QUANTUM COHOMOLOGY of PROJECTIVE BUNDLES OVER Pn

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 9, September 1998, Pages 3615–3638 S 0002-9947(98)01968-0

QUANTUM COHOMOLOGY OF PROJECTIVE BUNDLES OVER Pn

ZHENBO QIN AND YONGBIN RUAN

Abstract. In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space Pn. Using excessive intersection theory, we compute the leading coeﬃcients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev’s conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over Pn is partially veriﬁed. Moreover, relations between the quantum cohomology ring structure and Mori’s theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.

1. Introduction Quantum cohomology, proposed by Witten’s study [16] of two dimensional non- linear sigma models, plays a fundamental role in understanding the phenomenon of mirror symmetry for Calabi-Yau manifolds. This phenomenon was first observed by physicists motivated by topological field theory. A topological field theory starts with correlation functions. The correlation functions of the sigma models are linked with the intersection numbers of cycles in the moduli space of holomorphic maps from Riemann surfaces to manifolds. For some years, the mathematical construc- tion of these correlation functions has remained a difficult problem because the moduli spaces of holomorphic maps usually are not compact and may have the wrong dimension. The quantum cohomology theory was first put on a firm mathematical footing in [12], [13] for semi-positive symplectic manifolds (including Fano and Calabi-Yau manifolds), using the method of symplectic topology. Recently, an algebro-geometric approach has been taken by [8], [9]. The results of [12], [13] have been redone in the algebraic geometric setting for the case of homogeneous spaces. The advantage of homogeneous spaces is that the moduli spaces of holomorphic maps always have the expected dimension and their compactifications are nice. Beyond the homogeneous spaces, one cannot expect such nice properties for the moduli spaces. The projective bundles are perhaps the simplest examples. How- ever, by developing sophisticated excessive intersection theory, it is possible that the algebro-geometric method can work for any projective manifold. In turn, it may shed new light on removing the semi-positivity condition in the symplectic setting.

Received by the editors September 1, 1996. 1991 Mathematics Subject Classiﬁcation. Primary 58D99, 14J60; Secondary 14F05, 14J45. Both authors were partially supported by NSF grants. The second author also had a Sloan fellowship.

c 1998 American Mathematical Society

3615

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Although we have a solid foundation for the quantum cohomology theory at least for semi-positive symplectic manifolds, the calculation has remained a difficult task. So far, there are only a few examples which have been computed, e.g., Grassmannian [14], some rational surfaces [6], flag varieties [4], some complete intersections [3], and the moduli space of stable bundles over Riemann surfaces [15]. One common feature for these examples is that the relevant moduli spaces of rational curves have the expected dimension. This feature enables one to use the intersection theory. We should mention that there are many predications based on mathematically unjustified mirror symmetry (for Calabi-Yau 3-folds) and the linear sigma model (for toric varieties). In this paper, we attempt to determine the quantum cohomology of projective bundles over projective space Pn. In contrast to previous examples, the relevant moduli spaces in our case frequently do not have the expected dimensions. These moduli spaces make the calculations more difficult. We overcome this difficulty by using the excessive intersection theory. There are two main ingredients in our arguments. The first one is a result of Siebert and Tian (Theorem 2.2 in [14]), which says that if the ordinary cohomology H∗(X; Z) of a symplectic manifold X with the symplectic form ω is the ring gen- 1 t erated by α1,... ,αs with the relations f ,... ,f , then the quantum cohomology 1 t Hω∗(X; Z)ofXis the ring generated by α1,... ,αs with t new relations fω,... ,fω, i i where each new relation fω is just the relation f evaluated in the quantum cohomology ring structure. It was known that the quantum product α β is the deformation of the ordinary cup product by the lower order terms, called· quantum corrections. The second ingredient is that under certain numerical conditions, most of the quantum corrections vanish. Moreover, the nontrivial quantum corrections seem to come from Mori’s extremal rays. Let V be a rank-r bundle over Pn,andP(V) be the corresponding projective bundle. Let h and ξ be the cohomology classes of a hyperplane in Pn and the tautological line bundle in P(V ) respectively. For simplicity, we make no distinction n between h and π∗h,whereπ:P(V) P is the natural projection. Denote the → product of i copies of h and j copies of ξ in the ordinary cohomology ring by hiξj, and the product of i copies of h and j copies of ξ in the quantum cohomology ring by hi ξj .Fori=0,... ,r, put c (V )=c h for some integer c .Itiswellknown · i i· i i that KP(V ) =(n+1 c1)h+rξ, and the ordinary cohomology ring H∗(P(V ); Z) is the− ring generated by−h and ξ with the two relations r i (1.1) hn+1 =0 and ( 1) ci hiξr i =0. − · − i=0 X 2(n+r 2) In particular, H − (P(V ); Z) is generated by hn 1ξr 1 and hnξr 2, and its − − − Poincaré dual H2(P(V ); Z) is generated by (hn 1ξr 1) and (hnξr 2) ,wherefor − − ∗ − ∗ α H∗(P(V); Z), α stands for its Poincaré dual. We have ∈ ∗ (1.2) K (A)=a(n+1 c )+r ξ(A)=a(n+1 c )+r(ac + b) − P(V ) − 1 · − 1 1 for A =(ahn 1ξr 1 + bhnξr 2) H2(P(V ); Z). By definition,− −V is an ample− ∗ ∈ (respectively, nef) bundle if and only if the tautological class ξ is an ample (respectively, nef) divisor on P(V ). Assume that V is ample such that either c (n +1),orc (n+r)andV n( 1) is nef. 1 ≤ 1 ≤ ⊗OP − Then both ξ and KP(V ) are ample divisors. Thus, P(V ) is a Fano variety, and its quantum cohomology− ring is well-defined [13]. Here we choose the symplectic form 1 2 ω on P(V ) to be the Kahler form ω such that [ω]= K(V).Letfωand fω be − P

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the two relations in (1.1) evaluated in the quantum cohomology ring Hω∗(P(V ); Z). Then, by Theorem 2.2 in [14], the quantum cohomology Hω∗(P(V ); Z) is the ring 1 2 generated by h and ξ with the two relations fω and fω: 1 2 (1.3) Hω∗(P(V ); Z)=Z[h, ξ]/(fω,fω)

By Mori’s Cone Theorem [5], P(V ) has exactly two extremal rays, R1 and R2. Up to an order of R1 and R2, the integral generator A1 of R1 is represented by lines in the ﬁbers of the projection π. We shall show that under certain numerical conditions, the nontrivial homology classes A H2(P(V ); Z) which give nontrivial ∈ quantum corrections are A1 and A2,whereA2 is represented by some smooth rational curves in P(V ) which are isomorphic to lines in Pn via π. In general, it is unclear whether A2 generates the second extremal ray R2. However, we shall prove that under further restrictions on V , A2 generates the extremal ray R2.These analyses enable us to determine the quantum cohomology ring Hω∗(P(V ); Z). n r The simplest ample bundle over P is of the form V = n (mi), where i=1 OP mi > 0 for every i. Since we can twist V by n ( 1) without changing P(V ), we OP − L can assume that min m1,... ,mr =1.Inthiscase,P(V) is a special case of toric variety. Batyrev [2] conjectured{ a} general formula for the quantum cohomology of toric varieties. Furthermore, he computed the contributions from certain moduli spaces of holomorphic maps which have the expected dimensions. In our case, the contributions Batyrev computed are only part of the data to compute the quantum cohomology. As we explained earlier, the diﬃculty in our case lies precisely in computing the contributions from the moduli spaces with the wrong dimensions. Nevertheless, in our case, Batyrev’s formula (see also [1]) reads as follows.

r Batyrev’s Conjecture. Let V = n (m ),wherem >0for every i.Then i=1 OP i i the quantum cohomology ring Hω∗(P(V ); Z) is generated by h and ξ with two relations: L r r n+1 m 1 t(n+1+r r m ) tr h = (ξ m h) i− e− − i=1 i and (ξ m h)=e− . − i · − i i=1 P i=1 Y Y Our ﬁrst result partially veriﬁes Batyrev’s conjecture.

Theorem A. Batyrev’s conjecture holds if

r mi < min(2r, (n +1+2r)/2,(2n +2+r)/2). i=1 X Note that under the numerical condition of Theorem A, only extremal rational curves with fundamental classes A1 and A2 give contributions to the two relations in the quantum cohomology. The moduli space M(A2, 0) of rational curves with fundamental class A2 does not have the expected dimension in general, but it is compact. This fact greatly simplifies the excessive intersection theory involved. If the numerical condition is removed, then we have to consider other moduli spaces (for example M(kA2, 0) with k>1 and its excessive intersection theory). These moduli spaces are not compact in general. So we have extra difficulties in handing the compactification of these moduli spaces and the appropriate excessive intersection theory. These seem to be difficult problems, which we shall not pursue here.

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In general, ample bundles over Pn are not direct sums of line bundles. We can say much less about their quantum cohomology. However, we obtain some results about its general form and compute the leading coefficient. n Theorem B. (i) Let V be a rank-r ample bundle over P . Assume either c1 n, ≤ or c1 (n + r) and V n( 1) is nef, so that P(V ) is Fano. Then the quantum ≤ ⊗OP − cohomology Hω∗(P(V ); Z) is the ring generated by h and ξ with the two relations n+1 i j t(n+1 i j) h = a h ξ e− − − , i,j · · · i+j (c r) ≤X1− r i i r i tr i j t(r i j) ( 1) c h ξ − = e− + b h ξ e− − − , − i · · i,j · · · i=0 i+j (c n 1) X ≤ X1− − where the coefficients ai,j and bi,j are integers depending on V . (ii) If we further assume that c1 < 2r, then the leading coefficient a0,c r =1. 1− It is understood that when c1 n,thenthesummation in the ≤ i+j (c1 n 1) second relation in Theorem B (i) does not exist. In general, it≤ is− not− easy to P determine all the integers ai,j and bi,j in Theorem B (i). However, it is possible to compute these numbers when (c1 r) is relatively small. For instance, when − r (c r) = 0, then necessarily V = n (1)⊕ and it is well-known that the quantum 1 − OP cohomology Hω∗(P(V ); Z) is the ring generated by h and ξ with the two relations n+1 t(n+1) r i i r i tr h = e− and i=0( 1) ci h ξ − = e− .When(c1 r)=1andr

Proposition C. The quantum cohomology ring Hω∗(P(TPn ); Z) with n 2 is the ring generated by h and ξ with the two relations ≥ n n+1 tn i i n i n tn h = ξ e− and ( 1) c h ξ − =(1+( 1) ) e− . · − i · · − · i=0 X Recall that for an arbitrary projective bundle over a general manifold, its cohomology ring is a module over the cohomology ring of the base with the generator ξ and the second relation of (1.1). Naively, one may think that the quantum cohomology of a projective bundle is a module over the quantum cohomology of the base with the generator ξ and the quantanized second relation. Our calculation shows that one cannot expect such simplicity for its quantum cohomology ring. We hope that our results could shed some light on the quantum cohomology for general projective bundles, which we shall leave for future research. Our paper is organized as follows. In section 2, we discuss the extremal rays and extremal rational curves. In section 3, we review the deﬁnition of quantum product and compute some Gromov-Witten invariants. In the remaining three sections, we prove Theorem B, Theorem A, and Proposition C respectively. Acknowledgements We would like to thank Sheldon Katz, Yungang Ye, and Qi Zhang for valuable helps and stimulating discussions. In particular, we are grateful to Sheldon Katz for bringing Batyrev’s conjecture to our attention.

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2. Extremal rational curves Assume that V is ample such that either c (n +1),or c (n+r)and 1 ≤ 1 ≤ V n( 1) is nef. In this section, we study the extremal rays and extremal ⊗OP − rational curves in the Fano variety P(V ). By Mori’s Cone Theorem (p. 25 in [5]), P(V ) has precisely two extremal rays R1 = R 0 A1 and R2 = R 0 A2 ≥ · ≥ · such that the cone NE(P(V )) of curves in P(V )isequaltoR1+R2 and A1 and A2 are the homology classes of two rational curves E1 and E2 in P(V )with0< K(V)(Ai) dim(P(V ))+1. Up to orders of A1 and A2,wehaveA1 =(hnξr 2) , − P ≤ − ∗ that is, A1 is represented by lines in the ﬁbers of π. It is also well-known that if r V = n (mi) with m1 ... mr,thenA2 =[hn 1ξr 1+(m1 c1)hnξr 2] , i=1 OP ≤ ≤ − − − − ∗ which is represented by a smooth rational curve in P(V ) isomorphic to a line in Pn L via π. However, in general, it is not easy to determine the homology class A2 and the extremal rational curves representing A2. Assume that r (2.1) V = (m ) |` O` i i=1 n M for generic lines ` P , where we let m1 ... mr.SinceVis ample, m1 1. ⊂ ≤ ≤ ≥ Lemma 2.2. Let A =[hn 1ξr 1+(m1 c1)hnξr 2] .Then, − − − − ∗ (i) A is represented by a smooth rational curve isomorphic to a line in Pn; (ii) A2 = A if and only if (ξ m1h) is nef; (iii) A = A if 2c (n +1);− 2 1 ≤ (iv) A cannot be represented by reducible or nonreduced curves if m1 =1. n Proof. (i) Let ` P be a generic line. Then we have a natural projection V ` = r (m ) ⊂ (m). By Proposition 7.12 in Chapter II of [7], this surjective| i=1 O` i →O` 1 map V ` `(m1) 0 induces a morphism g : ` P(V ). Then g(`) is isomorphic L | →O → → to ` via the projection π.Sinceh([g(`)]) = 1 and ξ([g(`)]) = m1,wehave

[g(`)] = [hn 1ξr 1 +(m1 c1)hnξr 2] =A. − − − − ∗ (ii) First of all, if A2 =[hn 1ξr 1+(m1 c1)hnξr 2] , then for any curve E,[E]= − − − − ∗ a(hnξr 2) +b[hn 1ξr 1+(m1 c1)hnξr 2] for some nonnegative numbers a and b; so (ξ −m h∗)([E])− = a − 0; therefore− (ξ −m ∗h) is nef. Conversely, if (ξ m h)isnef, − 1 ≥ − 1 − 1 then 0 (ξ m1h)([E]) = ac1 + b am1 where [E]=(ahn 1ξr 1 + bhnξr 2) for ≤ − − − − − ∗ some curve E;thus[E]=(ac1 +b am1)(hnξr 2) +a[hn 1ξr 1+(m1 c1)hnξr 2] ; − − ∗ − − − − ∗ it follows that A2 =[hn 1ξr 1+(m1 c1)hnξr 2] =A. − − − − ∗ (iii) Let A2 =(ahn 1ξr 1+bhnξr 2) .SinceA1=(hnξr 2) and a = h(A2) 0, a 1. If a>1, then− since− 2c (n−+1),weseethat∗ − ∗ ≥ ≥ 1 ≤ K (A )=(n+1 c )a+r ξ(A ) 2(n +1 c )+r − P(V) 2 − 1 · 2 ≥ − 1 >n+r= dim(P(V )) + 1;

but this contradicts with K (V )(A2) dim(P(V )) + 1. Thus a =1andA2 = − P ≤ (hn 1ξr 1+bhnξr 2) .Now[π(E2)] = π (A2)=(hn 1).Soπ(E2) is a line in n − − −r ∗ ∗ n− ∗ P .SinceV`= i=1 `(mi) for a generic line ` P , it follows that V π(E2) = r | O ⊂ | (m0 ), where m0 m for every i.Thus,ξ(A) m,andsoc +b i=1 Oπ(E2) i L i ≥ 1 2 ≥ 1 1 ≥ m1. It follows that L A2 =[hn 1ξr 1+(m1 c1)hnξr 2] +(c1 +b m1) (hnξr 2) . − − − − ∗ − · − ∗ Therefore, A2 =[hn 1ξr 1+(m1 c1)hnξr 2] =A. − − − − ∗ (iv) Since ξ(A)=m1 =1andξis ample, the conclusion follows.

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Next, let M(A, 0) be the moduli space of morphisms f : P1 P(V ) with [Im(f)] = A. In the lemma below, we study the morphisms in M→(A, 0) when A =[hn 1ξr 1+(m c1)hnξr 2] .Notethatξ(A)=m. − − − − ∗ Lemma 2.3. Let A =[hn 1ξr 1+(m c1)hnξr 2] . − − − − ∗ (i) If M(A, 0) = ,thenm m1 and M(A, 0) consists of embeddings f : ` 6 ∅ ≥ n→ P(V ) induced by surjective maps V ` `(m) 0where ` are lines in P ; | →O → (ii) If m = m1 and m1 = ... = mk

In this section, we shall compute some Gromov-Witten invariants of P(V ). First of all, we recall that for two homogeneous elements α and β in H∗(P(V ); Z), the quantum product α β H∗(P(V ); Z) can be written as · ∈ t K (A) (3.1) α β = (α β) e · P(V ) , · · A · A H ( (V ); ) ∈ 2XP Z

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where (α β) has degree deg(α)+deg(β)+2K (A) and is deﬁned by · A P(V) (α β)A(γ )=Φ(A,0)(α, β, γ) · ∗ for a homogeneous cohomology class γ H∗(P(V ); Z) with ∈ (3.2) deg(γ)= 2K (A)+2(n+r 1) deg(α) deg(β). − P(V) − − − Furthermore, for higher quantum products, we have t K (A) (3.3) α α ... α = (α α ... α ) e · P(V) , 1 · 2 · · k 1 · 2 · · k A · A H ( (V); ) ∈ 2XP Z where (α1 α2 ... αk)A is deﬁned as (α1 α2 ... αk)A(γ )=Φ(A,0)(α1,α2,... ,αk,γ). Thus, α ·α · ...· α =α α ...α + (lower· · order· terms),∗ where α α ...α stands 1 · 2 · · k 1 2 k 1 2 k for the ordinary cohomology product of α1,α2,... ,αk,andthedegreeofalower

order term is dropped by 2KP(V )(A)forsomeA H2(P(V); Z) which is represented by a nonconstant effective rational curve. ∈ There are two explanations for the Gromov-Witten invariant Φ(A,0)(α, β, γ)defined by the second author [12]. Recall that the Gromov-Witten invariant is only defined for a generic almost complex structure and that M(A, 0) is the moduli space of morphisms f : P1 P(V ) with [Im(f)] = A. Assume the genericity conditions: → 1 (i) M(A, 0)/P SL(2; C) is smooth in the sense that h (Nf ) = 0 for every f ∈ M(A, 0), where Nf is the normal bundle, and (ii) the homology class A is only represented by irreducible and reduced curves. Then the complex structure is already generic, and one can use algebraic geometry to calculate the Gromov-Witten invariants. Moreover, M(A, 0)/P SL(2; C)is compact with the expected complex dimension (3.4) K (A)+(n+r 1) 3. − P(V ) − − The first explanation for Φ(A,0)(α, β, γ)isthatwhenα, β, γ are classes of subvarieties B,C,D of P(V ) in general position, Φ(A,0)(α, β, γ) is the number of rational curves E in P(V ) such that [E]=Aand E intersects with B,C,D (counted with suitable multiplicity). The second explanation for Φ(A,0)(α, β, γ)isthat

Φ (α, β, γ)= e∗(α) e∗(β) e∗(γ), (A,0) 0 · 1 · 2 ZM(A,0) where the evaluation map ei : M(A, 0) P(V ) is defined by ei(f)=f(i). → 1 Assume that the genericity condition (i) is not satisfied but h (Nf ) is independent of f M(A, 0) and M(A, 0)/P SL(2; C) is smooth with dimension ∈ K (A)+(n+r 1) 3+h1(N ). − P(V ) − − f 1 Then one can form an obstruction bundle COB of rank h (Nf ) over the moduli space M(A, 0). Moreover, if the genericity condition (ii) is satisfied, then by Proposition 5.7 in [11], we have

(3.5) Φ (α, β, γ)= e∗(α) e∗(β) e∗(γ) e(COB) (A,0) 0 · 1 · 2 · ZM(A,0) where e(COB) stands for the Euler class of the bundle COB. We remark that in general, the cohomology class hiξj may not be representable by a subvariety of P(V ). However, since ξ is ample, sξ is very ample for s 0. Thus, the multiple thiξj with t 0 can be represented by a subvariety of P(V ) n whose image in P is a linear subspace of codimension i. Since Φ(A,0)(α, β, hiξj )=

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1/t Φ(A,0)(α, β, t hiξj )forαand β in H∗(P(V ); Z), it follows that to compute · · Φ(A,0)(α, β, hiξj ), it suﬃces to compute Φ(A,0)(α, β, t hiξj ). In the proofs below, we shall assume implicitly that t = 1 for simplicity. ·

Now we compute the Gromov-Witten invariant Φ((hnξr 2) ,0)(ξ,ξr 1,hnξr 1). − ∗ − −

Lemma 3.6. Φ((hnξr 2) ,0)(ξ,ξr 1,hnξr 1)=1. − ∗ − −

Proof. First of all, we notice that A =(hnξr 2) can only be represented by lines ` in the ﬁbers of π. In particular, there is no reducible− ∗ or nonreduced eﬀective curve representing A.Thus,M(A, 0)/P SL(2; C) is compact and has dimension n dim(P )+dimG(2,r)=n+2(r 2) = n +2r 4, − − which is the expected dimension by (3.4) (here we use G(2,r)tostandforthe r 1 Grassmannian of lines in P − ). Next, we want to show that M(A, 0)/P SL(2; C)is 1 smooth. Let p = π(`). Then from the two inclusions ` π− (p) P(V ), we obtain an exact sequence relating normal bundles: ⊂ ⊂

0 N` π 1(p) N` (V ) (Nπ 1(p) (V )) ` 0. → | − → |P → − |P | → (r 2) Since N 1 = N r 1 = (1) and N 1 =(π 1 )T n ,the ` π− (p) ` P − ` ⊕ − π− (p) P(V ) π− (p) ∗ p,P previous| exact sequence| is simplifiedO into the exact sequence| | (r 2) 0 `(1)⊕ − N` (V ) (π `)∗Tp, n 0. →O → |P → | P → 1 It follows that H (`, N` (V ))=0.Thus,M(A, 0)/P SL(2; C)issmooth. |P Finally, the Poincaré dual of hnξr 1 is represented by a point q0 P(V ). If a − 1 ∈ line ` M(A, 0) intersects q0,then` π− (π(q0)). Since the restriction of ξ to ∈ 1 r 1 ⊂ r 1 the fiber π− (π(q0)) ∼= P − is the cohomology class of a hyperplane in P − ,we conclude that Φ((hnξr 2) ,0)(ξ,ξr 1,hnξr 1)=1. − ∗ − − Next we show the vanishing of some Gromov-Witten invariant.

Lemma 3.7. Let A = b(hnξr 2) with b 1 and α H∗(P(V ); Z).Then − ∗ ≥ ∈

Φ(A,0)(hp1 ξq1 ,hp2ξq2,α)=0,

if p1,q1,p2,q2 are nonnegative integers with (q1 + q2)

Proof. We may assume that α is a homogeneous class in H∗(P(V ); Z). By (3.2), 1 deg(α)=(n+r 1) K (A) (p + p + q + q ) 2 · − − P(V ) − 1 2 1 2 =(n+r+br 1) (p + p + q + q ). − − 1 2 1 2

Let α = h(n+r+br 1) (p +p +q +q +q )ξq3 with 0 q3 (r 1). Let B,C,D be the − − 1 2 1 2 3 ≤ ≤ − subvarieties of P(V ) in general position, whose homology classes are Poincar´e dual

to hp1 ξq1 ,hp2ξq2,α respectively. Then the homology classes of π(B),π(C),π(D)in n P are Poincaré dual to hp1 ,hp2,h(n+r+br 1) (p1+p2+q1+q2+q3) respectively. Since (q +q +q ) < (2r 1), we have p +p +[(−n+−r+br 1) (p +p +q +q +q )] = 1 2 3 − 1 2 − − 1 2 1 2 3 (n + r + br 1) (q1 + q2 + q3) >n.Thus,π(B) π(C) π(D)= . Notice that the genericity conditions− − (i) and (ii) mentioned earlier∩ in this∩ section∅ are not satisfied for b 2. However, we observe that these conditions can be relaxed by assuming: ≥1 (i0) h (N ) = 0 for every f M(A, 0) such that Im(f) intersects B,C,D,and f ∈ (ii0) there is no reducible or nonreduced effective (connected) curve E such that [E]=Aand E intersects B,C,D.

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In fact, we will show that there is no eﬀective connected curve E at all representing A and intersecting B,C,D. This obviously implies (i0), (ii0)and

Φ(A,0)(hp1 ξq1 ,hp2ξq2,α)=0.

Suppose that E = aiEi is such an eﬀective connected curve, where ai > 0andEi is irreducible and reduced. Then, ai[Ei]=[E]=A. Since (hnξr 2) generates P − ∗ an extremal ray for P(V ), [Ei]=bi(hnξr 2) for 0

Finally, we show that if c1 < 2r and A =[hn 1ξr 1+(1 c1)hnξr 2] ,then − − − − ∗ n Φ(A,0)(h, hn,hnξ2r c1 1) = 1. Since c1 < 2r, we see that for a generic line ` P , − − ⊂ k V = (1)⊕ (m ) ... (m ), |` O` ⊕O` k+1 ⊕ ⊕O` r where k 1and2 m ... m . We remark that even though the moduli ≥ ≤ k+1 ≤ ≤ r space M(A, 0)/P SL(2; C) is compact by Lemma 2.2 (iv), it may not have the correct dimension by Lemma 2.3 (ii). The proof is lengthy, but the basic idea is that we shall determine the obstruction bundle and use the formula (3.5).

n Lemma 3.8. Let V be a rank-r ample vector bundle over P satisfying c1 < 2r and the assumption of Theorem B(i).IfA=[hn 1ξr 1+(1 c1)hnξr 2] ,then − − − − ∗ Φ(A,0)(h, hn,hnξ2r c 1)=1. − 1− Proof. Note that by Lemma 2.2 (iv), the moduli space M(A, 0)/P SL(2; C)iscom- pact. Let B,C,D be the subvarieties of P(V ) in general position, whose homology

classes are Poincaré dual to h, hn,hnξ2r c1 1 respectively. Then the homology n − − classes of π(B),π(C), π(D)inP are Poincaré dual to h, hn,hn respectively. Thus n π(C)andπ(D) are two different points in P .Let`0be the unique line passing k through π(C)andπ(D). Let V = (1)⊕ (m ) ... (m ), where |`0 O`0 ⊕O`0 k+1 ⊕ ⊕O`0 r 2 mk+1 ... mr.Sincec1<2r,k 1. Let f : ` P(V ) be a morphism in ≤ ≤ ≤ n ≥ → M(A, 0) for some line ` P .IfIm(f) intersects with B,C,andD,then`=`0. ∈ As in the proof of Lemma 2.3 (ii), the morphisms f : `0 P(V )inM(A, 0) are k 1 → parameterized by P(Hom(V `0 , `0 (1))) = P − ; moreover, Im(f) are of the form | O ∼ k 1 k (3.9) `0 q `0 P − =P( `0(1)⊕ ) P(V `0 ) P(V ), ×{ }⊂ × O ⊂ | ⊂ k 1 r 1 1 where q stands for points in P − P − = π− (π(D)). Note that `0 q always ⊂ ∼ ×{ } intersects with B and C,andthatDis a dimension-(c1 r) linear subspace in r 1 1 − P − ∼= π− (π(D)). Thus, `0 q intersects with B,C,D simultaneously if and only if ` q intersects with×{D, and} if only only if 0 ×{ } c +k 2r def k 1 r 1 1 (3.10) q P 1 − = P − D P − = π− (π(D)). ∈ ∩ ⊂ ∼ c1+k 2r It follows that M/P SL(2; C) ∼= P − ,whereMconsists of morphisms f M(A, 0) such that Im(f) intersects with B,C,D simultaneously. ∈ If c1 + k 2r =0,thena0 =Φ(A,0)(h, hn,hnξ2r c 1) = 1. But in general, we − − 1− have c1 + k 2r 0. We shall use (3.5) to compute a0 =Φ(A,0)(h, hn,hnξ2r c1 1). − ≥ −1 − Let Nf = N`0 q (V ) be the normal bundle of Im(f)=`0 q in P(V ). If h (Nf ) ×{ }|P ×{ }

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is constant for every f M, then by (3.5), Φ(A,0)(h, hn,hnξ2r c1 1) is the Euler number e(COB) of the∈ rank-(c + k 2r) obstruction bundle COB− − over 1 − c1+k 2r M/P SL(2; C) ∼= P − . Thus we need to show that h1(N ) is constant for every f M. f ∈ First, we study the normal bundle N` c1+k 2r (V ). The three inclusions 0×P − |P

c1+k 2r k 1 k (3.11) `0 P − `0 P − = P( `0 (1)⊕ ) P(V `0 ) P(V ) × ⊂ × O ⊂ | ⊂ give rise to two exact sequences relating normal bundles:

0 N c +k 2r N c +k 2r N 0, `0 1 − (V ` ) `0 1 − (V ) P(V ` ) P(V ) → ×P |P | 0 → ×P |P → | 0 | →

c +k 2r k c +k 2r k 0 N` 1 ( (1) ) N` 1 (V ) N ( ` (1) ) (V ` ) 0. → 0×P − |P O`0 ⊕ → 0×P − |P |`0 → P O 0 ⊕ |P | 0 → (n 1) n Notice that N (V ) (V ) =(π (V ))∗(N` )= `0(1)⊕ − and that P |`0 |P |P |`0 0|P O

(2r c1 1) N` c1+k 2r ( (1) k) = N` c1+k 2r ` k 1 = c1+k 2r (1)⊕ − − . 0×P − |P O`0 ⊕ 0×P − | 0×P − OP − k r Since V = (1) (m ), it follows that ξ k 1 = (1) `0 `0 ⊕ i=k+1 `0 i `0 − `0 | O ⊕ O | ×P O ⊗ k 1 (1) and OP − L r

N ( (1) k) (V ) = `0 ( mi) ξ ` k 1 P O`0 ⊕ |P |`0 O − ⊗ | 0×P − i=k+1 Mr

= ` (1 mi) k 1(1). O 0 − ⊗OP − i=Mk+1 Thus the previous two exact sequences are simpliﬁed to

(n 1) (3.12) 0 N` c1+k 2r (V ) N` c1+k 2r (V ) `0(1)⊕ − 0, → 0×P − |P |`0 → 0×P − |P →O →

(2r c1 1) 0 c1+k 2r(1)⊕ − − N` c1+k 2r (V ) →OP − → 0×P − |P |`0

(3.13) ` (1 mi) c1+k 2r(1) 0. → O 0 − ⊗OP − → i=Mk+1 Now (3.13) splits, since for k +1 i r we have m 2and ≤ ≤ i ≥ 1 Ext ( ` (1 mi) c1+k 2r(1), c1+k 2r (1)) O 0 − ⊗OP − OP − 1 c1+k 2r = H (`0 P − , `0 (mi 1)) = 0. × O −

Thus, the normal bundle N` c1+k 2r (V ) is isomorphic to 0×P − |P |`0 r (2r c1 1) ` (1 mi) c1+k 2r(1) c1+k 2r(1)⊕ − − , O 0 − ⊗OP − ⊕OP − i=Mk+1 and the exact sequence (3.12) becomes the exact sequence

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r (2r c1 1) 0 `0 (1 mi) c1+k 2r(1) c1+k 2r(1)⊕ − − (3.14) → O − ⊗OP − ⊕OP − i=Mk+1 (n 1) N` c1+k 2r (V ) `0(1)⊕ − 0. → 0×P − |P →O →

Restricting (3.14) to `0 q and taking the long exact cohomology sequence, we get ×{ }

r 1 H ( `0 (1 mi)) c1+k 2r(1) q (3.15) O − ⊗OP − | i=Mk+1 1 c +k 2r H ((N` 1 (V )) `0 q ) 0. → 0×P − |P | ×{ } → 1 Next, we determine Nf and show that h (Nf ) c1 + k 2r. The two inclusions c1+k 2r ≤ − `0 q `0 P − P(V) give an exact sequence ×{ }⊂ × ⊂

c +k 2r c +k 2r 0 N` q ` 1 N`0 q (V ) (N` 1 (V )) `0 q 0. → 0×{ }| 0×P − → ×{ }|P → 0×P − |P | ×{ } →

Since N` q ` c1+k 2r = Tq, c1+k 2r , the above exact sequence becomes 0×{ }| 0×P − P −

c +k 2r c +k 2r (3.16) 0 Tq, 1 Nf (N` 1 (V )) `0 q 0. → P − → → 0×P − |P | ×{ } → 1 1 c +k 2r Thus, h (Nf )=h ((N` 1 (V )) `0 q ). By (3.15), we obtain 0×P − |P | ×{ } r 1 1 1 c +k 2r h (Nf )=h ((N` 1 (V )) `0 q ) h ( `0 (1 mi)) 0×P − |P | ×{ } ≤ O − i=k+1 r X = (m 2) = c + k 2r. i − 1 − i=Xk+1 1 1 Finally, we show that h (Nf )=c1+k 2r. It suﬃces to prove that h (Nf ) n − k r ≥ c1 + k 2r.Since`0is a generic line in P and V `0 = `0 (1)⊕ i=k+1 `0 (mi), − | 0 O ⊕ O dimM(A, 0) = (2n + k) by Lemma 2.3 (ii). Since h (Nf ) is the dimension of the 0 L Zariski tangent space of M(A, 0)/P SL(2; C)atf,h(Nf) (2n + k 3). Thus, ≥ − h1(N )=h0(N ) χ(N ) (2n + k 3) (2n +2r c 3) = k + c 2r. f f − f ≥ − − − 1 − 1 − 1 1 Therefore, h (Nf )=c1+k 2r. In particular, h (Nf ) is independent of f M. − c +k 2r ∈ To obtain the obstruction bundle COB over P 1 − , we notice that (3.15) gives r 1 1 c +k 2r c +k 2r H ( `0 (1 mi)) 1 (1) q = H ((N` 1 (V )) `0 q ). O − ⊗OP − | ∼ 0×P − |P | ×{ } i=Mk+1 Thus by the exact sequence (3.16), we conclude that

1 1 c +k 2r H (Nf ) = H ((N` 1 (V )) `0 q ) ∼ 0×P − |P | ×{ } r 1 (3.17) = H ( ` (1 mi)) c1+k 2r(1) q. ∼ O 0 − ⊗OP − | i=Mk+1

(c1+k 2r) It follows that COB = c1+k 2r (1)⊕ − . By (3.5), we obtain OP −

a0 =Φ(A,0)(h, hn,hnξ2r c 1)=e(COB)=1. − 1−

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4. Proof of Theorem B In this section, we prove Theorem B, which we restate below.

n Theorem 4.1. (i) Let V be a rank-r ample bundle over P . Assume either c1 n, ≤ or c1 (n + r) and V n( 1) is nef, so that P(V ) is Fano. Then the quantum ≤ ⊗OP − cohomology Hω∗(P(V ); Z) is the ring generated by h and ξ with the two relations n+1 i j t(n+1 i j) (4.2) h = a h ξ e− − − , i,j · · · i+j (c r) ≤X1− r i i r i tr i j t(r i j) (4.3) ( 1) c h ξ − = e− + b h ξ e− − − , − i · · i,j · · · i=0 i+j (c n 1) X ≤ X1− − where the coefficients ai,j and bi,j are integers depending on V . (ii) If we further assume that c1 < 2r, then the leading coefficient a0,c r =1. 1− 1 Proof. (i) First, we determine the first relation fω in (1.3). By Lemma 3.7, tK (A) (4.4) h h = h + (h h ) e P(V ) , · p p+1 · p A · A H X∈ 20 where p 1andH20 stands for H2(P(V ); Z) Z (hnξr 2) .Thus, ≥ − · − ∗ n p n p+1 n p tK (A) h− h = h − h h − (h h ) e P(V ) . · p+1 · p − · · p A · A H X∈ 20 If (h h ) =0,thenA=[E] for some effective curve E.Soa=h(A) 0. · p A 6 ≥ Since A H20 , a 1. We claim that KP(V )(A) (n +1 c1 +r), with equality ∈ ≥ − ≥def − if and only if A =[hn 1ξr 1+(1 c1)hnξr 2] = A2. Indeed, if c1 n,then K (A)=(n+1 −c )a−+r ξ(A−) (n+1− ∗c +r), with equality if≤ and only − P(V) − 1 · ≥ − 1 if a = ξ(A) = 1, that is, if and only if A = A2;ifc1 (n+r)and(ξ h)isnef, then again K (A)=(n+1+r c )a+r (ξ h≤)(A) (n +1 c−+r), with − P(V ) − 1 · − ≥ − 1 equality if and only if a =1and(ξ h)(A) = 0, that is, if and only if A = A2.Thus, − n p deg((h hp)A)=1+p+KP(V)(A) (p n+c1 r), and deg(h − (h hp)A) (c1 r). · ≤ − − tK· ·(A) ≤ − Using induction on p and keeping track of the exponential e P(V ) ,weobtain

n+1 i j t(n+1 i j) 0=h = h a h ξ e− − − . n+1 − i,j · · · i+j (c r) ≤X1− 1 Therefore, the ﬁrst relation fω for the quantum cohomology ring is n+1 i j t(n+1 i j) h = a h ξ e− − − . i,j · · · i+j (c r) ≤X1− 2 Next, we determine the second relation fω in (1.3). We need to compute the i r i quantum product h ξ − for 0 i r. First, we calculate the quantum product r · ≤ ≤ ξ .NotethatifA=(bhnξr 2) with b 1, then K (V )(A)=br r, with − ∗ P ≥ def − ≥ K (V )(A)=rif and only if A =(hnξr 2) = A1.Thus,forp 1, − P − ∗ ≥ tK (A) ξ + (ξ ξ ) e P(V ) , if p

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Note that (ξ ξr 1)A is of degree zero; by Lemma 3.6, we obtain (ξ ξr 1)A = · − 1 · − 1 Φ(A ,0)(ξ,ξr 1,hnξr 1) = 1. Therefore for p 1, 1 − − ≥ tK (A) ξ + (ξ ξ ) e P(V ) , if p

when i + j = r,(hiξj)A1 is of degreeP zero; by Lemma 3.7, we have (hi ξj)A1 = · Φ(A ,0)(hi,ξr i,hnξr 1) = 0. Therefore for i 1andj 1 with i + j r, 1 − − ≥ ≥ ≤ tK (A) (4.6) h ξ = h ξ + (h ξ ) e P(V ) . i · j i j i · j A · A H X∈ 20 1 From the proof of the ﬁrst relation fω,weseethatifαand β are homogeneous elements in H∗(P(V ); Z) with deg(α)+deg(β)=m r,thendeg((α β)A) m ≤ · ≤ − (n+1 c1+r)forA H20.Thusifγis a homogeneous element in H∗(P(V ); Z) with − ∈ r i deg(γ)=r m,thendeg(γ (ξ ξp)A) (c1 n 1). Since i=0( 1) ci hiξr i =0, it follows from− (4.4), (4.5),· and· (4.6)≤ that the− − second relation f 2−is · − P ω r i i r i tr i j t(r i j) ( 1) c h ξ − = e− + b h ξ e− − − . − i · · i,j · · · i=0 i+j (c n 1) X ≤ X1− − (ii) From the proof of the ﬁrst relation in (i), we see that K (A) (n +1 c +r) − P(V ) ≥ − 1 c1 r with equality if and only if A = A2; moreover, the term ξ − can only come from the quantum correction (h h ) .Now · n A2 c r 1− t(n+1 c1+r) (h hn)A2 =( ai0 hiξc1 r i) e− − , · − − · i=0 X where a0 =Φ(A2,0)(h, hn,hnξ2r c1 1). Since c1 < 2r,(c1 r)

It is understood that when c1 n, then the summations on the right-hand sides of the second relations (4.3) and≤ (4.9) below do not exist. Next, we shall sharpen the results in Theorem 4.1 by imposing additional con- n ditions on V .LetVbe a rank-r ample vector bundle over P .Thenc1 r.Thus ≥ if c1 < 2r and if either 2c1 (n + r)or2c1 (n+2r)andV Pn( 1) is nef, then the conditions in Theorem≤ 4.1 are satisﬁed.≤ ⊗O −

n Corollary 4.7. (i) Let V be a rank-r ample vector bundle over P with c1 < 2r. Assume that either 2c (n + r),or2c (n+2r) and V n( 1) is nef, so 1 ≤ 1 ≤ ⊗OP − that P(V ) is a Fano variety. Then the ﬁrst relation (4.2) is c r 1− n+1 i c r i t(n+1+r c ) (4.8) h = a h ξ 1− − e− − 1 , i · · · i=0 ! X where the integers ai depend on V . Moreover, a0 =1.

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n (ii) Let V be a rank-r ample vector bundle over P . Assume that 2c1 (2n+r+1) ≤ and V n( 1) is nef, so that P(V ) is Fano. Then the second relation (4.3) is ⊗OP − r c n 1 1− − i i r i tr i c n 1 i t(n+1+r c ) (4.9) ( 1) c h ξ − = e− + b h ξ 1− − − e− − 1 , − i · · i · · · i=0 i=0 X X where the integers bi depend on V . Proof. (i) From the proof of Theorem 4.1 (i), we notice that it suﬃces to show that

the only homology class A H20 = H2(P(V ); Z) Z (hnξr 2) which has nonzero contributions to the quantum∈ corrections in (4.4)− is · − ∗ def A =[hn 1ξr 1+(1 c1)hnξr 2] = A2. − − − − ∗ In other words, if A =(ahn 1ξr 1 +bhnξr 2) with a =0andifΦ(A,0)(h, hp,α)=0 − − − ∗ 6 6 for 1 p n and α H∗(P(V ); Z), then A = A2. First of all, we show that a =1. Suppose≤ a≤=1.Then∈a 2. By (3.2), 6 ≥ 1 deg(α)=(n+r 1) K (A) 1 p 2 · − − P(V ) − − =(n+r 1) + [(n +1 c )a+r ξ(A)] 1 p − − 1 · − − dim(P(V )) + [(n +1 c1)a+r ξ(A)] 1 n. ≥ − · − − If 2c (n + r), then c n,and 1 ≤ 1 ≤ [(n +1 c )a+r ξ(A)] 1 n 2(n +1 c )+r 1 n>0. − 1 · − − ≥ − 1 − − If 2c (n +2r)and(ξ h)isnef,thenc n+r,and 1 ≤ − 1≤ [(n +1 c )a+r ξ(A)] 1 n =[(n+1+r c )a+r (ξ h)(A)] 1 n − 1 · − − − 1 · − − − 2(n +1+r c ) 1 n>0. ≥ − 1 − − Thus, [(n +1 c1)a+r ξ(A)] 1 n>0, and so deg(α)/2 > dim(P(V )). But this − · − − is absurd. Next, we prove that b =(1 c1), or equivalently, ξ(A) = 1. Suppose ξ(A) =1.Thenξ(A) 2. By (3.2), − 6 ≥ 1 deg(α)=(n+r 1) + [(n +1 c )+r ξ(A)] 1 p 2 · − − 1 · − − dim(P(V )) + [(n +1 c1)+2r] 1 n ≥ − − − >dim(P(V )),

since c1 < 2r. But once again this is absurd. (ii) We follow the previous arguments for (i). Again it suﬃces to show that if A =(ahn 1ξr 1 + bhnξr 2) with a =0andifΦ(A,0)(α1,α2,α) =0forsome − − − ∗ 6 6 α1,α2,α H∗(P(V); Z) with deg(α1)+deg(α2) r,thenA=A2. Indeed, if a =1 ∈ ≤ 6 or if a = 1 but ξ(A) =1,thenwemusthavedeg(α)/2>dim(P(V )). But this is 6 impossible. Therefore, a =1andξ(A)=1.SoA=A2. Now we discuss the relation between the quantum corrections and the extremal rays of the Fano variety P(V ). Let V be a rank-r ample vector bundle over Pn with c < 2r and 2c (n + r). By (4.8) and (4.3), the quantum cohomology ring 1 1 ≤ Hω∗(P(V ); Z) is the ring generated by h and ξ with the two relations c r 1− n+1 i c r i t(n+1+r c ) (4.10) h = a h ξ 1− − e− − 1 , i · · · i=0 ! X

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r i i r i tr (4.11) ( 1) c h ξ − = e− . − i · · i=0 X From the proof of Theorem 4.1 (i), we notice that the quantum correction to the second relation (4.11) comes from the homology class A1 =(hnξr 2) which is − ∗ represented by the lines in the fibers of π : P(V ) Pn. Also, we notice from the proof of Corollary 4.7 (i) that the quantum correction→ to the first relation (4.10) comes from the homology class A2 =[hn 1ξr 1+(1 c1)hnξr 2] ; from the proof − − − − ∗ of Lemma 3.8, A2 can be represented by a smooth rational curve isomorphic to n lines in P via π.NowA1generates one of the two extremal rays of P(V ). It is unclear whether A2 generates the other extremal ray. By Lemma 2.2 (iii), if we further assume that 2c (n + 1), then indeed A generates the other extremal ray 1 ≤ 2 of P(V ). By Lemma 2.2 (ii), A2 generates the other extremal ray of P(V )ifand n only if (ξ h)isnef,thatis,V n( 1) is a nef vector bundle over P . − ⊗OP − 5. Direct sum of line bundles over Pn In this section, we partially verify Batyrev’s conjecture on the quantum cohomology of projective bundles associated to direct sum of line bundles over Pn.We shall use (3.5) to compute the necessary Gromov-Witten invariants. Our first step is to recall some standard materials for the Grassmannian G(2,n+ 1) from [3]. Then we determine a certain obstruction bundle and its Euler class. Finally we proceed to determine the first and second relations for the quantum cohomology. On the Grassmannian G(2,n+ 1), there exists a tautological exact sequence (n+1) (5.1) 0 S ( )⊕ Q 0, → → OG(2,n+1) → → where the sub- and quotient bundles S and Q are of rank 2 and (n 1) respectively. Let α and β be the virtual classes such that α + β = c (S)andαβ− = c (S). Then − 1 2 αp βp (5.2) cl( ` G(2,n+1)` h = )= − , { ∈ | ∩ p 6 ∅} α β − where cl( ) denotes the fundamental class and h stands for a fixed linear subspace · p of Pn of codimension p.IfP(α, β) is a symmetric homogeneous polynomial of degree (2n 2) (so that P (α, β) can be written as a polynomial of maximal degree in the Chern− classes of the bundle S), then we have 1 (5.3) P (α, β)= the coefficient of αnβn in (α β)2P (α, β) . − 2 − ZG(2,n+1) n Let Fn = (x, `) P G(2,n+1)x ` ,andπ1 and π2 be the two natural { ∈ ×n | ∈ } projections from Fn to P and G(2,n+ 1) respectively. Then Fn = P(S∗), where n S∗ is the dual bundle of S,and(π1∗ P (1)) Fn is the tautological line bundle over m O | F .LetSym (S∗)bethem-th symmetric product of S∗.Thenform 0, n ≥ m n (5.4) π2 (π1∗ (m) Fn ) = Sym (S∗). ∗ OP | ∼ By the duality theorem for higher direct image sheaves (see p. 253 in [7]), 1 n n R π2 (π1∗ ( m) Fn ) = (π2 (π1∗ (m 2) Fn ))∗ (detS∗)∗ ∗ OP − | ∼ ∗ OP − | ⊗ m 2 (5.5) = Sym − (S) (detS). ∼ ⊗ r Now, let V = n (m ), where 1 = m = ... = m

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generated by the two classes A1 =(hnξr 2) and A2 =[hn 1ξr 1+(1 c1)hnξr 2] . From the proof of Lemma 2.3 (ii), we see− that∗ − − − − ∗

k 1 (5.6) M(A2, 0)/P SL(2; C)=G(2,n+1) P − . × Let a morphism f M(A2, 0) be induced by some surjective map V ` `(1) 0 such that the image∈ Im(f)offis of the form | →O →

k 1 n k 1 Im(f)=` q ` P − P P − . ×{ }⊂ × ⊂ × Then by arguments similar to the proof of (3.17), we have

r 1 1 (5.7) H (Nf ) = H ( `(1 mu)) k 1(1) q. ∼ O − ⊗OP − | u=Mk+1 It follows that the obstruction bundle COB over M(A2, 0)/P SL(2; C)is r 1 n (5.8) COB = R π2 (π1∗ (1 mu) Fn ) k 1(1). ∼ ∗ OP − | ⊗OP − u=Mk+1 Since c (S)= (α+β)andc(S)=αβ, we obtain from (5.5) the following. 1 − 2 Lemma 5.9. The Euler class of the obstruction bundle COB is

r m 3 u− (5.10) e(COB)= [(1 + v)( α)+(m 2 v)( β)+h˜], − u − − − v=0 u=Yk+1 Y ˜ k 1 where h stands for the hyperplane class in P − .

Next assuming c1 < 2r, we shall compute the Gromov-Witten invariant

def (5.11) Wi =Φ(A ,0)(hn˜ ,hn+1 n˜ ,hn iξ2r c 1+i), 2 − − − 1− where 0 i (c r)and˜n= n+1 is the largest integer (n +1)/2. ≤ ≤ 1 − 2 ≤ Lemma 5.12. Assume c1 < min(2r,(n +1+2r)/2) and 0 i (c1 r).Then i ≤ ≤ − Wiis the coeﬃcient of t in the power series expansion of r m 2 (1 m t) u− . − u u=1 Y n k 1 k ˜ Proof. Note that the restriction of ξ to P P − = P( n (1)⊕ )is(h+h). Thus, × OP 2r c1 1+i − − 2r c1 1+i ˜ hn iξ2r c1 1+i n k 1 = − − hn i+jh2r c1 1+i j − − − |P ×P − j − − − − j=0 X i 2r c1 1+i ˜ = − − hn i+jh2r c 1+i j . j − − 1− − j=0 X So by (3.5) (replacing M(A2, 0) by M(A2, 0)/P SL(2; C)), (5.2), and Lemma 5.9, ˜ (5.13) Wi = P(α, β), G(2,n+1) k 1 Z ×P −

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where P˜(α, β) is the symmetric homogeneous polynomial of degree (2n 2)+(k 1): − − n˜ n˜ n+1 n˜ n+1 n˜ α β α − β − P˜(α, β)= − − α β · α β − − i n i+j n i+j 2r c1 1+i α − β − ˜ − − − h2r c 1+i j · j α β · − 1− − j=0 X − r m 3 u− [(1 + v)( α)+(m 2 v)( β)+h˜] · − u − − − v=0 u=Yk+1 Y i n+1 n+1 n˜ n˜ n˜ n+1 n˜ n+1 2r c 1+i α α − β α β − + β = − 1 − − − j (α β)2 j=0 X − n i+j 1 − − t n i+j 1 t ˜ α β − − − h2r c 1+i j · · − 1− − t=0 X r m 3 u− [(1 + v)( α)+(m 2 v)( β)+h˜]. · − u − − − v=0 u=Yk+1 Y By (5.3) and (5.13), we conclude from straightforward manipulations that

i 2r c1 1+i i j W = − − ( 1) − i j · − j=0 X r m 2 u ju − (mu 1) · ju − jk+1+...+jr =i j u=k+1 X − Y i 2r c 1+i = − 1 − ( 1)j i j · − j=0 X − r mu 2 − (m 1)ju . · j u − j +...+j =j u k+1 X r u=Yk+1 i Thus Wi is the coeﬃcient of t in the polynomial r 2r c 1+i m 2 (1 + t) − 1− [1 (m 1)t] u− · − u − u=k+1 rY 2r c 1+i m 2 =(1+t) − 1− [(1 + t) m t] u− · − u u=Yk+1 c 2r+k 1− 2r c 1+i =(1+t) − 1− · j=0 j +...+j =j X k+1 X r r m 2 u ju mu 2 ju − ( mut) (1 + t) − − · ju − · u=Yk+1 c 2r+k r 1− mu 2 j i+k 1 j = − ( m t) u (1 + t) − − j − u · j=0 j +...+j =j u X k+1 X r u=Yk+1

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since r (m 2 j )=c 2r+k j.SoW is the coeﬃcient of ti in u=k+1 u − − u 1− − i r + r P ∞ m 2 j + k 1 j m 2 1 (1 m t) u− − t = (1 m t) u− − u · k 1 − u · (1 t)k j=0 u=Yk+1 X − u=Yk+1 − r m 2 = (1 m t) u− . − u u=1 Y

r Proposition 5.14. Let V = n (m ),wherem 1for each i and i=1 OP i i≥ r L mi < min(2r, (n +1+2r)/2). i=1 X 1 Then the ﬁrst relation fω for the quantum cohomology ring Hω∗(P(V ); Z) is r n+1 m 1 t(n+1+r r m ) (5.15) h = (ξ m h) u− e− − i=1 i . − u · u=1 P Y Proof. We may assume that 1 = m1 = ... = mk

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where the right-hand side stands for the product in the ordinary cohomology. Thus r we need to show that (hn˜ hn+1 n˜ )A = (ξ muh)m 1,orequivalently, · − 2 u=1 − u− Q r (5.16) Φ(A ,0)(hn˜ ,hn+1 n˜ ,hn iξ2r c 1+i)= (ξ muh)m 1hn iξ2r c 1+i 2 − − − 1− − u− − − 1− u=1 Y for 0 i (c1 i). The left-hand side of (5.16) is computed in Lemma 5.12. ≤ ≤ − ˜ Denote the right-hand side of (5.16) by Wi.Letsibe the i-th Segre class of V . i r ju Then we have si =( 1) m and − · j1+...+jr =i u=1 u + r P∞ Q 1 (5.17) ( 1)is ti = . − i 1 m t i=0 u=1 u X Y − Moreover, from the second relation in (1.1) we obtain, for i r, ≥ i (r 1) ξi =( 1) − − si (r 1)ξr 1 + (terms with exponentials of ξ less than (r 1)). − − − − − ˜ It follows from the right-hand-side of (5.16) that Wi is equal to

c1 r r − mu 1 − ξm 1 j ( muh)j hn iξ2r c 1+i j u− − u − u − − 1− j=0 j +...+j =j u=1 u X 1 X r Y i r m 1 u ju = − ( mu) hn i+jξr 1+i j j − − − − j=0 j +...+j =j u=1 u X 1 X r Y i r m 1 i j u ju = ( 1) − si j − ( mu) . − − j − j=0 j +...+j =j u=1 u X 1 X r Y + ˜ i Therefore, the formal power series i=0∞ Wit is equal to + i r ∞ P m 1 i j i j u ju ( 1) − si jt − − ( mut) − − j − i=0 j=0 j +...+j =j u=1 u X X 1 X r Y + + r ∞ ∞ m 1 i j i j u ju = ( 1) − si jt − − ( mut) − − j − j=0 i=j j +...+j =j u=1 u X X 1 X r Y + + r ∞ ∞ mu 1 = ( 1)is ti − ( m t)ju − i j − u j=0 i=0 j +...+j =j u=1 u X X 1 X r Y + r r ∞ 1 mu 1 = − ( m t)ju 1 m t j − u j=0 u=1 u j +...+j =j u=1 u X Y − 1 X r Y r + r 1 ∞ mu 1 = − ( m t)ju 1 m t j − u u=1 u j=0 j +...+j =j u=1 u Y − X 1 X r Y r r 1 m 1 = (1 m t) u− 1 m t − u u=1 − u u=1 Yr Y m 2 = (1 m t) u− , − u u=1 Y

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˜ where we have applied (5.17) in the third equality. By Lemma 5.12, Wi = Wi for 0 i (c r). Hence the formulae (5.16) and (5.15) hold. ≤ ≤ 1 − It turns out that under certain conditions on the integers mi, the second relation 2 fω for the quantum cohomology ring Hω∗(P(V ); Z) is much easier to determine. Note that the second relation f 2 in (1.1) can be rewritten as r (5.18) (ξ m h)=0, − i i=1 Y where the left-hand side stands for the product in the ordinary cohomology ring. r Proposition 5.19. Let V = i=1 Pn (mi),wheremi 1for each i, mi =1for r O ≥ 2 some i,and mi < (2n+2+r)/2. Then the second relation f for the quantum i=1 L ω cohomology ring Hω∗(P(V ); Z) is P r tr (5.20) (ξ m h)=e− , − i i=1 Y where the left-hand side stands for the product in the quantum cohomology ring.

Proof. We may assume that 1 = m1 = ... = mk

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Then the quantum cohomology Hω∗(P(V ); Z) is generated by h and ξ with relations r r n+1 m 1 t(n+1+r r m ) tr h = (ξ m h) i− e− − i=1 i and (ξ m h)=e− . − i · − i i=1 P i=1 Y Y Proof. Follows immediately from Propositions 5.14 and 5.19.

6. Examples

In this section, we shall determine the quantum cohomology of P(V ) for am- n ple bundles V over P with 2 r n and c1 = r + 1. In these cases, V ` = (r 1) ≤ ≤ n | `(1)⊕ − `(2) for every line ` P . In particular, V is a uniform bundle. If O ⊕O ⊂ (r 1) r

Hω∗(P(TPn ); Z) is the ring generated by h and ξ with the two relations n n+1 tn i i n i tn (6.1) h =(a h+ξ) e− and ( 1) c h ξ − =(1+b ) e− . 1 · − i · · 0 · i=0 X More precisely, putting H20 = H2(P(V ); Z) Z (hnξn 2) , we see from the proof − · − ∗ of Corollary 4.7 (i) that the only homology class A H0 which has nonzero contri- ∈ 2 butions to the quantum corrections in (4.4) is A = A2. Thus by (4.4),

hp+1, if p n 2, tn ≤ − (6.2) h hp = hn + a10 e− , if p = n 1, · · tn − hn+1 +(a0h+a0ξ) e− , if p = n, 2 3 ·

where a10 =Φ(A2,0)(h, hn1,hnξn 1), a30 =Φ(A2,0)(h, hn,hnξn 2), and − − −

a20 =Φ(A2,0)(h, hn,hn 1ξn 1) c1a30 . − − − 1 By Lemma 3.8, a30 =1.Thusa1 =(a10 +a20)andtheﬁrstrelationfω in (6.1) is n+1 tn (6.3) h =((a0 +a0)h+ξ) e− . 1 2 · Similarly, from the proof of Corollary 4.7 (ii), we see that the only homology class

A H20 which has nonzero contributions to the quantum corrections in (4.5) and ∈ tn (4.6) is also A = A2. By (4.5), ξ ξp = ξp+1 if p

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(0) tn By (6.2), we have h hp = hp+1 if p 0, max(s, t) > 0,and(j+k+s+t)=n.Then,

Φ(A,0)(α, β, hnξn 1)=1. 2 −

Proof. (i) By Lemma 2.2 (iv), M(A2, 0)/P SL(2; C) is compact. By (3.17), we have 1 h (Nf ) = 0 for every f M(A2, 0). Thus, M(A2, 0)/P SL(2; C)isalsosmooth. n ∈ n n Fix a line `0 in P .Letg:`0 P(TP `0) P(TP) be the embedding induced → | (⊂n 1) by the natural projection T n = (1)⊕ − (2) (2) 0. Since P |`0 O`0 ⊕O`0 →O`0 → h([g(`0)]) = 1 and ξ([g(`0)]) = 2, we have [g(`0)] = [hn 1ξn 1 (n 1)hnξn 2] . − − − − − ∗ So hn 1ξn 1 =[g(`0)] +(n 1)hnξn 2,and − − ∗ − − Φ(A ,0)(h, hn,hn 1ξn 1)=Φ(A ,0)(h, hn, [g(`0)] )+(n 1)Φ(A ,0)(h, hn,hnξn 2). 2 − − 2 ∗ − 2 − By Lemma 3.8, it suffices to show that Φ(A ,0)(h, hn, [g(`0)] )=1.LetBand 2 ∗ C be the subvarieties of P(TPn ) in general position, whose homology classes are Poincaré dual to h and hn respectively. Then the homology classes of π(B)and n π(C)inP are Poincaré dual to h and hn respectively. Let f : ` P(T n )be → P a morphism in M(A2, 0) induced by a surjective map TPn ` `(1) 0for n | →O → some line ` P . If the image Im(f) intersects with B,C,andg(`0), then ` in- ⊂ tersects with π(B), π(C), and π(g(`0)) = `0. In other words, ` passes through the point π(C) and intersects with ` . Moreover, putting p = ` ` and noticing 0 ∩ 0 that every surjective map TPn ` `(1) 0 factors through the natural projection (n 1) | →O(n 1)→ TPn ` = `(1) − `(2) `(1) − , we conclude that the (n 1)-dimensional | O (n ⊕O1) →O − subspace ( (1) − ) in (T n ) = T n must contain the 1-dimensional sub- O` |p P |` |p p,P n n space ( `0 (2)) p in (TP `0 ) p = Tp,P . Conversely, let p `0 and let `p be the uniqueO line connecting| the| | two points π(C)andp.Ifthe(∈ n 1)-dimensional (n 1) − subspace ( (1) − ) in (T n ) = T n contains the 1-dimensional sub- O`p |p P |`p |p p,P space ( (2)) in (T n ) = T n , then there exists a unique surjective map O`0 |p P |`0 |p p,P n n TP `p `p(1) 0 such that the image of the induced morphism f : `p P(TP ) intersects| →Og(` )atthepoint→ g(p). Since there exists a unique point p ` →such that 0 ∈ 0

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(n 1) the (n 1)-dimensional subspace ( (1) − ) in (T n ) = T n contains the − O`p |p P |`p |p p,P 1-dimensional subspace ( (2)) in (T n ) = T n , it follows that O`0 |p P |`0 |p p,P Φ(A ,0)(h, hn, [g(`0)] )=1. 2 ∗ (ii) It is well-known (see p. 176 of [7]) that there is an exact sequence (n+1) (6.9) 0 n n(1)⊕ T n 0. →OP →OP → P → (n+1) The surjective map Pn (1)⊕ TPn 0 induces the inclusion φ : P(TPn ) n n O → → n n ⊂ P P such that ξ is the restriction of the (1, 1) class in P P .LetB,C,q0 be × × the subvarieties of P(TPn ) in general position, whose homology classes are Poincaré n dual to α, β, hnξn 1 respectively. Then q0 is a point. Put p0 = π(q0) P .Now − ∈ the morphisms in M(A2, 0) are of the forms f : ` P(TPn ) induced by surjective n → maps T n ` `(1) 0 for lines ` P . If the image Im(f) passes q0, then the P | →O → ⊂ line ` passes p0 and q0 is contained in the hyperplane n 2 (n 1) 1 n 1 n P − = P(( `(1)⊕ − ) p0 ) P((T `) p0 )=π− (p0)=P − . O | ⊂ P | | Conversely, if ` passes p0 and q0 is contained in the hyperplane n 2 (n 1) 1 n 1 n (6.10) P − = P(( `(1)⊕ − ) p0 ) P((T `) p0 )=π− (p0)=P − , O | ⊂ P | | then there exists a unique f M(A2, 0) of the form f : ` P(TPn ) such that ∈ n n → Im(f) passes q0; moreover, putting q0 =(p0,p0) P P such that π is the first n n ∈ n× n projection of P P ,thenIm(f)=` p0 P P . The set of all lines ` × ×{ 0}⊂ × passing p0 such that q0 is contained in the hyperplane (6.10) is parameterized by n 2 n n n an (n 2)-dimensional linear subspace P − in P (the first factor in P P ). It − × follows that the images Im(f) P(T n ) sweep a hyperplane ⊂ P def n 1 n (6.11) H = P − p0 P p0 . ×{ 0}⊂ ×{ 0} n n Since ξ is the restriction of the (1, 1) class in P P , ξ H is the hyperplane ˜ n 1 n 1 ×˜ | ˜ class h in H = P − p0 ∼=P − .ThusαH=hj+kand β H = hs+t.Since (j+k+s+t)=nand×{B and} C are in general| position, there is| a unique line in

H passing q0 =(p0,p0) and intersecting with B and C. Therefore,

Φ(A ,0)(α, β, hnξn 1)=1. 2 −

Finally, we summarize the above computations and prove the following.

Proposition 6.12. The quantum cohomology ring Hω∗(P(TPn ); Z) with n 2 is the ring generated by h and ξ with the two relations ≥ n n+1 tn i i n i n tn h = ξ e− and ( 1) c h ξ − =(1+( 1) ) e− . · − i · · − · i=0 X Proof. By Lemma 6.8 (ii), a10 = 1. By Lemma 3.8, a30 = 1. By Lemma 6.8 (i),

a20 =Φ(A2,0)(h, hn,hn 1ξn 1) c1a30 = 1. − − − − 1 n+1 tn (i) Thus by (6.3), the ﬁrst relation fω is h = ξ e− . By Lemma 6.8 (ii), b2 =1 · 2 n i i n i for 0 i n. By (6.7), the second relation fω is i=0( 1) ci h ξ − = ≤n ≤ i tn −n+1 · · (1 + i=0( 1) ci) e− . From the exact sequence (6.9), ci = i for 0 i n. − · 2 n i i n i P n tn ≤ ≤ Therefore, the relation f is ( 1) c h ξ − =(1+( 1) ) e− . P ω i=0 − i · · − · P

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Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078 E-mail address: [email protected] Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 E-mail address: [email protected]

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