Mirror symmetry for concavex vector bundles on projective spaces

Let $X\subset Y$ be smooth, projective manifolds. Assume that $X$ is the zero locus of a generic section of a direct sum $V+$ of positive line bundles on $\PP^n$. Furthermore assume that the normal bundle $N_{X/Y}$ is a direct sum $V-$ of negative line bundles. We show that a $V:=V+\oplus V-$-twisted Gromov-Witten theory of $\PP^n$ restricts to the Gromov-Witten theory of $X$ inherited form $Y$. The later one can be computed via a Mirror Theorem which we prove in this paper.


Introduction
Let V + = ⊕ i∈I O(k i ) and V − = ⊕ j∈J O(−l j ) be vector bundles on P s with k i and l j positive integers. Suppose X ι ֒→ P s is the zero locus of a generic section of V + and Y is a projective manifold such that X j ֒→ Y with normal bundle N X/Y = ι * (V − ). The relations between Gromov-Witten theories of X and Y are studied here by means of a suitably defined equivariant Gromov-Witten theory in P s . We apply mirror symmetry to the latter to evaluate the gravitational descendants of Y supported in X.
Section 2 is a collection of definitions and techniques that will be used throughout this paper. In section 3, using an idea from Kontsevich, we introduce a modified equivariant Gromov-Witten theory in P s corresponding to V = V + ⊕ V − . The corresponding D-module structure ( [6], [15], [29]) is computed in section 4. It is generated by a single functionJ V . In general, the equivariant quantum product does not have a nonequivariant limit. It is shown in Lemma 4.1.1 that the generatorJ V does have a limit J V which takes values in H * P m [[q, t]]. It is this limit that plays a crucial role in this work.
Let Y be a smooth, projective manifold. The generator J Y of the pure Dmodule structure of Y encodes one-pointed gravitational descendents of Y . It takes values in the completion of H * Y along the semigroup (Mori cone) of the rational curves of Y . The pullback map j * : H * Y → H * X extends to a map between the respective completions. In Theorem 4.2.2 we describe one aspect of the relation between pure Gromov-Witten theory of X j ֒→ Y and the modified Gromov-Witten theory of P s . Under natural restrictions, the pull back j * (J Y ) pushes forward to J V . It follows that although defined 1991 Mathematics Subject Classification. Primary 14N35. Secondary 14L30. 1 on P s , J V encodes the gravitational descendants of Y supported in X, hence the contribution of X to the Gromov-Witten invariants of Y .
The only way that X remembers the ambient variety Y in this context is by the normal bundle. Y can therefore be substituted by a local manifold. This suggests that there should be a local version of mirror symmetry (see the Remark at the end of section 4). This was first realized by Katz, Klemm, and Vafa [20]. The principle of local mirror symmetry in general has yet to be understood. Some interesting calculations that contribute toward this goal can be found in [8].
In section 5 we give a proof of the Mirror Theorem which allows us to compute J V . A hypergeometric series I V that corresponds to the total space of V is defined. The Mirror Theorem 5.0.3 states that I V = J V up to a change of variables. Hence, the gravitational descendants of Y supported on X can be computed in P s .
Two examples of local Calabi-Yau threefolds are considered in section 6. For X = P 1 and V = O(−1) ⊕ O(−1), we obtain the Aspinwall-Morrison formula for multiple covers. If X = P 2 and V = O(−3), the quantum product of Y pulls back to the modified quantum product in P 2 . The mirror theorem in this case yields the virtual number of plane curves on a Calabi-Yau threefold.
The rich history of mirror symmetry started in 1990 with a surprising conjecture by Candelas, de la Ossa, Green and Parkes ( [7]) which predicts the number n d of degree d rational curves on a quintic threefold. In [15], Givental presented a clever argument which, as shown later by Bini et al. in [6] and Pandharipande in [29], yields a proof of the mirror conjecture for Fano and Calabi-Yau (convex) complete intersections in projective spaces. Meanwhile, in a very well written paper [26], Lian, Liu and Yau used a different approach to obtain a complete proof of mirror theorem for concavex complete intersections on projective spaces. An alternative proof of the convex Mirror Theorem has been given by Bertram ([5]). In this paper we use Givental's approach to study the local nature of mirror symmetry and to present a proof of the concavex Mirror Theorem.
Acknowledgements. This work is part of author's Ph.D. thesis at Oklahoma State University. The author would like to thank Sheldon Katz for his passionate and tireless work in advising with this project. Special thanks to Bumsig Kim, Carel Faber, Ionut-Ciocan Fontanine and Zhenbo Qin who were very helpful throughout this work. At various times the author has benefited from conversations with Tom Graber, Rahul Pandharipande and Ravi Vakil, to whom the author is very grateful. We would like to thank also the referees whose help in improving this manuscript was invaluable.

Stable maps and localization
2.1. Genus zero stable maps. Let M 0,n (X, β) be the Deligne-Mumford moduli stack of pointed stable maps to X. For an excellent reference on the construction and its properties we refer the reader to [13]. We recall some of the features on M 0,n (X, β) and establish some notation. For each marking point x i let e i : M 0,n (X, β) → X be the evaluation map at x i and L i the cotangent line bundle at x i . The fiber of this line bundle over a moduli point (C, x 1 , ..., x n , f ) is the cotangent space of the curve C at x i . Let π k : M 0,n (X, β) → M 0,n−1 (X, β) be the morphism that forgets the k-th marked point. The obstruction theory of the moduli stack M 0,n (X, β) is described locally by the following exact sequence (Here and thereafter we are naming sheaves after their fibres). To understand the geometry behind this exact sequence we note that are respectively the tangent space and the obstruction space at the moduli point (C, describe respectively the infinitesimal automorphisms and infinitesimal deformations of the marked source curve. It follows that the expected dimension of M 0,n (X, β) is −K X · β + dimX + n − 3.
A smooth projective manifold X is called convex if H 1 (P 1 , f * T X) = 0 for any morphism f : P 1 → X. For a convex X, the obstruction bundle Υ vanishes and the moduli stack is unobstructed and of the expected dimension. Examples of convex varieties are homogeneous spaces G/P .
In general this moduli stack may behave badly and have components of larger dimensions. In this case, a Chow homology class of the expected dimension has been constructed [3] [27]. It is called the virtual fundamental class and denoted by [M 0,n (X, β)] virt . Although its construction is quite involved, we will be using mainly two relatively easy properties. The virtual fundamental class is preserved when pulled back by the forgetful map π n . A proof of this fact can be found in section 7.1.5 of [9]. If the obstruction sheaf Υ is free, the virtual fundamental class refines the top Chern class of Υ. This fact is proven in Proposition 5.6 of [3].

2.2.
Equivariant cohomology and localization theorem. The notion of equivariant cohomology and the localization theorem is valid for any compact connected Lie group. For a detailed exposition on this subject we suggest Chapter 9 of [9]. Below we state without proof the results that will be used in this work.
. For an arbitrary X, the equivariant cohomology Let U be a vector bundle over X. If the action of T on X can be lifted to an action on U which is linear on the fibers, U is an equivariant vector bundle and U T is a vector bundle over X T . The equivariant chern classes of E are c T k (U ) := c k (U T ). We will use E(U ) (E T (U )) to denote the nonequivariant (equivariant) top chern class of U . Let X T = ∪ j∈J X j be the decomposition of the fixed point locus into its connected components. X j is smooth for all j and the normal bundle N j of X j in X is equivariant. Let i j : X j → X be the inclusion. The following corollary of the localization theorem will be used extensively here: .
A basis for the characters of the torus is given by ε i (t 0 , ..., t s ) = t i . There is an isomorphism between the character group of the torus and H 2 (BT ) sending ε i to λ i . We will say that the weight of the character ε i is λ i .
For an equivariant vector bundle U over X it may happen that the restriction of U on a fixed point component X j is trivial (for example if X j is an isolated point). In that case U decomposes as a direct sum ⊕ m i=1 µ i of characters of the torus. If the weight of µ i is ρ i , then the restriction of c T k (U ) on X j is the symmetric polynomial σ k (ρ 1 , ..., ρ m ).
Our interest here is for X = P s . For any action of T on P s we will denote Consider the diagonal action of T = (C * ) s+1 on P s with weights (−λ 0 , ..., −λ s ) i.e.
There is an obvious lifting of the action of T on the tautological line bundle O(−1). It follows that O(k) is equivariant for all k. Let p = c T 1 (O P s (1)) be the equivariant hyperplane class. We obtain P = C[λ, p]/ i (p − λ i ) and R = C(λ)[p]/ i (p − λ i ). The locus of the fixed points consists of points p j for j = 0, 1, ..., s where p j is the point whose j-th coordinate is 1 and all the other ones are 0. On the level of the cohomology the map i * j sends p to λ j . A basis for R as a C(λ)-vector space is given by φ j = k =j (p − λ k ) for j = 0, 1, ..., s. Also .
By translating the target of a stable map we get an action of T on M 0,n (P s , d).
In There is a finite group of automorphisms G Γ acting on M Γ [9] [16]. The order of the automorphism group G Γ is The fixed point component corresponding to the decorated graph Γ is Let i Γ : M Γ ֒→ M 0,n (P s , d) be the inclusion of the fixed point component corresponding to Γ and N Γ its normal bundle. This bundle is T -equivariant. Let α be an equivariant cohomology class in H * T (M 0,n (P s , d)) and α Γ := i * Γ (α). Theorem 2.2.1 says: .
Explicit formulas for Euler T (N Γ ) in terms of chern classes of cotangent line bundles in H * T (M Γ ) have been found by Kontsevich in [24].

2.3.
Linear and nonlinear sigma models for a projective space. Two compactifications of the space of degree d maps P 1 → P s will be very important in this paper. M d := M 0,0 (P s × P 1 , (d, 1)) is called the degree d nonlinear sigma model of P s and N d := P(H 0 (P 1 , O P 1 (d)) s+1 ) is called the degree d linear sigma model of the projective space P s . An element in H 0 (P 1 , O P 1 (d)) s+1 is an s + 1-tuple of degree d homogeneous polynomials in two variables w 0 and w 1 . As a vector space, H 0 (P 1 , O P 1 (d)) s+1 is generated by the vectors v ir = (0, ..., 0, w r 0 w d−r 1 , 0..., 0) for i = 0, 1, ..., s and r = 0, 1, ..., d. The only nonzero component of v ir is the i-th one.
The action of T ′ := T × C * in P s × P 1 with weights (−λ 0 , ..., −λ s ) in the P s factor and (− , 0) in the P 1 factor gives rise to an action of T ′ in M d by translation of maps. T ′ also acts in N d as follows: fort = (t 0 , ..., t s ) ∈ T and t ∈ C * Here is a set-theoretical description of this map (for a proof that it is a morphism see [15] or [25]). Let q i for i = 1, 2 be the projection maps on P s × P 1 . For a stable map (C, f ) ∈ M d let C 0 be the unique component of C such that q 2 • f : C 0 → P 1 is an isomorphism. Let C 1 , ..., C n be the irreducible components of C − C 0 and d i the degree of the restriction of q 1 • f on C i . Choose coordinates on C 0 ∼ = P 1 such that q 2 • f (y 0 , y 1 ) = (y 1 , y 0 ). Let C 0 ∩ C i = (a i , b i ) and q 1 • f = [f 0 : f 1 : ... : f s ] : C 0 → P s . Then (14) ψ Let p ir be the points of N d corresponding to the vectors v ir . The fixed point loci of the T ′ -action on N d consists of the points p ir . We write κ for the equivariant hyperplane class of N d . The restriction of κ at the fixed point p ir is λ i + r . The restriction of the equivariant Euler class of the tangent space T N d at p ir is [25] (15) Fixed point components of M d are obtained as follows. Let Γ i d j be the graph of a T -fixed point component in M 0,1 (P s , d j ) where the marking is mapped to p i and d 1 + d 2 = d. Let (d 1 , d 2 ) be a partition of d. We identify . Let C be the nodal curve obtained by gluing C 1 with P 1 at x 1 and 0 ∈ P 1 and C 2 with P 1 at x 2 and ∞ ∈ P 1 . Let f : C → P s × P 1 map C 1 to the slice P s × ∞ by means of f 1 and C 2 to P s × 0 by means of f 2 . Finally f maps P 1 to p i × P 1 by permuting coordinates. ψ maps M i d 1 d 2 to p id 2 ∈ N d , hence the equivariant of this component in the above identification can be found by splitting it in five pieces: smoothing the nodes x 1 and x 2 and deforming the restriction of the map to C 1 , C 2 , P 1 . Using Kontsevich's calculations, Givental obtained [15] where c j , j = 1, 2 is the first Chern class of the cotangent line bundle on

A Gromov-Witten theory induced by a vector bundle
3.1. The obstruction class of a concavex vector bundle. The notion of concavex vector bundle is due to Lian, Liu and Yau [25] and is central to this work.
A direct sum of convex and concave line bundles on X is called a concavex vector bundle.
A concavex vector bundle V in a projective space P s has the form where k i and l j are positive numbers. Denote E + := E(V + ) and The obstruction class corresponding to V is defined to be: For a T -action on P s that lifts to a linear action on the fibers of V = Consider the trivial action of Let p denotes the equivariant hyperplane class. The T -action lifts to a linear action on the fibers of V with weights ((−λ i ) i∈I , (−λ j ) j∈J ). Let q i and q j denote the projection maps on M 0,n (P s , d) The equivariant obstruction class is The modified equivariant integral for the trivial action of T on P s gives rise to a modified perfect pairing in R Let T 0 = 1, T 1 = p, ..., T s = p s be a basis of R as a C(λ)-vector space. The intersection matrix (g rt ) := ( T r , T t V ) has an inverse (g rt ). Let T i = s j=0 g ij T j be the dual basis with respect to this pairing. Clearly Proof. For simplicity we will consider the case V = O(k) ⊕ O(−l) and k = n. The general case is similar. Let M k = M 0,k (P s , d) and M n,n = M n × M n−1 M n . Consider the following equivariant commutative diagram: We compute: By the projection formula Since the map µ is birational and Substituting in (24) and applying base extension properties (π n is flat) yields For the case of a negative line bundle we have We now use the spectral sequence The map µ is birational. If we think of M n as the universal map of M n−1 , then the map µ has nontrivial fibers only over pairs of stable maps in M n that represent the same special point (i.e. node or marked point) of a stable map in M n−1 . These nontrivial fibers are isomorphic to P 1 . Since F = e * n+1 O(−l) we obtain R q µ * F = 0 for q > 0. It follows that this spectral sequence degenerates, giving Now we proceed as in (26) to conclude The lemma is proven. † Remark 3.1.1. The previous lemma justifies the omission of n from the notation of the obstruction class.

Modified equivariant correlators and quantum cohomology.
Let γ i ∈ R for i = 1, ..., n and d > 0. Introduce the following modified equivariant Gromow-Witten invariants: Now M 0,n (P s , 0) = M 0,n × P s and all the evaluation maps equal the projection q 2 to the second factor. The integrals in this case are defined as follows The modified equivariant gravitational descendants are defined similarly to Gromov-Witten invariants: (34) Lemma (3.1.1) is essential in proving that the modified correlators satisfy the same properties, such as fundamental class property, divisor property, point mapping axiom etc., that the usual Gromov-Witten invariants do. The proofs are similar to the ones in pure Gromov-Witten theory. As an illustration, we prove one of these properties.
Fundamental class property. Let γ n = 1 and d = 0. The forgetful morphism π n : M 0,n (P s , d) → M 0,n−1 (P s , d) is equivariant. Using Lemma 3.1.1 we obtain: . Therefore: The last equality is because the fibers of π n are positive dimensional. If d = 0, by the point mapping property we know that the integral is zero unless n = 3. In that case:Ĩ 0 (γ 1 , γ 2 , 1) = γ 1 , γ 2 . † We will now prove a technical lemma which will be very useful later. Let A ∪ B be a partition of the set of markings and d = d 1 + d 2 . Let D = D(A, B, d 1 , d 2 ) be the closure in M 0,n (P s , d) of stable maps of the following type. The source curve is a union C = C 1 ∪ C 2 of two lines meeting at a node x. The marked points corresponding to A are on C 1 and those corresponding to B are on C 2 . The restriction of the map f on C i has degree d i for i = 1, 2. D is a boundary divisor in M 0,n (P s , d). Let M 1 := M 0,|A|+1 (P s , d 1 ) and M 2 := M 0,|B|+1 (P s , d 2 ). Let e x andẽ x be the evaluation maps at the additional marking in M 1 and M 2 and µ := (e x ,ẽ x ). The boundary divisor D is obtained from the following fibre diagram where ν is the "evaluation map at the node x" and δ is the diagonal map.
Proof. This lemma is the analogue of the Lemma 16 in [13]. The proof needs a minor modification. Let α : D → M 0,n (P s , d). Consider the normalization sequence at x: Twisting it by f * (V + ) and f * (V − ) and taking the cohomology sequence yields the following identities on D: . and . By combining equations (36) and (37) we obtain the restriction of E d in the divisor D: Using formula (23) we obtain The lemma is proven. † The same proof can be used to show that the previous splitting lemma is true for gravitational descendants as well.
Definition 3.2.1. The modified, equivariant quantum product on R is defined to be the linear extension of is a commutative, associative algebra with unit T 0 .
Proof. A simple calculation shows that: The commutativity of the modified, equivariant quantum product follows from the symmetry of the new integrals. T 0 is the unit due to the fundamental class property for the modified Gromov-Witten invariants. To proving the associativity we proceed as in Theorem 4 in [12]. LetΦ ijk = ∂ 3Φ ∂t i ∂t j ∂t k .
We compute Since the matrix (g ld ) is nonsingular, Equation (44) Here q = e t . We extend this product to R ⊗ C C[[q]] to obtain the small equivariant quantum cohomology ring SQH * V P s T . We will use * V to denote both the small and the big quantum product. The difference will be clear from the context.
where c is a formal symbol that stands for c 1 T (L 1 ) and T a − c should be expanded in powers of c .
Proof. On one hand On the other hand The theorem follows from the topological recursion relations (41). † Restrictionss a of the sections s a to γ ∈ H 0 (P m ) ⊕ H 2 (P m ) are solutions of ∂ ∂t i = T i * V : i = 0, 1. Recall that e 1 : M 0,2 (P s , d) → P s is the evaluation map at the first marked point and c is the chern class of the cotangent line bundle at the first marked point. Substituting (50) in (51) and using the projection formula we obtain: In the above expression P D : is the Poincaré duality isomorphism. It will be convenient for us to work with the moduli space of one pointed stable maps. To that end we note that This identity follows easily from the fact that if π 2 : M 0,2 (P s , d) → M 0,1 (P s , d) forgets the second marked point and D is the image of the universal section of π induced by the marked point, then c = π 2 * (c) + D and E d = π 2 * (E d ). The final expression forJ V is From this presentation we see that the presence of the equivariant class E + in the denominator ofJ V is a potential problem for the existence of the nonequivariant limit. Proof. Let V ′ d be the subbundle of V + d whose fiber consists of those sections of H 0 (C, f * (V + )) that vanish at the marked point. Let There is an exact sequence of equivariant bundles on M 0,1 (P s , d): Taking the top chern classes we obtain It is now visible from this presentation thatJ V ∈ P[[q]] and The lemma is proven.
has nonpositive degree in P s , we intersect with an ample divisor in Y to see that is impossible. Otherwise, we intersect with [P s ] to get the same contradiction. Hence I 2 is empty and all the curves C i lie in P s . It follows that f factors through P s and therefore (C ′ , x 1 , ...x n , f ) ∈ M 0,n (P s , d). On the other hand M 0,n (P s , d) is a component of M 0,n (Y, j * ([C])) (see for example section 7.4.4 in [9]). These two arguments imply the lemma. † Denote M 0,n (Y, d) := M 0,n (Y, j * ([C])), where C is any rational curve of degree d in P s . The following Lemma is a special case of a conjecture by Cox, Katz and Lee in [10] which was proved in [23]. We extend the map j * : H * Y → H * P s to a homomorphism by defining j * (t i ) = 0 for i > 1 and j * (q β ) = q β for β ∈ j * (H 2 (P s , Z)) and j * (q β ) = 0 for β ∈ H 2 (Y, Z)− j * (H 2 (P s , Z)). The following results show that J-function is local.
Proof. Notice that Consider the following fiber diagram [12] and the previous lemma The theorem follows easily. † on P s . Let ι : X ֒→ P s be the zero locus of a generic section of V + . Assume that X is smooth and dim X > 2. Let Y be a smooth projective variety such that j : ∈ M X then all the irreducible components C i of C satisfy C i ⊂ X. Let j * be the map constructed as in (62). Let J Y be the generator of the pure D-module of Y ( [14]). Then where ι ! is the Gysin map on cohomology.
Proof. Since dimX > 2 it follows that H 2 X is generated by ι * (H). Let β 1 be the Poincaré dual to ι * (H) and let D 1 , D 2 , ..., D r be a set of generators of H 2 (Y, Q). We may assume that j * (D 1 ) = ι * (H) and j * (D i ) = 0 for i > 1. Let tD := t 1 D 1 + ... + t r D r . Now Consider the following diagram The square on the left is a fibre diagram. We repeatedly use the projection formula:

MIRROR SYMMETRY FOR CONCAVEX VECTOR BUNDLES ON PROJECTIVE SPACES 19
(65) The equality in the second row follows from excess intersection theory in the left square. An argument similar to Lemma 4.2.1 implies that There are two obstruction theories in this moduli stack corresponding to the moduli problems of maps to X and Y respectively. They differ exactly by the bundle R 1 π 2 * e * 2 (N ) where is the map that forgets the second marked point and N = N X/Y . It follows that: Consider the following commutative diagram: We compute: There is the following fibre square: We apply Proposition 9.3 in [18] to obtain: . Therefore: On the other hand, Proposition 11.2.3 of [9] says that : Substituting (67) and (68) in (65) we obtain . Substituting this in (69) and using the projection formula we obtain The theorem is proven. † Remark 4.2.1. This naturally leads to local mirror symmetry. For example, let Y be a Calabi-Yau threefold that contains X = P 2 . By adjunction formula, the normal bundle of P 2 in X is K P 2 = O P 2 (−3). The last theorem asserts that the restriction of J Y in X depends only on V = O P 2 (−3) i.e. in a neighborhood of X in Y . Hence J V encodes Gromov-Witten correlators of the total space of O P 2 (−3) which is a local Calabi-Yau. In the next section we will see that mirror symmetry can be applied to J V establishing that mirror symmetry is local at least on the A-side. Interesting calculations in this direction can be found in [8].

Mirror Theorem
In this section we will formulate and prove The Mirror Theorem which computes the generator J V . Recall that V = (⊕ i∈I O(k i )) ⊕ (⊕ j∈J O(−l j )) = V + ⊕ V − with k i , l j > 0 for all i ∈ I and j ∈ J. Consider the H * P s -valued hypergeometric series (71) Theorem 5.0.3. (Mirror theorem). Assume that i∈I k i + j∈J l j ≤ s + 1 and that J is nonempty. If |J| > 1 or i∈I k i + j∈J l j < s + 1 then J V = I V . Otherwise, there exists a power series I 1 of q such that J V (t 0 , t 1 + I 1 ) = I V (t 0 , t 1 ) as power series of q.
Remark 5.0.2. The case in which J is empty has been treated in [5], [6], [15], [29]. It was suggested by Givental that his techniques should apply in the case in which J is nonempty.

The Equivariant Mirror Theorem.
We use Givental's approach for complete intersections in projective spaces [15] to prove an equivariant version of the theorem. For the remainder of this paper we will use the standard diagonal action of T = (C * ) s+1 on P s with weights (−λ 0 , ..., −λ s ). Recall from section 2.
and I eq V := exp Obviously the nonequivariant limits of J eq V and I eq V are respectively J V and I V . The mirror theorem will follow as a nonequivariant limit of the following Remark 5.1.1. As the reader will see, the central part of the proof of the Mirror Theorem (up to section 5.5) involves lengthy formulaes and algebraic manipulations. To simplify the presentation, we will assume during this part that V = O(k) ⊕ O(−l). The general case is similar. We will return to the general case V = V + ⊕ V − in section 5.5.
Recall that the equivariant Thom classes φ i of the fixed points p i form a basis of R as a C(λ)-vector space. Let S i and S ′ i be the restrictions of S and S ′ at the fixed point p i . By the localization theorem in P s they determine S and S ′ . By the projection formula The proof of the equivariant mirror theorem is based on exhibiting similar properties of the correlators S i and S ′ i . The extra property S i = 1 + o( −2 ) determines S i uniquely. After the change of variables that property is satisfied by S ′ i as well which implies S i = S ′ i . We now proceed with displaying properties of the correlators S i and S ′ i .

Linear recursion relations. The first property is given by this
Lemma 5.2.1. The correlators S i satisfy the following linear recursion relations: (75) .
Proof. We will see during the proof that S j is regular at = λ j − λ i d . The integrals that appear in the formula for S i can be evaluated using localization theorem (77) There are three types of fixed point components M Γ of M T 0,1 (P s , d). The first one consists of those M Γ where the component of the curve that contains the marked point is collapsed to p i . We denote the set of these components by F i 1,d . Let F i 2,d be the set of those M Γ in which the marked point is mapped at p i and its incident component is a multiple cover of the line p i , p j for some j = i. Finally let F i 0,d be the rest of the fixed point components. Notice first that: Indeed, let Γ j ∈ F i 0,d represent a fixed point component with the marked point mapped to the fixed point p j for some j = i. Since (e * 1 (φ i )) Γ j = 0 we are done. Next, in each fixed point component that belongs to F i 1,d the class c is nilpotent. Indeed, if Γ is the decorated graph that represents such a fixed point component, let M 0,k correspond to the vertex of Γ that contains the marked point. Then k ≤ d + 1. There is a morphism: is a polynomial of c in M Γ . Hence (79) We now consider the fixed point components in F i 2,d . Again let Γ represent such a component. For a stable map (C, x 1 , f ) in Γ let C ′ be the component of C containing x 1 , C ′′ the rest of the curve, x = C ′ ∩ C ′′ and f (x) = p j for some j = i. Let d ′ be the degree of the map f on the component C ′ and . Denote its decorated graph by Γ ′′ . Choose the coordinates on C ′ such that the restriction of f on C ′ is given by f (y 0 , y 1 ) = (0, ..., z i = y d ′ 0 , ..., z j = y d ′ 1 , ..., 0). As Γ moves in F i 2,d , the set of all such Γ ′′ exhausts all the fixed points in M 0,1 (P s , d ′′ ) where the first marked point is not mapped to p i . Since Aut(Γ) = Aut(Γ ′′ ) it follows from (10) that: The local coordinate at p i on the component C ′ is z = y 1 y 0 . The weight of the T -action on y l is λ l d ′ for l = 0, 1. It follows that the weight of the action on the coordinate z and hence on T * p i C ′ is can be split in three pieces: smoothing the node x and deforming the maps f | C ′′ and f | C ′ . It follows [15] Twisting it by f * (V + ) and f * (V − ) respectively and taking the cohomology sequence yields It allows us to compute: Therefore we have: A basis for H 0 (C ′ , f * (V + )) = H 0 (O P 1 (kd ′ )) consists of monomials y s 0 y kd ′ −s 1 for s = 0, ...kd ′ . It can be used to calculate We pause here to show that S i is regular at = λ j − λ i d for any j = i and any d > 0. It follows from (78) and (79) that: .
From this representation of S i it is clear that the coefficients of the power and R id has poles only at = 0 therefore S i is regular at = λ i − λ j d . We use the equations (81), (86) and (85) to compute The lemma follows by substituting the above identity into (87) †. Proof. We know that Note that S ′ id ∈ C(λ, ) is a proper rational expression of . It has multiple poles at = 0 and simple poles at = λ r − λ i m for any r = i and any 1 ≤ m ≤ d. Applying calculus of residues in the -variable yields: for some polynomials R id ∈ C(λ)[ −1 ] such that R id (0) = 0. Substitute equation (90) in (89) to obtain: Changing the order of summation in the last equation yields: The lemma follows from the identity 5.3. Double polynomiality. Recall from Section 3.1 that V induces a modified equivariant integral ω V : R → C(λ) defined as follows: As one can see, in the case V = O(k) ⊕ O(−l), this modified equivariant integral simplifies via We have chosen not to simplify this integral in the proof of the following Lemma so that it is easier to see how to proceed in the general case.
Lemma 5.3.1. If z is a variable, the expression: Proof. In Section 2.3 we introduced the action of T ′ = T × C * on P s × P 1 with weights (−λ 0 , ..., −λ s ) on the first factor and (− , 0) in the second factor. Consider the following T ′ -equivariant diagram ) The lemma will follow from the identity: where ψ and κ were defined in Section 2.3. The localization formula for the diagonal action of T on P s applied to the left side gives We recall from identity (74) To compute the integrals on the right side of (96) we will use localization for the action of T ′ on M d . In Section 2.3 we found that the components of the fixed point loci have the form M i for some i = 0, 1, ..., s and a splitting d = d 1 + d 2 . We first compute the restriction of E T ′ (W d ) in such a component. Consider the following normalization sequence Twist (99) by f * (O(−l) ⊗ O P 1 ) and take the corresponding long exact cohomology sequence. We obtain The first piece is trivial since it comes from the isomorphism The left hand side is generated by 1 z i l therefore the weight of that piece is −lλ i . It follows that Similarly, twisting the normalization sequence (99) by f * (O(k) ⊗ O P 1 ) and taking the corresponding cohomology sequence we obtain: We now use the localization theorem to calculate the integrals on the right side of (96 If we use localization to compute S i in (98) and then substitute in (97) we obtain the right side of the last equation. † Lemma 5.3.2. If z is a variable, the expression: Proof. The lemma will follow from the identity  The lemma is proven. † 5.4. Mirror transformation and uniqueness. The following two theorems carry over from [15]. The first lemma deals with uniqueness. Then S = S ′ .
The second lemma describes a transformation which preserves the properties of lemma 5.4.1.
Lemma 5.4.2. Let I 1 be a power series in q whose first term is zero. Then exp( I 1 p )S(qe I 1 , ) satisfies conditions 1, 2, 3 of Lemma 5.4.1.
5.5. The conclusion of the proof of Mirror Theorem. Recall that .
We are assuming that there is at least one negative line bundle. We expand the second factor of I eq V as a polynomial of −1 . Each negative line bundle produces a factor of p . For example, in the case V = O(k) ⊕ O(−l) the expansion yields: I eq V = exp If V contains two or more negative line bundles it follows that I eq V = exp Lemma (5.4.1) and (5.4.2) imply that J eq V = I eq V . If i∈I k i + j∈J l j < s+1 the presence of 1 d(s+1−k−l) in the above expansion shows that again I eq V = exp hence J eq = I eq . We may assume that i∈I k i + j∈J l j = s+1 and |J| = 1.
In this case I eq V = exp where I 1 is a power series of q whose first term is zero. For example if V = O(k) ⊕ O(−l) the power series I 1 is Recall that S = 1 + o( −2 ). Therefore Multiplying both sides of this identity by exp t 0 + pt yields (107) J eq V (t 0 , t + I 1 ) = I eq V (t 0 , t). This completes the proof. † is called the virtual number of degree d rational curves in X. As promised in Remark 3.2.1, we will show that the modified equivariant quantum product in this case has a nonequivariant limit. We will also use The Mirror Theorem to calculate these numbers N d .
For example, using the divisor axiom we obtain The mirror theorem for this case says that J(t 0 , t 1 + I 1 ) = I V (t 0 , t 1 ). This theorem allows us to compute the virtual number of rational plane curves in the Calabi-Yau X. The first few numbers are 3, −45 8 , 244 9 .