Samuel Boissi Ère

Étienne Mann

Fabio Perroni

COMPUTING CERTAIN GROMOV-WITTEN INVARIANTS OF THE CREPANT RESOLUTION OF P(1, 3, 4, 4)

Keywords: Mathematics Subject Classification. 53D45, 14J33, 14M25, 14E46

We prove a formula computing the Gromov-Witten invariants of genus zero with three marked points of the resolution of the transversal A3singularity of the weighted projective space P(1, 3, 4, 4) using the theory of deformations of surfaces with An-singularities. We use this result to check Ruan's conjecture for the stack P(1, 3, 4, 4).

§1. Introduction

The main results of this paper concern the weighted projective space P [START_REF] Abramovich | Compactifying the space of stable maps[END_REF][START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF]. The singular locus of its coarse moduli space |P [START_REF] Abramovich | Compactifying the space of stable maps[END_REF][START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF]| is the disjoint union of an isolated singularity of type (1/3)(1, 1, 1) (we use Reid's notation in [START_REF] Reid | Young person's guide to canonical singularities[END_REF]) and a transversal A 3 -singularity. Using toric methods, we construct a crepant resolution Z of |P [START_REF] Abramovich | Compactifying the space of stable maps[END_REF][START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF]|. In Theorem 3.3.1, we determine a formula for certain Gromov-Witten invariants of Z over the A 3 -singularity using the theory of deformations of surfaces with rational double points and the deformation invariance property of Gromov-Witten invariants. We then apply this result in Theorem 5.2.1 to construct a ring isomorphism-predicted in [START_REF] Ruan | The cohomology ring of crepant resolutions of orbifolds[END_REF] by Ruan's cohomological crepant resolution conjecture -between the quantum corrected cohomology ring of Z and the Chen-Ruan orbifold cohomology of P [START_REF] Abramovich | Compactifying the space of stable maps[END_REF][START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF], after evaluating the quantum parameters related to the transversal A 3 -singularity to a fourth root of the unity and putting the last parameter to zero. This last evaluation is quite surprising (in [START_REF] Boissière | The cohomological crepant resolution conjecture for P(1,3,4,4)[END_REF], we show that this parameter can be evaluated to 1). To confirm this property, we show in Proposition 5.3.1 that, for all weighted projective spaces P(1,...,1,n)w i t hn weights equal to 1 which have only one isolated singular point (1/n)(1,...,1), the predicted ring isomorphism can be obtained simply by putting the quantum parameter to zero. §2. Weighted projective spaces Let n ≥ 1 be an integer, and let w =(w 0 ,...,w n ) be a sequence of integers greater than or equal to 1. The multiplicative group C ⋆ acts on C n+1 \{0} by λ • (x 0 ,...,x n ):=(λ w 0 x 0 ,...,λ wn x n ).

The weighted projective space P(w) is defined as the quotient stack [C n+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 Borisov, Chen, and Smith [START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF], P(w) is a toric stack associated to the stacky fan ( 1)

N := Z n+1 / n i=0 w i v i ,β : Z n+1 → N, Σ ,
where v 0 ,...,v n is the standard basis of Z n+1 , β is the canonical projection, and Σ ⊂ N ⊗ Z Q is the fan whose cones are generated by any proper subset of the set {β(v 0 ) ⊗ 1,...,β(v n ) ⊗ 1}.

The weighted projective space P(w) comes with a natural invertible sheaf O P(w) (1) defined as follows: for any scheme Y and any morphism Y → P(w) giv enb yaprincipalC ⋆ -bundle P → Y and a C ⋆ -equivariant morphism P → C n+1 \{0}, O P(w) [START_REF] Abramovich | Compactifying the space of stable maps[END_REF] Y is the sheaf of sections of the associated line bundle of P .

Recall that an orbifold is by definition a smooth Deligne-Mumford stack over C with generically trivial stabilizers. A Gorenstein orbifold is an orbifold such that, at each point, the stabilizer acts with determinant 1 on the tangent space. This implies that the coarse moduli space is a Gorenstein variety, but that it is not equivalent. For instance, the variety |P(1, 3)| is Gorenstein (in fact, smooth, isomorphic to P 1 ) but P(1, 3) is not a Gorenstein orbifold, as is easily seen using the following classical result. Proposition 2.0.1. We have the following.

(1) The Deligne-Mumford stack P(w) is an orbifold if and only if the greatest common divisor of w 0 ,...,w n is 1. (2) An orbifold P(w) is Gorenstein if and only if w i divides n j=0 w j for any i.

In dimension n, the problem of determining all Gorenstein orbifolds P(w) is equivalent to the problem of Egyptian fractions, that is, the number of [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF]). Hence, there is a finite number of such P(w)( o n l yP 1 in dimension 1, three in dimension 2, and 14 in dimension 3). All weighted projective spaces that we will be considering satisfy these two conditions. §3. Gromov-Witten invariants of the resolution of |P(1, 3, 4, 4)|

solutions of 1 = 1/x 0 + •••+1/x n with 1 ≤ x 0 ≤•••≤x n (see

The Mori cone

Let X be a Gorenstein orbifold with coarse moduli space X. Recall that a resolution of singularities ρ :

Z → X is called crepant if ρ * K X ∼ = K Z . Assume furthermore that X and Z are projective. Let N + (Z) ⊂ A 1 (Z; Z)b et h e cone of effective 1-cycles in Z,a n ds e tM ρ (Z):=Ker(ρ ⋆ ) ∩ N + (Z), where ρ ⋆ :A ⋆ (Z; Z) → A ⋆ (X; Z)
is the morphism of Chow groups induced by ρ. The set M ρ (Z) is called the Mori cone of contracted effective curves. Lemma 3.1.1. Let P(w) be a Gorenstein orbifold, and let ρ : Z →|P(w)| be a toric crepant resolution associated to a subdivision Σ ′ of Σ and the identity morphism of N . Then the cone M ρ (Z) is polyhedral.

Proof. Let Σ ′ (n -1) be the set of (n -1)-dimensional cones of Σ ′ .T h e n M ρ (Z)= ν∈Σ ′ (n-1) γ ν [V(ν)] γ ν ∈ N,ρ ⋆ ν∈Σ ′ (n-1) γ ν [V(ν)] =0 ,
where, for any ν ∈ Σ ′ (n -1), V(ν) denotes the rational curve in Z stable under the torus action which is associated to ν,a n dw h e r e[ V ( ν)] is the induced Chow class (see Fulton [START_REF] Fulton | Introduction to Toric Varieties[END_REF]). Now let L ∈ Pic(|P(w)|)b ea na m p l e line bundle. From standard intersection theory, we have (see, e.g., [START_REF]Intersection Theory[END_REF]) [START_REF] Abramovich | Compactifying the space of stable maps[END_REF][START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF]| The coarse moduli space of P(1, 3, 4, 4) has a transversal A 3 -singularity on the line {[0 : 0 : x 2 : x 3 ]} ∼ = P 1 and an isolated singularity of type (1/3)(1, 1, 1) at the point [0 : 1 : 0 : 0]. We identify the stacky fan (N,β,Σ) with Z 3 , {Λ(β(v i ))} i∈{0,1,2,3} , Σ ,w h e r et h ev i are defined in [START_REF] Abramovich | Compactifying the space of stable maps[END_REF] and where Λ : N → Z 3 is the isomorphism defined by sending v 0 → (-3, -4, -4), v 1 → (1, 0, 0), v 2 → (0, 1, 0), and v 3 → (0, 0, 1). A crepant resolution of |P [START_REF] Abramovich | Compactifying the space of stable maps[END_REF][START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF][START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF]| can be constructed using standard methods in toric geometry. Consider the integral points

ρ ⋆ ν∈Σ ′ (n-1) γ ν [V(ν)] = 0 if and only if c 1 (ρ ⋆ L) ∩ ν∈Σ ′ (n-1) γ ν [V(ν)] =0. Since c 1 (ρ ⋆ L) ∩ [V(ν)] ≥ 0 for any ν, it follows that M ρ (Z)= ν∈Σ ′ (n-1) γ ν [V(ν)] γ ν ∈ N,ρ ⋆ ([V(ν)]) = 0 , hence the claim.

Crepant resolution of |P

P 1 := (0, -1, -1) = 3 4 Λ β(v 1 ) + 1 4 Λ β(v 0 ) , P 2 := (-1, -2, -2) = 1 2 Λ β(v 1 ) + 1 2 Λ β(v 0 ) , P 3 := (-2, -3, -3) = 1 4 Λ β(v 1 ) + 3 4 Λ β(v 0 ) , P 4 := (-1, -1, -1) = 1 3 Λ β(v 0 ) + 1 3 Λ β(v 2 ) + 1 3 Λ β(v 3 ) ,
and subdivide Σ by inserting the rays generated by P 1 ,P 2 ,P 3 ,a n dP 4 as showninFigure1.LetΣ ′ be the fan obtained after this subdivision, let Z be Figure 1: Polytope of P(1, 3, 4, 4) and of the crepant resolution Z the associated toric variety, and let ρ : Z →|P(1, 3, 4, 4)| be the birational morphism associated to the identity on Z 3 . One checks easily that Z is smooth and ρ is crepant. This follows from the existence of a continuous piecewise linear function |Σ|→R, which is linear when restricted to each cone of Σ and associates the value -1 to the minimal lattice points of the rays of Σ ′ (see [START_REF] Fulton | Introduction to Toric Varieties[END_REF]Section 3.4]).

Remark 3.2.1. In [2, Proposition 2.2], we showed that the coarse moduli space |P(1, 3, 4, 4)| admits a unique crepant resolution, up to isomorphism. So the toric crepant resolution Z constructed above is unique.

Statement of the main result

By Lemma 3.1.1, the cone M ρ (Z) can be directly determined from the combinatorial data Σ and Σ ′ .Inourcase,M ρ (Z) is generated by four curves: an A 3 -chain Γ 1 ,Γ 2 ,Γ 3 over the transversal A 3 -singularity and one curve Γ 4 over the isolated singularity (see Section 5.2 for the toric equations of these curves). We compute the Gromov-Witten invariants of the crepant resolution Z of genus zero, homology class Γ = d 1 Γ 1 + d 2 Γ 2 + d 3 Γ 3 , and without marked points. We denote by M 0,0 (Z, Γ) the corresponding moduli space. Note that the expected dimension is zero. Our result confirms Perroni's conjecture [16, of |P(1, 3, 4, 4)|, we construct an explicit deformation of V ,a n dt h e nw e construct a simultaneous resolution. This gives a deformation of ρ -1 (V ), a neighborhood of the component of the exceptional divisor which lies over |P(4, 4)|. We will denote this deformation by Graph(μ) t , t ∈ Δ. Then we relate the Gromov-Witten invariants of Z we are interested in with some Gromov-Witten invariants of Graph(μ) t that we can explicitly compute.

The neighborhood

By abuse of notation, for any a ∈ Z,w ed e n o t eb yt h es a m es y m b o l O(a)thesheafO P 1 (a) and the corresponding vector bundle, and we identify O(a) ⊗O(b)w i t hO(a + b) using the canonical isomorphism. For any vector bundle E,w ed e n o t eb y0 E its zero section.

The transversal

A 3 -singularity of |P(1, 3, 4, 4)| is identified with P 1 by the morphism [z 0 : z 1 ] → [0 : 0 : z 0 : z 1 ]. We set V i := [x 0 : x 1 : x 2 : x 3 ] ∈|P(1, 3, 4, 4)| such that x i =0 for any i ∈{0, 1, 2, 3},a n dw es e tV := V 2 ∪ V 3 ⊂|P(1, 3, 4, 4)|.
Consider the bundle morphism

ψ : O(1) ⊕O(3) ⊕O(1) -→ O (4) (ξ,η,ζ) -→ ξ ⊗ η -ζ ⊗4 ,
and consider the inverse image under ψ of the zero section of O( 4)

ψ -1 Im(0 O(4) ) ⊂O(1) ⊕O(3) ⊕O(1). Lemma 4.1.1. The variety V is isomorphic to ψ -1 (Im(0 O(4) )).
Proof. Let U 4 ⊂ C ⋆ be the group of fourth roots of the unity acting linearly on C 3 with weights (1, 3, 0). We have the identification

C 3 /U 4 -→ V 2 (2) [(x 0 ,x 1 ,x 3 )] -→ [x 0 : x 1 :1:x 3 ],
where (x 0 ,x 1 ,x 3 ) are coordinates on C 3 and where [(x 0 ,x 1 ,x 3 )] denotes the equivalence class of the corresponding point. On the other hand, we have the isomorphism

(3) C 3 /U 4 -→ Spec C[s, u, v, w]/(uv -w 4 )
g i v e nb ys e t t i n gs := x 3 ,u := x 4 0 ,v := x 4 1 ,a n dw := x 0 x 1 . The composition of the inverse of ( 2) with (3) gives the isomorphism V 2 ≃ Spec(C[s, u, v, w]/ (uv -w 4 )). In the same way, by setting t := x 2 ,x := x 4 0 ,y := x 4 1 ,a n dz :=

x 0 x 1 , we have the isomorphism

V 3 ≃ Spec(C[t, x, y, z]/(xy -z 4 )).
The affine open subvarieties V 2 ,V 3 ⊂ V glue together by means of the ring isomorphism

C[s, 1 s ,u,v,w] (uv -w 4 ) -→ C[t, 1 t ,x,y,z] (xy -z 4 ) s -→ 1 t u -→ 1 t x v -→ 1 t 3 y w -→ 1 t z.
On the other hand, consider a trivialization of the bundle O(1)⊕O( 3)⊕O( 1)

on W 0 = {[z 0 ,z 1 ] ∈ P 1 | z 0 =0}
. On such a trivialization, the morphism ψ is given by

W 0 × C 3 -→ W 0 × C (s, v 1 ,v 2 ,v 3 ) -→ (s, v 1 v 2 -v 4 3 ). Hencew eha v ethat,o v erW 0 , ψ -1 (Im(0 O(4) )) is Spec(C[s, v 1 ,v 2 ,v 3 ]/(v 1 v 2 - v 4
3 )). If we do the same over

W 1 = {[z 0 ,z 1 ] ∈ P 1 | z 1 =0}, we deduce that V and ψ -1 (Im(0 O(4)
)) are a union of the same affine varieties with the same gluing. This proves that they are isomorphic.

The deformation

We now construct a deformation of V . The construction is inspired by the theory of deformations of surfaces with A n -singularities (see Tyurina [START_REF] Tyurina | Resolution of singularities of plane deformations of double rational points[END_REF]). Consider the fibration f : V → P 1 defined as the composition of the isomorphism 1) ⊕O( 3) ⊕O(1), and then the bundle map. The mor- phism f : V → P 1 exhibits V as a 3-fold fibered over P 1 with fibers isomorphic to a surface A 3 -singularity. Furthermore, the fibration is locally trivial.

V ∼ -→ ψ -1 (Im(0 O(4) )) in Lemma 4.1.1, followed by the inclusion ψ -1 (Im(0 O(4) )) ⊂O(
The aim is to extend some of the results of Tyurina [START_REF] Tyurina | Resolution of singularities of plane deformations of double rational points[END_REF] to V ,w h e n viewed as a family of such surfaces with respect to f : V → P 1 . Consider the bundle morphism

χ : O(1) ⊕4 -→ O (1) (δ 1 ,...,δ 4 ) -→ δ 1 + •••+ δ 4 ,
and set F := χ -1 (Im(0 O(1) )). Then consider the bundle morphism

π : O(1) ⊕O(3) ⊕O(1) ⊕F -→O(4) (ξ,η,ζ,δ 1 ,...,δ 4 ) -→ ξ ⊗ η - 4 i=1 (ζ + δ i ),
and set

V F := π -1 (Im(0 O(4) )). We obtain a Cartesian diagram V f V F F P 1 0 F F
where the arrow F is the composition of the inclusion V F →O(1) ⊕O(3) ⊕ O(1) ⊕F followed by the projection O(1) ⊕O(3) ⊕O(1) ⊕F →F. Note that F : V F →F is a family of surfaces. We now construct a simultaneous resolution. Consider the rational map

μ : V F P O(1) ⊕O(1) × P O(1) ⊕O(2) × P O(1) ⊕O(3) (ξ,η,ζ,δ 1 ,...,δ 4 ) -→ (ξ,ζ + δ 1 ) × ξ,(ζ + δ 1 ) ⊗ (ζ + δ 2 ) × ξ, 3 i=1 (ζ + δ i ) ,
and let Graph(μ) be the graph of μ. We denote by Graph(μ) the closure of Graph(μ)i nV

F × × 3 i=1 P(O(1) ⊕O(i))
.L e tR : Graph(μ) →V F be the composition of the inclusion Graph(μ) →V F × × 3 i=1 P(O(1) ⊕O(i))

followed by the projection on the first factor

V F × × 3 i=1 P(O(1) ⊕O(i)) → V F . Lemma 4.2.1. The diagram (4) Graph(μ) R F •R V F F F id F is a simultaneous resolution of F : V F →F.
Proof. The property of being a simultaneous resolution is local in F . Diagram ( 4) is fibered over P 1 . If we restrict it to an open subset of P 1 where O(1) is trivial, then the assertion is exactly the result obtained by Brieskorn [START_REF] Brieskorn | Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen[END_REF].

Set Δ := C. For any section θ ∈ H 0 (P 1 , F ), we get a deformation of V parameterized by Δ

V f V θ f θ V F F P 1 P 1 × Δ Θ F
where Θ :

P 1 × Δ →F sends ([z 0 : z 1 ],t)t ot • θ([z 0 : z 1 ])
, where V θ is defined by the requirement that the diagram is Cartesian, and where the map

P 1 → P 1 × Δ is the inclusion [z 0 : z 1 ] → ([z 0 : z 1 ], 0). The pullback of diagram (4)
with respect to Θ gives the diagram ( 5)

Graph(μ) θ ρ θ V θ f θ P 1 × Δ id P 1 × Δ
where ρ θ is the pullback of R in (4). Observe that ( 5) is a simultaneous resolution of V θ over P 1 × Δ.

Computation of the invariants

We specialize the previous construction in the case where θ is given as follows. Let δ ∈ H 0 (P 1 , O(1)) be a nonzero section, set

δ ℓ := exp (2ℓ +1)πi 4 • δ, ℓ ∈{1,...,4},
and define θ := (δ 1 ,...,δ 4 ) ∈ H 0 (P 1 , F ). For any t ∈ Δ, we set V t := f -1 θ (P 1 × {t}), f t : V t → P 1 ×{t} as the restriction of f θ ,a n dw es e tGraph(μ) t := ρ -1 θ (V t ), ρ t : Graph(μ) t →V t as the restriction of ρ θ . We have the following commutative diagram

ρ -1 (V )= Graph(μ) 0 ρ 0 =ρ Graph(μ) θ ρ θ Graph(μ) t ρt V f 0 =f V θ f θ V t ft P 1 ×{0} P 1 × Δ P 1 ×{t} Lemma 4.3.1.
Let δ be a global section of O(1) → P 1 that vanishes only at one point. Then, for t =0, the variety Graph(μ) t has only one connected nodal complete curve of genus zero whose dual graph is of type A 3 and which is contracted by ρ t (see the diagram above).

Proof. Without lost of generality, we can assume that δ vanishes only at the point [1 : 0]. Let W 0 := {[z 0 : z 1 ] ∈ P 1 | z 0 =0}. As our bundles are trivial over W 0 , the restriction of V over W 0 is given by W 0 × V(xyz 4 ) ⊂ W 0 × C 3 . The choice of the δ ℓ implies that the 3-fold V t is given by

W 0 × V xy - 4 ℓ=1 (z + δ ℓ t) = W 0 × V xy -z 4 -(tδ) 4 ⊂ W 0 × C 3 .
By means of f t , V t is viewed as a family of surfaces parameterized by P 1 . As t = 0 and δ([1 : 0]) = 0, the only singular surface of the family is the surface f -1 t ([1 : 0] ×{t}), which is a surface with an isolated A 3 -singularity. Since ρ t : Graph(μ) t →V t is a simultaneous resolution over P 1 ×{t},t h e fiber Graph(μ) ([1:0],t) is a smooth surface with only one complete connected curve of genus zero whose dual graph is of type A 3 and which is contracted by ρ t . For any [z 0 :

z 1 ] = [1 : 0], the fiber Graph(μ) ([z 0 :z 1 ],t) is isomorphic to the smooth surface f -1 t ([z 0 : z 1 ] ×{t}).
Hence, the exceptional locus of the resolution ρ t : Graph(μ) t →V t has only one connected nodal complete curve of genus zero whose dual graph is of type A 3 and which is contracted by ρ t .

Let Γ 1 , Γ 2 , Γ 3 ∈ H 2 (Graph(μ) t ; Z) be the homology classes of the compo- nents of the connected nodal complete curve of genus zero whose dual graph is of type A 3 and which is contracted by ρ t . Let us assume that they are numbered in such a way that if Γ i is the class of Γi , then the intersection Γi ∩ Γj is empty if |i -j| > 1. Then Lemma 4.3.1 implies that, for t =0, Graph(μ) t satisfies the hypothesis of [START_REF] B R Y A N | Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds[END_REF]Proposition 2.10]. Therefore, we deduce the following formula:

deg M 0,0 (Graph(μ) t , Γ) vir (6) = 1/d 3 if Γ = d(Γ μ +Γ μ+1 + •••+Γ ν ), for μ ≤ ν,
0o t h e r w i s e .

This formula together with Lemma 4.3.2 completes the proof of Theorem 3.3.1.

Lemma 4.3.2. Let Γ be as in the statement of Theorem 3.3.1. For any t ∈ Δ, the following equality holds:

deg M 0,0 Graph(μ) t , Γ vir =deg[M 0,0 (Z, Γ)] vir .
Proof. Since Γ is the homology class of a contracted curve, we have an isomorphism of moduli stacks (see [START_REF] Perroni | Chen-Ruan cohomology of ADE singularities[END_REF]Lemma 7.1]) [START_REF] Bryan | The orbifold quantum cohomology of C 2 /Z3 and Hurwitz-Hodge integrals[END_REF] M 0,0 (Z, Γ) ≃ M 0,0 ρ -1 (V ), Γ .

In particular, the right-hand side moduli stack is proper with projective coarse moduli space. The isomorphism (7) identifies the tangent-obstruction theories used to define the Gromov-Witten invariants, hence the virtual fundamental classes [M 0,0 (ρ -1 (V ), Γ)] vir and [M 0,0 (Z, Γ)] vir have the same degree. Then it is enough to prove that, for any t ∈ Δ,

(8) deg M 0,0 (Graph(μ) t , Γ) vir =deg M 0,0 (ρ -1 (V ), Γ) vir .
Gromov-Witten invariants of projective varieties are invariant under deformation of the target variety. We now explain why this result holds for ρ -1 (V )a n dGraph(μ) t , even if they are not projective.

Let q θ : Graph(μ) θ → Δ be the composition of f θ • ρ θ in (5), followed by the projection P 1 × Δ → Δ. The morphism q θ is smooth, as it is a composition of smooth morphisms. Moreover, q θ factorizes through an embedding followed by a projective morphism. To see this, it is enough to prove the same statement for the morphism F •R: Graph(μ) →F in [START_REF] Borisov | The orbifold Chow ring of toric Deligne-Mumford stacks[END_REF]. By construction, Graph(μ) is embedded in (O(1) ⊕O( 3 3) ⊕O(1) ⊕F →F. Because it has a vector bundle structure over F , then it can be seen as a subbundle of the projective bundle P(O(1) ⊕O( 3) ⊕O(1) ⊕F⊕O F ) →F, and therefore we have that F •R factorizes as the composition of an embedding followed by a projective morphism.

To finish the proof, consider the moduli stack which parameterizes relative stable maps to q θ : Graph(μ) θ → Δ of homology class Γ and genus zero. We denote it by M 0,0 (Graph(μ) θ /Δ, Γ). As Γ is the class of curves which are contracted by the resolution ρ θ and q θ : Graph(μ) θ → Δf a c t o r - izes through an embedding followed by a projective morphism, Abramovich and Vistoli's theorem [1, Theorem 1.4.1] implies that the moduli space M 0,0 (Graph(μ) θ /Δ, Γ) is a proper Deligne-Mumford stack. Since the class Γ is contracted by ρ θ , for any t ∈ Δ, the fiber at t of the natural morphism M 0,0 (Graph(μ) θ /Δ, Γ) → Δ is the proper Deligne-Mumford stack M 0,0 (Graph(μ) t , Γ). Then, by applying the same proof as in [14, Theorem 4.2] to this situation, we get [START_REF] B R Y A N | Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds[END_REF]. §5. Application to the cohomological crepant resolution conjecture 5.1. The cohomological crepant resolution conjecture Ruan's crepant resolution conjecture states that when ρ : Z → X is a crepant resolution of the coarse moduli space X of a Gorenstein orbifold X , the (orbifold) quantum cohomology of X and Z are related by analytic continuation in the quantum parameters. This conjecture was formulated more precisely by Bryan and Graber [START_REF] Bryan | The crepant resolution conjecture[END_REF] as an isomorphism of Frobenius manifolds (under some condition), and then further interpreted in full generality by Coates, Iritani, and Tseng [START_REF] Coates | Wall-crossings in toric Gromov-Witten theory, I: Crepant examples[END_REF] as a symplectic transformation between the Givental spaces associated to X and Z. This symplectic transformation encodes all information on the relationships between the genus zero Gromov-Witten theories of X and Z. We refer to Iritani [START_REF] Iritani | An integral structure in quantum cohomology and mirror symmetry for toric orbifolds[END_REF] for details and references on this still-evolving conjecture. At a lower level, the conjecture implies the cohomological crepant resolution conjecture; that is, the quantum corrected cohomology ring of Z ( d e f o r m e db yG r o m o v -W i t t e ni n v a r i a n t sc o m p u t e d on curves contracted by ρ) is isomorphic to the orbifold (Chen-Ruan) cohomology ring of X , after evaluation of the quantum parameters to roots of the unity. In the next sections, we check this conjecture on some weighted projective spaces.

We briefly recall the definition of the quantum corrected cohomology ring (see [START_REF] Ruan | The cohomology ring of crepant resolutions of orbifolds[END_REF]). Assume that M ρ (Z) is generated by a finite number of classes of rational curves which are linearly independent over Q. This will be the case for weighted projective spaces by Lemma 3.1.1. Fix a set of such generators Γ 1 ,...,Γ m . Then, any Γ ∈ M ρ (Z)c a nb ew r i t t e ni nau n i q u ew a ya sΓ= m ℓ=1 d ℓ Γ ℓ for some nonnegative integers d ℓ . Assign a formal variable q ℓ for each Γ ℓ so that Γ = m ℓ=1 d ℓ Γ ℓ ∈ M ρ (Z) corresponds to the monomial

q d 1 1 •••q dm m .T h equantum 3-points function is by definition (9) (α 1 ,α 2 ,α 3 ) q (q 1 ,...,q m ):= d 1 ,...,dm>0 Ψ Z Γ (α 1 ,α 2 ,α 3 )q d 1 1 •••q dm m ,
where

α 1 ,α 2 ,α 3 ∈ H ⋆ (X, C)andwhereΨ Z Γ (α 1 ,α 2 ,α 3
) is the Gromov-Witten invariant of Z of genus zero, homology class Γ, and three marked points. One makes the assumption that ( 9) defines an analytic function of the variables q 1 ,...,q m on some region of the complex space C m .T h equantum corrected cup product α 1 * ρ α 2 of two classes α 1 ,α 2 ∈ H ⋆ (Z; C) is then defined by requiring that, for all α 3 ∈ H ⋆ (Z; C), one has

Z (α 1 * ρ α 2 )α 3 = Z α 1 α 2 α 3 +(α 1 ,α 2 ,α 3 ) q (q 1 ,...,q m ).
The resulting associative, skew-symmetric, graded ring (H ⋆ (Z; C), * ρ )isthe quantum corrected cohomology ring with quantum parameters specialized at (q 1 ,...,q m ). It is also denoted by H ⋆ ρ (Z; C)(q 1 ,...,q m ). Let us fix the notation used for the computations below. Let ρ : Z → |P(w)| be a crepant resolution defined by a subdivision Σ ′ of the fan Σ of |P(w)|.S e tH := c 1 (O P(w) (1)) ∈ H 2 (P(w); C)a n dh := ρ ⋆ H ∈ H 2 (Z; C). For i ∈{0,...,n}, we denote by b i ∈ H 2 (Z; C) the first Chern class of the line bundle associated to the torus-invariant divisor corresponding to the ray of Σ ′ generated by β(v i ), and similarly e 1 ,...,e d ∈ H 2 (Z; C) for the rays in Σ ′ (1) \ Σ(1). Since ρ is crepant, we have h =(1/ n i=0 w i )( n i=0 b i + d j=1 e j ) (see Fulton [START_REF] Fulton | Introduction to Toric Varieties[END_REF]). Since H is an ample line bundle (see [START_REF] Fulton | Introduction to Toric Varieties[END_REF]Section 3.4]), Lemma 3.1.1 shows that the Mori cone M ρ (Z) is generated by the set [START_REF] Coates | Wall-crossings in toric Gromov-Witten theory, I: Crepant examples[END_REF] [ Theorem 5.2.1. For (q 1 ,q 2 ,q 3 ,q 4 ) ∈{(i, i, i, 0), (-i, -i, -i, 0)}, there is an explicit ring isomorphism

V (ν)] h ∩ [V (ν)]=0,ν ∈ Σ ′ (n -1) \ Σ(n -1) .
H ⋆ ρ (Z; C)(q 1 ,q 2 ,q 3 ,q 4 ) ∼ = H ⋆ CR P(1, 3, 4, 4); C
which is an isometry with respect to the Poincaré pairing on H ⋆ ρ (Z; C)(q 1 ,q 2 , q 3 ,q 4 ) and with respect to the Chen-Ruan pairing on H ⋆ CR (P(1, 3, 4, 4); C).

Proof. A toric computation shows that the cohomology ring of Z is isomorphic to the quotient of the polynomial ring C[h, e 1 ,e 2 ,e 3 ,e 4 ]bytheideal generated by 3he 4 ,e 1 e 3 ,e 1 e 4 ,e 2 e 4 ,e 3 e 4 , e 2 1 -10he

1 -4he 2 -2he 3 +24h 2 , e 1 e 2 +3he 1 +2he 2 + he 3 -12h 2 , e 2 2 -6he 1 -12he 2 -2he 3 +24h 2 , e 2 e 3 +3he 1 +6he 2 + he 3 -12h 2 , e 2 3 -6he 1 -12he 2 -14he 3 +24h 2 ,
16h 2 e 1 , 16h 2 e 2 , 16h 2 e 3 , 16h 3 -1 27 e 3 4 .

We fix the following basis of the vector space H ⋆ (Z; C):

1,h,e 1 ,e 2 ,e 3 ,e 4 ,h 2 ,he 1 ,he 2 ,he 3 ,e 2 4 ,h 3 .

Let Γ 1 , Γ 2 , Γ 3 , Γ 4 ∈ M ρ (Z) be the generators defined in Section 3.3 and whose equations are Γ 1 := PD(4he 1 ), Γ 2 := PD(4he 2 ), Γ 3 := PD(4he 3 ), and Γ 4 := PD(-(1/3)e 2 4 ). We now give a presentation of the quantum corrected cohomology ring H ⋆ ρ (Z; C)(q 1 ,q 2 ,q 3 , 0). First, notice that any curve of homology class d 4 Γ 4 is disjoint from any other curve of class

d 1 Γ 1 + d 2 Γ 2 + d 3 Γ 3 ;i no t h e rw o r d s ,M 0,0 (Z, Γ) is empty if Γ = 4 ℓ=1 d ℓ Γ ℓ with d 4 • (d 1 + d 2 + d 3 ) = 0.
From the degree axiom, it follows that we need to consider only Gromov-Witten invariants Ψ Z Γ (α 1 ,α 2 ,α 3 )w i t hα i ∈ H 2 (Z; C), i ∈{1, 2, 3}. Finally, by applying the divisor axiom, we deduce the following expression for the quantum 3-points function:

α 1 ,α 2 ,α 3 q (q 1 ,q 2 ,q 3 ,q 4 ) = d 1 ,d 2 ,d 3 >0 3 i=1 3 ℓ=1 d ℓ Γ ℓ α i deg M 0,0 Z, 3 ℓ=1 d ℓ Γ ℓ vir q d 1 1 q d 2 2 q d 3 3 + d 4 >0 3
i=1 d 4 Γ 4 α i deg[M 0,0 (Z, d 4 Γ 4 )] vir q d 4 4 .
Since Γ ℓ h =0 for an y ℓ ∈{1, 2, 3, 4}, one has h * ρ α = hα for any α ∈ H ⋆ (Z; C), and similarly

e i * ρ e 4 = e i e 4 =0 if i =4, ǫ(q 4 )e 2 4
otherwise, for some function ǫ(q 4 ) such that ǫ(0) = 1. Since in the isomorphism of rings that we will define we put q 4 =0, we only consider classes Γ =

d 1 Γ 1 + d 2 Γ 2 + d 3 Γ 3 for d i ∈ N.W es e tΓ μν := Γ μ + •••+Γ ν for 1 ≤ μ ≤ ν ≤ 3. Using Theorem 3.3.1, we get deg[M 0,0 (Z, Γ)] vir = 1/d 3 if Γ = dΓ μν for 1 ≤ μ ≤ ν ≤ 3a n dd ∈ N * ,
0o t h e r w i s e .

Hence, the remaining part of the multiplicative table of H ⋆ ρ (Z; C)(q 1 ,q 2 ,q 3 , 0) is as follows: e 1 * ρ e 1 = -24h 2 + 10 + 16 q 1 1 -q 1 +4 q 1 q 2 1 -q 1 q 2 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 1 + 4+4 q 2 1 -q 2 +4 q 1 q 2 1 -q 1 q 2 +4 q 2 q 3 1 -q 2 q 3 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 2 + 2+4 q 2 q 3 1 -q 2 q 3 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 3 , e 1 * ρ e 2 =12h 2 + -3 -8 q 1 1 -q 1 +4 q 1 q 2 1 -q 1 q 2 he 1 + -2 -8 q 2 1 -q 2 +4 q 1 q 2 1 -q 1 q 2 -4 q 2 q 3 1 -q 2 q 3 he 2 + -1 -4 q 2 q 3 1 -q 2 q 3 he 3 , P(w I(g) ), where w I(g) =(w i ) i∈I(g) . The inertia stack is the disjoint union of the twisted sectors IX = g∈T P(w I(g) ).

As a vector space, the Chen-Ruan cohomology is the cohomology of the inertia stack; that is, the graded structure is obtained by shifting the degree of the cohomology of any twisted sector by twice the corresponding age. We have

H p CR (X ; C)= g∈T H p-
H, E 1 ,E 2 ,E 3 ,E 4 ∈ H ⋆ CR (X ; C) as the image of c 1 (O X (1)) ∈ H 2 (X ; C), 1 ∈ H 0 (X (exp(πi/2)) ; C), 1 ∈
H 0 (X (exp(πi)) ; C), 1 ∈ H 0 (X (exp(πi3/2)) ; C), and 1 ∈ H 0 (X (exp(2πi/3)) ; C), respectively, under the inclusion H ⋆-2 age(g) (P(w I(g) )) → H ⋆ CR (X ) determined by the decomposition [START_REF] Fulton | Introduction to Toric Varieties[END_REF]. As a C-algebra, the Chen-Ruan cohomology ring is generated by H, E 1 ,E 2 ,E 3 ,E 4 with the following relations (see [START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF]):

HE 4 ,E 1 E 1 -3HE 2 ,E 1 E 2 -3HE 3 ,E 1 E 3 -3H 2 , E 2 E 2 -3H 2 ,E 2 E 3 -HE 1 ,E 3 E 3 -HE 2 , 16H 3 -E 3 4 , H 2 E 1 ,H 2 E 2 ,H 2 E 3 ,E 1 E 4 ,E 2 E 4 ,E 3 E 4 .
We see that the following elements form a basis of H ⋆ CR (X ; C) which we fix for the rest of the proof:

1,H,E 1 ,E 2 ,E 3 ,E 4 ,H 2 ,HE 1 ,HE 2 ,HE 3 ,E 2 4 ,H 3 .
(Note that the elements of our basis are different from those used in [START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF] by a combinatorial factor.) For cohomology classes α 1 and α 2 , the product α 1 * ρ α 2 ∈ H ⋆ ρ (Z; C)(q 1 ,q 2 , q 3 , 0) differs from the usual cup product only when α 1 ,α 2 ∈{e 1 ,e 2 ,e 3 }.W e now set q 1 = q 2 = q 3 = i, and we compute e i * ρ e j in the choosen basis of H ⋆ (Z; C)t og e t

e 1 * ρ e 1 = -24h 2 +(-2 + 6i)he 1 -4he 2 +(-2 -2i)he 3 , e 1 * ρ e 2 =12h 2 +(-1 -4i)he 1 +(2-4i)he 2 + he 3 , e 1 * ρ e 3 = -2ihe 1 -2ihe 3 , e 2 * ρ e 2 = -24h 2 + (2 + 2i)he 1 +8ihe 2 +(-2 + 2i)he 3 , e 2 * ρ e 3 =12h 2 -he 1 +(-2 -4i)he 2 +(1-4i)he 3 , e 3 * ρ e 3 = -24h 2 +(2-2i)he 1 +4he 2 + (2 + 6i)he 3 .
We now define a linear map ( 12)

H ⋆ ρ (Z; C)(i, i, i, 0) → H ⋆ CR P(1, 3, 4, 4); C as follows. We send ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ h e 1 e 2 e 3 e 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ -→ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 10 0 0 0 0 - √ 2 -2i √ 20 0 -i √ 22 i-i √ 20 0 √ 2 -2i - √ 20 00 0 03 e x p ( 2πi 3 ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ H E 1 E 2 E 3 E 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(the image of the other elements of the basis is uniquely determined by requiring that (12) be a ring isomorphism). A direct computation shows that ( 12) is a ring isomorphism and that it is an isometry with respect to the inner products given by the Poincaré duality and the Chen-Ruan pairing, respectively. The case q 1 = q 2 = q 3 = -ia n dq 4 = 0 is analogous to the previous one. We define a linear map (13)

H ⋆ ρ (Z; C)(-i, -i, -i, 0) → H ⋆ CR P(1, 3, 4, 4); C by sending ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ h e 1 e 2 e 3 e 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ -→ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 10 0 0 0 0 - √ 22 i √ 20 0i √ 2 -2i i √ 20 0 √ 22 i - √ 20 00 0 03 e x p ( 2πi 3 ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ H E 1 E 2 E 3 E 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and extending to the remaining part of the basis (in the unique way) such that the resulting map is a ring isomorphism. Also, in this case a direct computation shows that ( 13) is a ring isomorphism and that it respects the inner pairings.

Remark 5.2.2. The values q 1 = q 2 = q 3 = ±i were proposed in [16, Conjecture 1.9]: thus Theorem 5.2.1 agrees with this conjecture. The change of variables is inspired by those of Nahm and Wendland [START_REF] Nahm | Mirror symmetry on Kummer type K3 surfaces[END_REF] (see also [START_REF] Bryan | The orbifold quantum cohomology of C 2 /Z3 and Hurwitz-Hodge integrals[END_REF], [START_REF] Coates | The crepant resolution conjecture for type A surface singularities[END_REF]).

The fact that one quantum parameter can be put to zero to get the conjectured isomorphism is strange in regard to the conjecture, and somehow unsatisfactory. The problem concerns the computation of the function ǫ(q 4 ) in the proof. In [START_REF] Boissière | The cohomological crepant resolution conjecture for P(1,3,4,4)[END_REF], we solve this problem so that the fourth quantum parameter can be put to 1.

The case of |P(1,...,1,n)|

The quantum parameter put to zero in the case of P(1, 3, 4, 4) corresponds to the isolated singularity (1/3)(1, 1, 1). Surprisingly, this phenomenon can be observed in any dimension by considering the n-dimensional weighted projective space P(1,...,1,n) whose coarse moduli space has an isolated singularity of type (1/n)(1,...,1) at the point [0 : ...:0:1].Inthisexample, a crepant resolution can be constructed for any n.

We identify the stacky fan (N,β,Σ) defined in (1) with Z n , {Λ(β(v i ))} n i=0 , Σ by means of the isomorphism Λ : N → Z n defined by sending v 0 to (-1,...,-1, -n)a n dv i to the ith vector of the standard basis of Z n ,f o r i ∈{1,...,n}. The crepant resolution is defined as follows. Consider the ray generated by P:=(0,...,0, -1) = 1 n

n-1 i=0 Λ β(v i ) ,
and let Σ ′ be the fan obtained from Σ (by refinement) by replacing the cone generated by Λ(β(v 0 )),...,Λ(β(v n-1 )) with the cones generated by P and the rays Λ(β(v 0 )),..., Λ(β(v i )),...,Λ(β(v n-1 )) for any i ∈{0,...,n-1}.W e draw as an example the polytope for the case n = 3 in Figure 2. Define Z to be the toric variety associated to Σ ′ , and define ρ : Z →|P(1,...,1,n)| to be the morphism associated to the identity in Z n .W eh a v et h ef o l l o w i n g . The Mori cone M ρ (Z) is generated by one class Γ 1 := PD((h -(e/n)) n-2 e).

We will set the quantum parameter q 1 t oz e r os ot h a tw ed on o th a v et o compute any nontrivial Gromov-Witten invariants.

Concerning the Chen-Ruan cohomology ring of X := P(1,...,1,n), the t w i s t e ds e c t o r sa r ei n d e x e db yt h es e tT={exp((2πik/n)) | k ∈{0,...,n-1}}.F o rg ∈ T \{1}, one has X (g) ∼ = P(n), whereas X (1) ∼ = X . As a vector space, we have [START_REF] Li | Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties[END_REF] H ⋆ CR (X ; C):= g∈T H ⋆-2 age(g) (X (g) ).

Let H, E 1 ∈ H 2 CR (X ; C) be the image of c 1 (O X (1)) ∈ H 2 (X ; C)a n d1∈ H 0 (X (exp(2πi/n)) ; C), respectively, with respect to the inclusion H ⋆-2 age(g) (X (g) ) → H ⋆ CR (X ) determined by [START_REF] Li | Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties[END_REF]. Then we have the following presentation:

1

 1 Conjecture 5.1]. Theorem 3.3.1. Let ρ : Z →|P(1, 3, 4, 4)| be the crepant resolution of |P(1, 3, 4, 4)| defined in Section 3.2, and let Γ=d 1 Γ 1 + d 2 Γ 2 + d 3 Γ 3 .T h e n deg[M 0,0 (Z, Γ)] vir = 1/d 3 if Γ=d ν i=μ Γ i , with 1 ≤ μ ≤ ν ≤ 3 and d ∈ N * , 0 otherwise. §4. Proof of Theorem 3.3.To prove Theorem 3.3.1, we use the deformation invariance property of the Gromov-Witten invariants. We define an open neighborhood V of the singular locus [0 : 0 : x 2 : x 3 ] ∈|P(1, 3, 4, 4)| ∼ = P 1

  ) ⊕O(1) ⊕F) × × 3 i=1 P(O(1) ⊕ O(i)) ;m o r e o v e r ,F •R is the restriction of the projection (O(1) ⊕O(3) ⊕ O(1) ⊕F) × × 3 i=1 P(O(1) ⊕O(i)) →F. Let us now consider the projection O(1) ⊕O(

5. 2 .

 2 The case of |P(1, 3, 4, 4)| Consider the crepant resolution ρ : Z →|P(1, 3, 4, 4)| defined in Section 3.2.

Proposition 5 . 3 . 1 .

 531 For any n ≥ 2, there is an explicit ring isomorphismH ⋆ (Z; C) ∼ = H ⋆ CR P(1,...,1,n); C .

Figure 2 :

 2 Figure 2: Polytope of P(1, 1, 1, 3) and a crepant resolution

  H ⋆ CR P(1,...,1,n); C ∼ = C[H, E 1 ]/ H n -(E 1 ) n ,HE 1 .The ring isomorphismH ⋆ CR P(1,...,1,n); C ∼ -→ H ⋆ (Z; C)is obtained by mapping H → h and E 1 → -exp(iπ/n)(e/n).

  ⊕ H p-2 P(4, 4); C ⊕ H p-2 P(4, 4); C ⊕ H p-2 P(4, 4); C .

		2 age(g) P(w I(g) ); C
	(11)	= H p P(1, 3, 4, 4); C ⊕ H p-2 P(3); C ⊕ H p-4 P(3); C
	A basis of H ⋆ CR (X ; C) is easily obtained in the following way, set

Acknowledgments. Part of this work was done during the visit of the first and third authors to the Scuola Internazionale Superiore di Studi Avanzati (SISSA) in Trieste in September 2006, and we gratefully acknowledge that hospitality and support. Particular thanks go to Barbara Fantechi and Yongbin Ruan for very useful discussions.

Perroni's work was partially supported by Schweizerischer Nationalfonds zur Förderung grant 200020-107464/1.

e 1 * ρ e 3 = -4 q 1 q 2 1 -q 1 q 2 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 1 + 4 q 2 1 -q 2 -4 q 1 q 2 1 -q 1 q 2 -4 q 2 q 3 1 -q 2 q 3 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 2 + -4 q 2 q 3 1 -q 2 q 3 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 3 ,

+4 q 2 q 3 1 -q 2 q 3 he 3 , e 3 * ρ e 3 = -24h 2 + 6+4 q 1 q 2 1 -q 1 q 2 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 1 + 12 + 4 q 2 1 -q 2 +4 q 1 q 2 1 -q 1 q 2 +4 q 2 q 3 1 -q 2 q 3 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 2 + 14 + 16 q 3 1 -q 3 +4 q 2 q 3 1 -q 2 q 3 +4 q 1 q 2 q 3 1 -q 1 q 2 q 3 he 3 .

To compute the Chen-Ruan cohomology ring H ⋆ CR (X ; C)(hereX = P(1, 3, 4, 4)), we follow Boissière, Mann, and Perroni [START_REF]A model for the orbifold Chow ring of weighted projective spaces[END_REF]. The twisted sectors are indexed by the set T := exp(2πiγ) | γ ∈{0, 1/3, 2/3, 1/4, 1/2, 3/4} .F o r g ∈ T , written g =exp(2πiγ)w i t hγ ∈{0, 1/3, 2/3, 1/4, 1/2, 3/4},t h eage of g is given by the formula age(g)={γ} + {3γ} + {4γ} + {4γ}, where {•} denotes the fractional part. For any g ∈ T, X (g) is a weighted projective space. Setting I(g):={i ∈{0, 1, 2, 3}|g w i =1}, one has X (g) =


F Delarue

F Flandoli

The transition point in the zero noise limit for a 1D Peano example

Keywords: Stochastic differential equations, zero-noise limit, Peano phenomena, non-uniqueness. MSC: 60H10, 34A12, 60J65

 is revisited. The original proof was based on explicit solutions to the elliptic equations for probabilities of exit times. The new proof given here is purely dynamical, based on a direct analysis of the SDE and the relative importance of noise and drift terms. The transition point between noisy behavior and escaping behavior due to the drift is identified.

Introduction

Given a probability space (Ω, A, P ), consider the SDE

dX x,ε t = b (X x,ε t ) dt + εdW t , X x,ε 0 = x (1)
in R d with b ∈ C α R d , R d and at most linear growth at infinity, (W t ) t≥0 being a Brownian motion in R d . When ε > 0, a strong solution exists and it is pathwise unique, hence also unique in law (see [START_REF] Veretennikov | On strong solution and explicit formulas for solutions of stochastic integral equations[END_REF], [START_REF] Krylov | Strong solutions of stochastic equations with singular time dependent drift[END_REF], [START_REF] Flandoli | Well posedness of the transport equation by stochastic perturbation[END_REF], [START_REF] Flandoli | Random Perturbation of PDEs and Fluid Dynamic Models, Saint Flour summer school lectures[END_REF]); we denote its law by P ε x . When ε = 0, namely for the deterministic Cauchy problem

dX x t dt = b (X x t ) , X x 0 = x, (2)
a solution exists by Peano theorem but uniqueness may fail, like in the example studied below. In absence of uniqueness, a relevant question is selection: does the limit of P ε x as ε → 0 concentrate on special solutions, which, because of this property, may be considered more "physical" than others?

Under the previous assumptions on b, the family {P ε x } ε∈(0,1) is tight and each limit measure µ is supported by the non empty closed set C of continuous solutions (X x t ) t≥0 to the ODE [START_REF] Borkar | A New Markov Selection Procedure for Degenerate Diffusions[END_REF]. As we just said, the set C is then expected to contain more than one element. The question is then to determine the limit measures µ of the family {P ε

x } ε∈(0,1) together with the elements of C which are selected by these limit measures.

Precise general results on this problem, in R d , are not known, except for structural facts as in [START_REF] Borkar | A New Markov Selection Procedure for Degenerate Diffusions[END_REF], [START_REF] Buckdahn | On limiting values of stochastic differential equations with small noise intensity tending to zero[END_REF] (however, see [START_REF] Delarue | Noise prevents collapse of Vlasov-Poisson point charges[END_REF] for a special 2D example). The main and striking paper is [START_REF] Bafico | Small random perturbations of Peano phenomena[END_REF], which gives a very general and explicit solution in the case d = 1, b having one singular point x 0 (b (x 0 ) = 0, b being not locally Lipschitz around x 0 ). The proof makes essential use of explicit solutions to elliptic equations for the exit time from an interval and for ruin type probabilities. This powerful analytic approach allows [START_REF] Bafico | Small random perturbations of Peano phenomena[END_REF] to treat a large variety of examples. However, it is also the main restriction in the attempt to extend the theory to higher dimensions. Indeed, the elliptic equations in d = 1 are second order linear ordinary differential equations, hence explicitly solvable. In higher dimensions this approach fails, unless special symmetries allow to reduce the dimension. Concerning one of the main examples of [START_REF] Bafico | Small random perturbations of Peano phenomena[END_REF], namely (3) studied below, we also refer to the very interesting works [START_REF] Gradinaru | A singular large deviations phenomenon[END_REF] and [START_REF] Herrmann | Phénomène de Peano et grandes déviations[END_REF], where large deviations are investigated and atypical results are discovered.

It would be thus desirable to have other approaches to the same problem. This motivated us to develop the new proof given here. Let us however insist that these remarks are only a general motivation for this research, since we do not solve the problem of a generalization to higher dimensions; we only give a new proof based on dynamical considerations.

We give here a new proof of the result of Bafico and Baldi [START_REF] Bafico | Small random perturbations of Peano phenomena[END_REF], in the particular case (always d = 1) of the drift

b (x) = A + |x| α for x ≥ 0 -A -|x| α for x < 0 (3)
for some α ∈ (0, 1) and A + , A -> 0. The result is that P ε 0 , the law of the solution of the SDE with initial condition x = 0, weakly converges to a combination of delta Dirac masses at the extreme solutions of the ODE:

Theorem 1 P ε 0 → p + δ x + • + p -δ x -
• , in the weak sense, as ε → 0, where

p + = (A -) -1 1+α (A + ) -1 1+α + (A -) -1 1+α , p -= (A + ) -1 1+α (A + ) -1 1+α + (A -) -1 1+α
, and

x + t = C α,A + t 1 1-α , x - t = -C α,A -t 1 1-α , with C α,A ± = (A ± (1 -α)) 1/(1-α) .
The point of the new proof given here is its dynamical character. The identification of the paths which are selected is not based on some auxiliary PDEs as in [START_REF] Bafico | Small random perturbations of Peano phenomena[END_REF] but only on the dynamical properties of the SDE [START_REF] Bafico | Small random perturbations of Peano phenomena[END_REF]. Only the computation of the precise weights in the combination of the Dirac masses involves a PDE through a martingale argument. The new proof thus gives some new insight into the actual behavior of solutions, which is not visible in the PDE approach. Precisely, we identify the existence of two regimes. At the beginning of time, the solution which started from x = 0 behaves like the Brownian motion (εW t ) t≥0 , although ε is very small, because the drift is much smaller (this happens also for a Lipschitz drift). But close to the time-space points

(t ε , x ε ) := ε 2(1-α) 1+α , x ± tε = ε 2(1-α) 1+α , ±C α,A ± ε 2 1+α
(4) a transition occurs: the drift becomes much stronger than the noise (the usual fluctuations of the noise do not contrast the drift anymore) and pushes the trajectories far away from the neighborhood of x = 0, roughly along one of the trajectories (x ± t ) t≥0 . The easiest way to identify heuristically the transition time-space point (4) is to compare x + t (or x - t ) with εW t . Forgetting the scale constants C α,A ± in the definition of x + t and x - t , we thus claim that the typical time t at which transition is given by the solution of the equation

t 1 1-α = εt 1 2 .
(5)

3

We then get t ε := ε

2(1-α) 1+α
as typical transition time. Accordingly, we then define

x ε := t 1 1-α ε = ε 2 1+α
as the typical space scale for observing the transition between the two regimes. This simple intuition is confirmed by the proofs below.

We emphasize that the same method would permit to handle completely asymmetric cases for which b(x) has the form b(x) = A + |x| α + for x ≥ 0 and b(x) = -A -|x| α -for x < 0 with α + = α -. In such cases, the two extremal paths do not generate the same transition times so that only one of them is selected, namely the one driven by the smallest exponent. Intuitively, this amounts to letting the ratio A + /A -tend to 0 or +∞ in our case and thus letting the weights in the Dirac combination tend to (0, 1) or (1, 0).

We feel appropriate to state, right here in the introduction, our two main results concerning the transition point, because they are our main contribution to the understanding of this dynamical problem. For every r > 0, let us denote by τ r the exit time from (-r, r), defined on the canonical space C ([0, +∞); R), as

τ r (ξ) = inf {t > 0 : |ξ t | ≥ r} , ξ ∈ C ([0, +∞); R) ,
when this set is not empty, τ r (ξ) = +∞ otherwise. We denote by P ε x the law of X x,ε and by E ε

x the corresponding expectation.

Proposition 2 For every function t ε > t ε such that lim ε→0 t ε /t ε = +∞, we have lim

ε→0 sup x∈(-xε,xε) P ε x τ xε > t ε = 0.
The proposition states that with high probability the system reaches ±x ε in a time just a little greater than t ε , when it starts inside (-x ε , x ε ). This fact is mainly due to the fluctuations of the noise, the drift playing a negligible role on a time interval of length of the same order as t ε . Put it differently, until time t ε , the noise dominates. We prove this result in any dimension, under quite general conditions.

Then we show that, starting from x ε or above, the solution remains above x + t forever, with probability larger than some λ > 0 (similarly if it starts from -x ε ). More precisely, it remains above (1 -γ) x + t for any arbitrarily prescribed γ ∈ (0, 1). The result is complemented by the following fact: for every function x ε > x ε such that lim ε→0 x ε /x ε = +∞, if we start above x ε then the solution remains above x + t forever, with probability close to one. The formal statement is: Theorem 3 Let γ ∈ (0, 1) be given. Then there exists a constant λ γ > 0, independent of ε ∈ (0, 1), such that

inf x≥xε P X x,ε t ≥ (1 -γ) x + t , ∀t ≥ 0 ≥ λ γ > 0.
Moreover, for every function x ε > x ε such that lim ε→0 x ε /x ε = +∞, we have

lim ε→0 inf x≥ xε P X x,ε t ≥ (1 -γ) x + t , ∀t ≥ 0 = 1.
Similar results hold for (x - t ) t≥0 .

This fact will imply our final result. We emphasize that it occurs because of the drift. In particular, a Lipschitz continuous drift would not give such a result: Observe for instance that the transition time, as defined by ( 5), tends to 1 as α tends to 1, saying that no transition occurs in small time when α = 1.

These two steps lead to the solution of our problem. Indeed, with large probability, we reach ±x ε in a very short time of order t ε (Proposition 2). Then, iterating the argument of escape with probability larger than λ γ , we reach a prescribed ± x ε with probability close to one, again in a short time (Corollary 8). Finally, restarting from ± x ε , we escape above or below (1 -γ) x ± t forever (Theorem 3). This shows that only the extremal paths are selected at the limit. The weights in the combination of the delta Dirac masses appearing in the statement of Theorem 1 are then computed by a martingale argument.

Our hope is that we have identified new ideas behind the zero-noise limit problem which may be extended to other examples and higher dimension, but for the time being we need to restrict ourselves to the special 1D case above.

Proof of Proposition 2

We give two proofs. The first one, which is purely dynamical, is true in any dimension under moderate assumptions. This may be an indication that the dynamical proof presented here is promising for generalizations. The second one is a refinement of the first one; it relies on the exact scaling properties of the SDE at hand.

Multidimensional result

Let b : R d → R d be a function such that |b (x)| ≤ M r α ∀x ∈ R d , ∀r > 0, with |x| ≤ r,
for some M > 0, α ∈ (0, 1). For a given x ∈ R d and every ε ∈ (0, 1), let (X x,ε t ) t≥0 be the solution of

dX x,ε t = b (X x,ε t ) dt + εdW t , X x,ε 0 =
x where (W t ) t≥0 is a Brownian motion in R d and let P ε

x be its law on the space of continuous paths. As above, set with the usual convention for the infimum of an empty set.

r ε := ε 2 1+α , t ε := ε 2(1-α)
Proposition 4 One has

sup |x|≤rε P ε x (τ rε > t ε ) ≤ θ := P (|Z| ≤ 2 + M )
where Z ∼ N (0, Id).

Proof. If τ rε (X x,ε ) > t ε then |X x,ε t | ≤ r ε for t ∈ [0, t ε ]. Hence, from X x,ε t = x + t 0 b (X x,ε s ) ds + εW t ,
with |x| ≤ r ε , we get

ε |W t | ≤ |X x,ε t | + |x| + t 0 |b (X x,ε s )| ds ≤ 2r ε + tM r α ε ≤ 2r ε + t ε M r α ε for t ∈ [0, t ε ]. Moreover, t ε r α ε = r ε . Hence ε |W t | ≤ (2 + M ) r ε for t ∈ [0, t ε ] .
In particular, this implies ε |W tε | ≤ (2 + M ) r ε . Since the random vector

Z := t -1/2 ε
W tε is N (0, Id), we deduce that

P ε |W tε | ≤ (2 + M ) r ε = P t -1/2 ε W tε ≤ (2 + M ) r ε εt 1/2 ε = P (|Z| ≤ 2 + M ) =: θ < 1.
We have used the identity

r ε = εt 1/2 ε . We have proved P ε x (τ rε > t ε ) ≤ θ. The proof is complete.
Corollary 5 Assume that, for every ε ∈ (0, 1), the solutions (X x,ε t ) t≥0 are a strong Markov family w.r.t. the initial condition. Then, for any integer n ≥ 1, sup

|x|≤rε P ε x (τ rε > nt ε ) ≤ θ n .
In particular, for every function t ε > t ε such that lim ε→0 t ε /t ε = +∞, we have lim

ε→0 sup |x|≤rε P ε x τ rε > t ε = 0.
Proof. We know that

P ε x (τ rε > t ε ) ≤ θ = P (|Z| ≤ 2 + M ) .
We consider the sequence of stopping times

τ (n) rε := (nt ε ) ∧ τ rε
for n = 0, 1, ... and apply strong Markov property to get

P ε x (τ rε > nt ε ) = P ε x τ rε > nt ε |ξ τ (n-1) rε | < r ε P ε x |ξ τ (n-1) rε | < r ε + P ε x τ rε > nt ε |ξ τ (n-1) rε | = r ε P ε x |ξ τ (n-1) rε | = r ε ≤ θP ε x (τ rε > (n -1) t ε ) , because P ε x τ rε > nt ε |ξ τ (n-1) rε | = r ε = 0 and P ε x τ rε > nt ε |ξ τ (n-1) rε | < r ε ≤ θ. Therefore P ε x (τ rε > nt ε ) ≤ θ n . The proof is complete.

Proof by exact scaling

We also provide this proof because it provides the rate of convergence to zero of P ε x τ xε > t ε . However, it is restricted to the 1D case, since it relies on the perfect scaling property of the drift b in (3).

Notice, by standard recurrence properties of the 1D Brownian motion, that E 1

• [τ x 1 ] ∞ < ∞ Lemma 6
Consider the 1D case with b given by (3) and recall that

x ε = ε 2 1+α and t ε = ε 2(1-α)
1+α . Then, for all x ∈ (-x ε , x ε ), the law of τ xε under P ε

x is the same as the law of t ε τ x 1 under P 1

x -1 ε x . In particular,

P ε x τ xε > tε = P 1 x -1 ε x τ 1 > tε t ε ≤ t ε t ε E 1 • [τ x 1 ] ∞ .
Proof. We notice that

d x -1 ε X tεt = x -1 ε t ε b(X tεt )dt + x -1 ε t 1 2 ε εd Ŵ ε t , for another Brownian motion ( Ŵ ε t ) t≥0 . It is well-checked that x -1 ε t 1 2 ε ε = 1.
Moreover, by scaling property of b,

x -1 ε t ε b(X tεt ) = x α-1 ε t ε b x -1 ε X tεt = b x -1 ε X tεt .
Therefore, (x -1 ε X x,ε tεt ) t≥0 has the same law as (X x -1

ε x,1 t ) t≥0 .

Proof of Theorem 3

Let τ γ,x + be the random time, defined on the canonical space C ([0, +∞); R):

τ γ,x + (ξ) = inf t > 0 : ξ (t) < (1 -γ) x + t (6)
(equal to +∞ if this event never happens).

For an initial condition x ≥ x ε , we have X x,ε 0 > (1 -γ) x + 0 = 0 at time zero. Since the processes are continuous, P ε

x (τ γ,x

+ > 0) = 1. For s ∈ [0, τ γ,x + (X x,ε )] (all s ≥ 0 if τ γ,x + (X x,ε ) = +∞), we have X x,ε s ≥ (1 -γ) x + s , so that (by definition of b) b (X x,ε s ) ≥ (1 -γ) α b x + s .
Therefore, for every t ∈ [0, τ γ,x + (X x,ε )],

X x,ε t = x + t 0 b (X x,ε s ) ds + εW t ≥ x + (1 -γ) α t 0 b x + s ds + εW t = x + (1 -γ) α x + t + εW t , because x + t = t 0 b (x + s )
ds. Now consider η ∈ (0, 1) such that 1 -η is the mid point between (1 -γ) and (1 -γ) α . We have (with equal distance)

(1 -γ) < 1 -η < (1 -γ) α .
We rewrite the inequality above in the form

X x,ε t ≥ (1 -η) x + t + R x,ε,γ t , R x,ε,γ t := x + (γ -η) x + t + εW t ,
which, we recall, holds for every t ∈ [0,

τ γ,x + (X x,ε )]. Letting A (x, ε, γ) be the event A (x, ε, γ) = {R x,ε,γ t ≥ 0, ∀t ≥ 0} ,
we deduce from next lemma that

inf x≥xε P (A (x, ε, γ)) ≥ λ γ > 0 lim ε→0 inf x≥g(ε)xε P (A (x, ε, γ)) = 1,
whenever g(ε) → +∞ as ε → 0. This implies the two claims of the theorem. Indeed, on the event A (x, ε, γ), it holds

X x,ε t ≥ (1 -η) x + t > (1 -γ) x + t for every t ∈ [0, τ γ,x + (X x,ε )]. But this is compatible only with the statement τ γ,x + (X x,ε ) = +∞. Hence A (x, ε, γ) ⊂ {τ γ,x + (X x,ε ) = +∞}. The proof of Theorem 3 is complete.
We now prove Lemma 7 Given A > 0, there is a constant λ A , independent of ε, such that

P ε 2 1+α + At 1 1-α + εW t > 0, for all t ≥ 0 ≥ λ A > 0.
Moreover, given g (ε) such that lim ε→0 g (ε) = +∞,

lim ε→0 P ε 2 1+α g (ε) + At 1 1-α + εW t > 0, for all t ≥ 0 = 1.
Proof. Since the process (W t ) t≥0 has the same law as (β -1/2 W βt ) t≥0 for every β > 0, we have

P ε 2 1+α g (ε) + At 1 1-α + εW t > 0, for all t ≥ 0 = P β 1 2 ε 2 1+α -1 g (ε) + β 1 2 At 1 1-α ε -1 + W βt > 0, for all t ≥ 0 . Choose β = β ε so that β 1 2 ε ε 2 1+α -1 = 1, namely β 1 2 ε = ε -1-α 1+α . Then, since ε = β -1 2 1+α 1-α ε , β 1 2 ε t 1 1-α ε -1 = β 1 2 1+α 1-α ε β 1 2 ε t 1 1-α = β 1 1-α ε t 1 1-α = (β ε t) 1 1-α . Therefore P β 1 2 ε ε 2 1+α -1 g (ε) + β 1 2 ε At 1 1-α ε -1 + W βεt > 0, for all t ≥ 0 = P g (ε) + A (β ε t) 1 1-α + W βεt > 0, for all t ≥ 0 = P g (ε) + As 1 1-α + W s > 0, for all s ≥ 0 .
Whenever g(ε) = 1, the latter probability is positive and independent of ε; we call it λ A and the first claim of the lemma is proved. Whenever lim ε→0 g(ε) = +∞, the latter probability tends to one. The proof is complete.

Proof of Theorem 1 4.1 Selection of the extremal paths

Throughout the proof, we are given a number γ ∈ (0, 1). As an easy consequence of Theorem 3 we have: Corollary 8 Let λ γ > 0 be the constant, independent of ε ∈ (0, 1), given by Theorem 3. Given an infinitesimal function x ε > x ε such that lim ε→0 x ε /x ε = +∞, there exists an infinitesimal function t ε > 0 such that

inf |x|≥xε P ε x τ xε ≤ t ε ≥ λ γ > 0.
Moreover, there exists an infinitesimal function t ′ ε > 0 such that

lim ε→0 P ε ±xε τ xε ≤ t ′ ε = 1.
Here we call an infinitesimal function a function of ε which tends to 0 with ε.

Proof. We know from Theorem 3 that

inf x≥xε P X x,ε t ≥ (1 -γ) x + t , ∀t ≥ 0 ≥ λ γ > 0.
The function (1 -γ) x + t is equal to x ε at some time t ε , and t ε is infinitesimal if x ε is so, that is t ε → 0 if x ε → 0 with ε. Then, by continuity of the trajectories of (X x,ε t ) t≥0 ,

P ε x τ xε ≤ t ε ≥ P X x,ε t ≥ (1 -γ) x + t , ∀t ≥ 0 ≥ λ γ , for x ≥ x ε .
Using in a similar argument for initial conditions in (-∞, -x ε ], the first assertion is proved.

We now prove the second assertion. We mimic the proof of Proposition 2. For any integer n ≥ 1, we compute

max η=± P ε ηxε τ xε ≥ 2n t ε ≤ max η=± P ε ηxε τ xε ≥ 2(n -1) t ε sup |x|≥xε P ε x τ xε ≥ 2 t ε + max η=± P ε ηxε τ xε ≥ 2(n -1) t ε sup |x|≤xε P ε x τ xε ≥ 2 t ε := T 1 + T 2 .
By the first assertion of the statement, the first term in the right-hand side is less than

T 1 ≤ (1 -λ γ ) max η=± P ε ηxε τ xε ≥ 2(n -
1) t ε . Now, by Proposition 2, the second term is less than

T 2 ≤ max η=± P ε ηxε τ xε ≥ 2(n -1) t ε sup |x|≤xε P ε x τ xε > t ε max η=±1 P ε ηxε τ xε ≥ t ε ≤ (1 -λ γ ) 2 ,
for ε small enough. This proves that there exists a constant c > 0, independent of ε, such that for ε small enough

max η=± P ε ηxε τ xε ≥ 2n t ε ≤ c n . Choosing n = t -1/2 ε
, we complete the proof.

Now we can prove that the limit measures of {P ε 0 } ε∈(0,1) only charge x + and x -. With the same notations as above we define the event

B ε = X 0,ε 2 tε+t ≥ (1 -γ) x + t for all t ≥ 0 ∪ X 0,ε 2 tε+t ≤ (1 -γ) x - t for all t ≥ 0 = max τ γ,x + (X 0,ε 2 tε+• ), τ γ,x + (X 0,ε 2 tε+• ) = +∞ ,
with the same definition as in [START_REF] Flandoli | Well posedness of the transport equation by stochastic perturbation[END_REF]. By strong Markov property and Proposition 2, there exists an infinitesimal function δ ε such that

P B ε ≥ (1 -δ ε ) inf η=± P max τ γ,x + (X ηxε,ε tε+• ), τ γ,x -(X ηxε,ε tε+• ) = +∞ .
Using the second assertion in Corollary 8 and modifying δ ε , we deduce that

P B ε ≥ (1 -δ ε ) inf η=± P τ γ,x η (X η xε,ε ) = +∞ .
By the second claim of Theorem 3, we deduce that that

lim ε→0 P (B ε ) = 1.
This implies that any weak limit µ of P ε 0 is concentrated on the extremal solutions ±x + • .

Weights of the extremal paths

The computation of the weights relies on the following martingale property Lemma 9 Define the functions

v(x) = exp - 2 1 + α |x| 1+α , V (x)
U ε (x) =        A + ε 2 -1 1+α V A + ε 2 1 1+α , x ≥ 0, A - ε 2 -1 1+α V A - ε 2 1 1+α
, x < 0, Then, the process

M ε t = U ε X 0,ε t t≥0
is a martingale.

Before we prove Lemma 9, we first explain how it applies to the end of the proof of Theorem 1.

As in the statement of Theorem 3, we consider an infinitesimal function x ε such that x ε /x ε → +∞ as ε → 0. We know that τ xε (X 0,ε ) is finite and that (M ε t ) t≥0 is a bounded martingale (V is obviously bounded). By Doob's theorem, we deduce that E M ε τ xε (X 0,ε ) = 0, so that U ε ( x ε )P X 0,ε

τ xε = x ε + U ε (-x ε )P X 0,ε τ xε = -x ε = 0, that is A + -1 1+α V A + 1 1+α x ε x ε P X 0,ε τ xε = x ε + A --1 1+α V -A -1 1+α x ε x ε P X 0,ε τ xε = -x ε = 0.
By Theorem 3, we can let ε tend to zero and then deduce that any limit measure µ must satisfy

A + -1 1+α V (+∞)µ(x + • ) + A --1 1+α V (-∞)µ(x - • ) = 0,
which completes the proof since V (+∞) = V (-∞).

It now remains to prove Lemma 9: Proof. We notice that the function U ε is continuously differentiable with

U ′ ε (x) =        v A + ε 2 1 1+α x , x ≥ 0, v A - ε 2 1 1+α x , x < 0.
The function v is also continuously differentiable with v ′ (x) = 2|x| α v(x), x ≥ 0, -2|x| α v(x), x < 0.

Therefore, the function U ε is continuously differentiable on (-∞, 0) and (0, +∞) and

U ′ ε (x) =        2 A + ε 2 |x| α U ε (x), x > 0, - 2
A - ε 2 |x| α U ε (x), x < 0, which proves that, for any x = 0,

ε 2 2 U ′′ ε (x) + b(x)U ′ ε (x) = 0.
The martingale property follows from Itô's formula.

  1+α , and τ r (ξ) := inf {t > 0 : |ξ t | ≥ r} for any r > 0 and ξ ∈ C([0, +∞); R d ),

= x 0 v

 0 (r)dr, for x ∈ R, together with

Acknowledgment

Part of this work was achieved when the authors were visiting the Centre Interfacultaire Bernoulli of the École Polytechnique Fédérale de Lausanne during the semester Stochastic Analysis and Applications organized in 2012. The authors are very grateful to the organizers and to the staff, with special thanks to R. Dalang.