ. As a base change of the smooth morphism Graph(µ) → F, the morphism p∆ is smooth. Show that p∆ factorizes through an embedding followed by a projective morphism. By base change, it is enough to prove this property for the morphism Graph(µ) → F. The following commutative diagram implies the desired property for Graph(µ) → F, hence for p∆ . Graph(µ) ⊂ O(1) ⊕ O(3) ⊕ O(1) ⊕ F

(id,1)

P(O(1) ⊕ O(3) ⊕ O(1) ⊕ F ⊕ OF ) F

To finish the proof of the Lemma, we consider the moduli stack which parameterizes stable maps in Graph(µ)∆ of homology class Γ, relative to ∆. We denote it by M0,0 (Graph(µ)∆ /∆, Γ). As Γ is the class of curves which are contracted by the resolution p∆ and p∆ : Graph(µ)∆ → ∆ factorizes through an embedding followed by a projective morphism, Theorem 1.4.1 of [AV02] implies that the moduli space M0,0 (Graph(µ)∆ /∆, Γ) is a proper Deligne-Mumford stack. Since the class Γ is contracted by p∆ , for any t ∈ ∆ the fiber at t of the natural morphism M0,0 (Graph(µ)∆ /∆, Γ) → ∆ is the proper Deligne-Mumford stack M0,0 (Graph(µ)t , Γ). With the previous properties and the smoothness of p∆ : Graph(µ)∆ → ∆, we can apply Theorem 4.2 of [LT98] or Proposition 7.2 of [BF97] to the relative version of [Beh97] and we deduce Formula (5.8) which finishes the proof. 6. P(1, . . . , 1, n) In this section, we will prove the following proposition.


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Proposition 6.1. Let n be an integer greater or equal to 2. Then there is a ring isomorphism A∗ (Z; C) ∼ = A∗orb (P(1, . . . , 1, n)). | {z } n

Proof. We follow the same strategy that in Section 4. The coarse moduli space |P(1, . . . , 1, n)| has a singularity n1 (1, . . . , 1) at the point [0 : . . . : 0 : 1]. (1) The stacky fan Σ of P(1, . . . , 1, n) is given in the lattice Zn by the vectors b0 = (−1, . . . , −1, −n), bi = (0, . . . , 0, 1i , 0, . . . , 0) for i = 1, . . . , n. To resolve the Pn−1 singularity, we add the ray generated by e = n1 i=0 bi = (0, . . . , 0, −1). The fan Σ0 of Z is obtained from Σ by removing the cone (b0 , . . . , bn−1 ) and replacing it by the cones (b0 , . . . , bbi , . . . , bn−1 , e) for i = 0, . . . , n − 1 (see Figure (2) in case n = 3). A direct computation show that the toric variety Z obtained from the fan Σ0 is smooth. b1

e b2

PSfrag replacements

b0

b3 Figure 2. Polytope of P(1, 1, 1, 3) and a crepant resolution (2) We have A? (Z) ∼ = C[b0 , . . . , bn , e]/I where I is generated by: −b0 + bi

for 1 ≤ i ≤ n − 1,

−nb0 − e + bn , ebn , b0 · · · bn−1 .

+ . . . + bn + e) = b0 + n1 e, one gets: e n A? (Z) ∼ , hei. = C[h, e]/hhn + (−1)n n (3) The new effective curves in Σ0 are PD(b0 · · · bbi · · · bbj · · · e) for1 ≤ i 6= j ≤ n−1. n−2 e . With the relations this gives only one contracted curve Γ := PD h − ne

Using h =

1 2n (b0

(4) For d > 0 and α1 , α2 , α3 ∈ A? (Z), one has ΨZ dΓ (α1 , α2 , α3 ) 6= 0 if: |α1 | + |α2 | + |α3 | = dim[M0,3 (Z, dΓ)]vir = n.

Take the basis 1, h, . . . , hn , e, . . . , en−1 of A? (Z). From [Per] we know that if one αi is some hj , then ΨZ dΓ (α1 , α2 , α3 ) = 0. It remains to compute all Gromov-Witten invariants for αi ∈ {e, . . . , en−1 }, i = 1, 2, 3. We do not compute these invariants: in fact, the ring isomorphism will work without quantum correction, although the Gromov-Witten invariants are a priori not zero.


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(5) Setting H := c1 (OP(w) (1)) and E := 11/n one gets the presentation: A?orb (P(1, . . . , 1, n)) ∼ = C[H, E]/hH n − E n , HEi. (6) The ring isomorphism: ∼

A?orb (P(1, . . . , 1, n)) − → A? (Z) e is obtained by mapping H 7→ h and E 7→ − exp iπ n n.

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[Per] [QW02]

[Rei83] [Rei87]

[Rua06]

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Samuel Boissi` ere, Laboratoire J.A.Dieudonn´ e UMR CNRS 6621, Universit´ e de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice E-mail address: [email protected] ´ Etienne Mann, SISSA, Via Beirut 2-4, 34014 Trieste, Italy E-mail address: [email protected] ¨r Mathematik, Universita ¨ t Zu ¨rich, Winterthurerstrasse Fabio Perroni, Institut fu ¨rich, Switzerland 190, 8057 Zu E-mail address: [email protected]