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Journal ArticleDOI

Quantum cohomology of projective bundles over $\mathbb P^n$

01 Jan 1998-Transactions of the American Mathematical Society (American Mathematical Society, AMS)-Vol. 350, Iss: 9, pp 3615-3638
About: This article is published in Transactions of the American Mathematical Society.The article was published on 1998-01-01 and is currently open access. It has received 29 citations till now. The article focuses on the topics: Equivariant cohomology & Group cohomology.

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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 P
n
ZHENBO QIN AND YONGBIN RUAN
Abstract. In this paper we study the quantum cohomology ring of certain
projective bundles over the complex projective space P
n
. Using excessive inter-
section 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 P
n
is partially verified. Moreover, rela-
tions 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 mathe-
matical 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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3616 ZHENBO QIN AND YONGBIN RUAN
Although we have a solid foundation for the quantum cohomology theory at
least for semi-positive symplectic manifolds, the calculation has remained a diffi-
cult 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 in-
tersections [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 ra-
tional 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 P
n
. 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-
erated by α
1
,...
s
with the relations f
1
,... ,f
t
, then the quantum cohomology
H
ω
(X; Z)ofXis the ring generated by α
1
,...
s
with t new relations f
1
ω
,... ,f
t
ω
,
where each new relation f
i
ω
is just the relation f
i
evaluated in the quantum co-
homology 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 P
n
,andP(V) be the corresponding projective
bundle. Let h and ξ be the cohomology classes of a hyperplane in P
n
and the
tautological line bundle in P(V ) respectively. For simplicity, we make no distinction
between h and π
h,whereπ:P(V)P
n
is the natural projection. Denote the
product of i copies of h and j copies of ξ in the ordinary cohomology ring by h
i
ξ
j
,
and the product of i copies of h and j copies of ξ in the quantum cohomology ring
by h
i
· ξ
j
.Fori=0,... ,r, put c
i
(V )=c
i
·h
i
for some integer c
i
.Itiswellknown
that K
P(V )
=(n+1c
1
)h+, and the ordinary cohomology ring H
(P(V ); Z)
is the ring generated by h and ξ with the two relations
h
n+1
=0 and
r
X
i=0
(1)
i
c
i
· h
i
ξ
ri
=0.(1.1)
In particular, H
2(n+r2)
(P(V ); Z) is generated by h
n1
ξ
r1
and h
n
ξ
r2
, and its
Poincae dual H
2
(P(V ); Z) is generated by (h
n1
ξ
r1
)
and (h
n
ξ
r2
)
,wherefor
αH
(P(V); Z), α
stands for its Poincar´e dual. We have
K
P(V )
(A)=a(n+1c
1
)+r·ξ(A)=a(n+1c
1
)+r(ac
1
+ b)(1.2)
for A =(ah
n1
ξ
r1
+ bh
n
ξ
r2
)
H
2
(P(V ); Z).
By definition, V is an ample (respectively, nef) bundle if and only if the tau-
tological class ξ is an ample (respectively, nef) divisor on P(V ). Assume that V
is ample such that either c
1
(n +1), or c
1
(n+r)andV⊗O
P
n
(1) is nef.
Then both ξ and K
P(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
ω on P(V ) to be the Kahler form ω such that [ω]=K
P(V)
.Letf
1
ω
and f
2
ω
be
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QUANTUM COHOMOLOGY OF PROJECTIVE BUNDLES OVER P
n
3617
the two relations in (1.1) evaluated in the quantum cohomology ring H
ω
(P(V ); Z).
Then, by Theorem 2.2 in [14], the quantum cohomology H
ω
(P(V ); Z) is the ring
generated by h and ξ with the two relations f
1
ω
and f
2
ω
:
H
ω
(P(V ); Z)=Z[h, ξ]/(f
1
ω
,f
2
ω
)(1.3)
By Mori’s Cone Theorem [5], P(V ) has exactly two extremal rays, R
1
and R
2
.
Up to an order of R
1
and R
2
, the integral generator A
1
of R
1
is represented by
lines in the fibers of the projection π. We shall show that under certain numerical
conditions, the nontrivial homology classes A H
2
(P(V ); Z) which give nontrivial
quantum corrections are A
1
and A
2
,whereA
2
is represented by some smooth
rational curves in P(V ) which are isomorphic to lines in P
n
via π. In general, it is
unclear whether A
2
generates the second extremal ray R
2
. However, we shall prove
that under further restrictions on V , A
2
generates the extremal ray R
2
.These
analyses enable us to determine the quantum cohomology ring H
ω
(P(V ); Z).
The simplest ample bundle over P
n
is of the form V =
L
r
i=1
O
P
n
(m
i
), where
m
i
> 0 for every i. Since we can twist V by O
P
n
(1) without changing P(V ), we
can assume that min{m
1
,... ,m
r
}=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.
Batyrev’s Conjecture. Let V =
L
r
i=1
O
P
n
(m
i
),wherem
i
>0for every i.Then
the quantum cohomology ring H
ω
(P(V ); Z) is generated by h and ξ with two rela-
tions:
h
n+1
=
r
Y
i=1
(ξ m
i
h)
m
i
1
· e
t(n+1+r
P
r
i=1
m
i
)
and
r
Y
i=1
(ξ m
i
h)=e
tr
.
Our first result partially verifies Batyrev’s conjecture.
Theorem A. Batyrev’s conjecture holds if
r
X
i=1
m
i
< min(2r, (n +1+2r)/2,(2n +2+r)/2).
Note that under the numerical condition of Theorem A, only extremal rational
curves with fundamental classes A
1
and A
2
give contributions to the two relations
in the quantum cohomology. The moduli space M(A
2
, 0) of rational curves with
fundamental class A
2
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(kA
2
, 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 inter-
section theory. These seem to be difficult problems, which we shall not pursue
here.
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3618 ZHENBO QIN AND YONGBIN RUAN
In general, ample bundles over P
n
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.
Theorem B. (i) Let V be a rank-r ample bundle over P
n
. Assume either c
1
n,
or c
1
(n + r) and V ⊗O
P
n
(1) is nef, so that P(V ) is Fano. Then the quantum
cohomology H
ω
(P(V ); Z) is the ring generated by h and ξ with the two relations
h
n+1
=
X
i+j(c
1
r)
a
i,j
· h
i
· ξ
j
· e
t(n+1ij)
,
r
X
i=0
(1)
i
c
i
· h
i
· ξ
ri
= e
tr
+
X
i+j(c
1
n1)
b
i,j
· h
i
· ξ
j
· e
t(rij)
,
where the coefficients a
i,j
and b
i,j
are integers depending on V .
(ii) If we further assume that c
1
< 2r, then the leading coefficient a
0,c
1
r
=1.
It is understood that when c
1
n,thenthesummation
P
i+j(c
1
n1)
in the
second relation in Theorem B (i) does not exist. In general, it is not easy to
determine all the integers a
i,j
and b
i,j
in Theorem B (i). However, it is possible
to compute these numbers when (c
1
r) is relatively small. For instance, when
(c
1
r) = 0, then necessarily V = O
P
n
(1)
r
and it is well-known that the quantum
cohomology H
ω
(P(V ); Z) is the ring generated by h and ξ with the two relations
h
n+1
= e
t(n+1)
and
P
r
i=0
(1)
i
c
i
· h
i
· ξ
ri
= e
tr
.When(c
1
r)=1andr<n,
then necessarily V = O
P
n
(1)
(r1)
⊕O
P
n
(2). When (c
1
r)=1andr=n,then
V =O
P
n
(1)
(r1)
⊕O
P
n
(2) or V = T
P
n
, the tangent bundle of P
n
. In these cases,
V ⊗O
P
n
(1) is nef. In particular, the direct sum cases have been computed by
Theorem A. We shall prove the following.
Proposition C. The quantum cohomology ring H
ω
(P(T
P
n
); Z) with n 2 is the
ring generated by h and ξ with the two relations
h
n+1
= ξ · e
tn
and
n
X
i=0
(1)
i
c
i
· h
i
· ξ
ni
=(1+(1)
n
) · e
tn
.
Recall that for an arbitrary projective bundle over a general manifold, its coho-
mology 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 co-
homology 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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QUANTUM COHOMOLOGY OF PROJECTIVE BUNDLES OVER P
n
3619
2. Extremal rational curves
Assume that V is ample such that either c
1
(n +1), or c
1
(n+r)and
V⊗O
P
n
(1) is nef. In this section, we study the extremal rays and extremal
rational curves in the Fano variety P(V ). By Mori’s Cone Theorem (p. 25 in
[5]), P(V ) has precisely two extremal rays R
1
= R
0
· A
1
and R
2
= R
0
· A
2
such that the cone NE(P(V )) of curves in P(V )isequaltoR
1
+R
2
and A
1
and
A
2
are the homology classes of two rational curves E
1
and E
2
in P(V )with0<
K
P(V)
(A
i
)dim(P(V ))+1. Up to orders of A
1
and A
2
,wehaveA
1
=(h
n
ξ
r2
)
,
that is, A
1
is represented by lines in the fibers of π. It is also well-known that if
V =
L
r
i=1
O
P
n
(m
i
) with m
1
...m
r
,thenA
2
=[h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
,
which is represented by a smooth rational curve in P(V ) isomorphic to a line in P
n
via π. However, in general, it is not easy to determine the homology class A
2
and
the extremal rational curves representing A
2
. Assume that
V |
`
=
r
M
i=1
O
`
(m
i
)(2.1)
for generic lines ` P
n
, where we let m
1
...m
r
.SinceVis ample, m
1
1.
Lemma 2.2. Let A =[h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
.Then,
(i) A is represented by a smooth rational curve isomorphic to a line in P
n
;
(ii) A
2
= A if and only if (ξ m
1
h) is nef;
(iii) A
2
= A if 2c
1
(n +1);
(iv) A cannot be represented by reducible or nonreduced curves if m
1
=1.
Proof. (i) Let ` P
n
be a generic line. Then we have a natural projection V |
`
=
L
r
i=1
O
`
(m
i
) →O
`
(m
1
). By Proposition 7.12 in Chapter II of [7], this surjective
map V |
`
→O
`
(m
1
)0 induces a morphism g : ` P(V ). Then g(`) is isomorphic
to ` via the projection π.Sinceh([g(`)]) = 1 and ξ([g(`)]) = m
1
,wehave
[g(`)] = [h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
=A.
(ii) First of all, if A
2
=[h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
, then for any curve E,[E]=
a(h
n
ξ
r2
)
+b[h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
for some nonnegative numbers a and b;
so (ξm
1
h)([E]) = a 0; therefore (ξm
1
h) is nef. Conversely, if (ξm
1
h)isnef,
then 0 (ξ m
1
h)([E]) = ac
1
+ b am
1
where [E]=(ah
n1
ξ
r1
+ bh
n
ξ
r2
)
for
some curve E;thus[E]=(ac
1
+bam
1
)(h
n
ξ
r2
)
+a[h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
;
it follows that A
2
=[h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
=A.
(iii) Let A
2
=(ah
n1
ξ
r1
+bh
n
ξ
r2
)
.SinceA
1
=(h
n
ξ
r2
)
and a = h(A
2
) 0,
a 1. If a>1, then since 2c
1
(n +1),weseethat
K
P(V)
(A
2
)=(n+1c
1
)a+r·ξ(A
2
)2(n +1c
1
)+r
>n+r= dim(P(V )) + 1;
but this contradicts with K
P(V )
(A
2
) dim(P(V )) + 1. Thus a =1andA
2
=
(h
n1
ξ
r1
+bh
n
ξ
r2
)
.Now[π(E
2
)] = π
(A
2
)=(h
n1
)
.Soπ(E
2
) is a line in
P
n
.SinceV|
`
=
L
r
i=1
O
`
(m
i
) for a generic line ` P
n
, it follows that V |
π(E
2
)
=
L
r
i=1
O
π(E
2
)
(m
0
i
), where m
0
i
m
1
for every i.Thus,ξ(A
2
)m
1
,andsoc
1
+b
m
1
. It follows that
A
2
=[h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
+(c
1
+bm
1
)·(h
n
ξ
r2
)
.
Therefore, A
2
=[h
n1
ξ
r1
+(m
1
c
1
)h
n
ξ
r2
]
=A.
(iv) Since ξ(A)=m
1
=1andξis ample, the conclusion follows.
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