- Email: [email protected]
The Gromov-Witten invariants of symplectic toric manifolds, and their quantum cohomology ring
C. R. Acad. Sci. Paris, t. 329, SCrie I, p. 699-704, Ckomhtrie algCbrique/Algebraic Geometry (Topologie/ Topology)
The Gromov-Witten manifolds, and their Holger
invariants quantum
of symplectic toric cohomology ring
SPIELBERG
Max Planck Institute for Mathematics E-mail: [email protected] (Recu le 1”’ septembre
Abstract.
1999
in the Sciences, InselstraSe
1999, accept6 le 8 septembre
22-26,
04103 Leipzig,
Allemagne
1999)
In this Note, we are interested in the Gromov-Witten invariants of symplectic toric manifolds. These are invariants of the symplectic structure of the manifold. We announce an explicit formula for their computation that uses in particular a localization argument in the corresponding equivariant theory. As an application we give an example of a (non-Fano) toric manifold for which Batyrev’s abstract computation of the quantum cohomology ring does not apply. 0 1999Academicdesscienceshditions
scientifiqueset mkdicalesElsevier SAS Les invariants symplectiques
R&urn&
de Gromov-Witten des vari6tbs toriques et leur anneau de cohomologie quantique
Nous ROUSintt+essons, dam cette Note, aux invariants de Gromov-Witten des vari&% toriques symplectiques, invariants de leurs structures symplectiques. Nous annonFons une formule explicite pour leur calcul qui utilise notamment la technique de localisation dans la thiorie Lquivariante correspondante. A l’aide de cette formule nous donnons un exemple d’une varikte’ torique (non-Fano) & laquelle le calcul abstrait de la cohomologie quantique de Batyrev ne s’applique pas. 0 1999 AcadCmiedes scienceshditions
scientifiqueset m&dicalesElsevier SAS
Version frangaise
abrbgke
Les invariants de Gromov-Witten sont des invariants de la classe de dkformation de la forme symplectique des variCtts lisses admettant une telle structure. 11sentrent Cgalementcomme fonctions de corrClation dans la definition de l’anneau de cohomologie quantique d’une telle variCt6, objet 6tudit pour la premi&re fois par Witten [12] en physique thCorique. Ruan et Tian [9] ont Ct6 les premiers ti dtfinir rigoureusementces invariants en termes mathCmatiques pour les variCt& symplectiques faiblement monotones, utilisant des structures presque complexes CC gCntriques >>.Siebert [lo], Li et Tian [7], Fukaya et Ono [4] ainsi que d’autres ont Ctendu cette dCfinition B toutes les variCt& symplectiques en construisant une classe fondamentale virtuelle dans l’espace de modules des applications stables, ainsi que Behrend et Fantechi [2] pour les va&Cs algebriques projectives. Note prBsent6e par Jean-Pierre 0764~4442/99/03290699 Tous droits r&erv&.
DEMAILLY.
0 1999 Acadkmie
des sciencesfiditions
scientifiques
et mkdicales
Elsevier
SAS.
699
H. Spielberg
Bien que les bases theoriques soient desormais disponibles, il est pourtant toujours tres difficile de calculer effectivement ces invariants, les deux constructions Ctant tres compliquees. Pour la grande classe des variCtts toriques projectives, on se trouve par contre dans une situation plus avantageuse puisque : (a) ce sont des varietes algebriques : on peut done travailler avec des courbes holomorphes que I’on connait beaucoup mieux que leurs pendants pseudo-holomorphes ; (b) on a une action d’un <>sur la variete, induisant une action sur l’espace de modules. En utilisant la technique de localisation de Graber et Pandharipande [6], on peut alors calculer les invariants comme un produit d’intersection sur les composantes des points fixes, espaces qui sont beaucoup plus simples que l’espace de modules de depart. Cela nous permet de proposer une formule explicite pour le calcul des invariants de Gromov-Witten de genre 0 pour toutes les varietts toriques projectives. Soient XC une varitte torique lisse projective de dimension d donnee par un &entail C, et A le polytope dual de C qui est Cgalement le polytope de moment de XC. Soient 21; . . . , 2, les diviseurs invariants sous l’action du tore T” sur X C. Ces classes engendrent l’anneau cohomologique de XC et nous utilisons des multi-indices pour d&ire une classe cohomologique: 2’ = Zfl . . Z$. THI~ORI~ME 1. - Soient m E N un entier et A E Ha (Xc, Z) une classehomologique de degre’2. Alors, les invariants de Gromov-Witten a m points dans la classeA sont donnespar :
06 la sommeest sur tous graphes connexes de dimension 1 et salts cycle sur le I-squelette du polytope A, representant la classe A. Soient Vert = { nr, . . . , 0,) l’ensemble des sommetsde I?, et Edge celui des a&es. Les termes Sr et Ti- sont don&s par :
sr = fi (kwf;i;’ .W&))> i=l
Tr =
rI &Edge
(2)
j=l
(-l)dd2d (d!)2(w:;)2d
rI UZ#-P~l
ae={ol,DZ)
(3)
03 les diffe’rentsw de’crivent despoids ou des termesde poids ’ de 1‘action du tore Td sur XC. Dans (l), Ar est le groupe d’automorphismesde la composantedes points&es de r. Cette formule nous permet notamment de calculer dans [l l] tous les invariants de Gromov-Witten (a trois points) de genre 0 de la variCtC Pcp2(O(2) $ l), en accord avec les resultats de Batyrev [l] et de Givental ’ [5]. En revanche, les invariants de la varitte torique (non-Fano) Pcp2(O(3) $ l), que nous avons calcults a la fin de cette Note, nous laissent deduire que l’anneau propose par Batyrev [l] n’est pas la cohomologie quantique de cette variete (Corollary 1).
700
Gromov-Witten
invariants
of symplectic
toric
manifolds
1. Introduction Gromov-Witten invariants are invariants of the symplectic deformation class of a manifold admitting a symplectic structure. They have first been studied by Witten [12] in the context of a-models. In fact, the quantum cohomology ring of a symplectic manifold is an example of such a a-model, and Gromov-Witten invariants are basically the structure constants of this ring. Mathematically rigorous foundations have been laid by now, first by Ruan and Tian [9] for weakly monotone symplectic manifolds 3, and later by different authors for general symplectic manifolds, for example Siebert [lo], Li and Tian [7], and Fukaya and Ono [4]. Behrend and Fantechi [2], and Li and Tian [8] have also given a compatible definition for projective algebraic manifolds that works entirely in the algebraic category. However, all these definitions of the invariants use very complicated technical constructions: Ruan and Tian’s approach relies on the use of “generic” almost-complex structures J, the invariants being defined as the number of certain J-holomorphic curves; though we still understand very little about these curves in the generic case. The other approaches are based on the construction of a virtual fundamental class in the moduli space of (J)-holomorphic (stable) curves. This is necessary since the moduli space is very often “too big” for a general symplectic manifold. Moreover, the space has singularities and the codimension of the virtual class, that is a cycle representing this class, is in general not constant. In the case of symplectic toric manifold, though, we can apply a localization argument due to Graber and Pandharipande [6] to greatly simplify this construction, and actually to find an explicit formula for the genus-0 Gromov-Witten invariants of these manifolds.
2. Toric
manifolds
Toric manifolds are (compact connected) manifolds that contain an algebraic torus (C*)d as open and dense subset. The free action of the torus on itself extends to an effective action on the entire toric manifold. We find it useful to consider toric manifolds as a compactification of the big torus (C*)d by lower dimensional tori (and points which we regard as tori of dimension zero). These tori of lower dimension are in fact the submanifolds of the toric manifold that are invariant under the torus action, and therefore correspond to faces of the moment polytope of the toric manifold in case it is symplectic. In fact, Delzant [3] has proved that symplectic toric manifolds are in one-to-one correspondence with a certain class of polytopes, called Delzant polytopes. There also exists a dual combinatorial description of toric manifolds by fans (see e.g. [l] and the references therein). Let Xc be a symplectic toric manifold, A its moment polytope, and C be the corresponding complete regular fan. The combinatorial data of the fan C can in fact be used to obtain XC by symplectic reduction (or a similar construction in the algebraic category) from 15’” with the action of a subtorus (C*)caBd) c (C*)n, the latter acting on C” in the standard way. Here, 12 is the number of one-dimensional cones in C. Let WI,... , w, E (t”)* be the (constant) weights of the standard action on C”. The projection of the (C*)d-action to the quotient of (C*)n induces a map ( td)* + (t”)*; we can therefore express the weights of the (C*)d-action on XC in terms of the wi. Each maximal cone CJE C defines a chart of Xc, and on these charts the weights of the (C*)d-action are again constant. One can define coordinates indexed by neighboring cones (T’ E C, i.e. cones U’ that have a facet in common with c (we will write 0 o (T’). We will call the weights with respect to this coordinate system w;,. We will denote by wtDota, the product of these d weights for each such CT:
701
H. Spielberg
We will now describe the fixed point components of the induced action on the moduli space of stable genus-zero curves in XC, and in particular the notation in equations (l), (2), (3). First note, that fixed points in XC are given by the maximal cones in C, or dually by the comer points of the moment polytope A. Similarly, irreducible invariant one-dimensional subvarieties of Xc are in one-to-one correspondence with (d - l)-dimensional cones of C, or edges of the moment polytope A. The image of fixed points in the moduli space are invariant one-dimensional subvarieties of Xc, their irreducible components are thus sent via the moment map to comer points or edges of the polytope. Since we only consider genus-zero curves, they are characterized by the comer point (a constant map), or the edge e and the multiplicity d, of the map. Note, that special points of the stable curve have to be mapped to fixed points in XC. We can therefore describe the fixed point components in the moduli space of stable curves in XC by connected graphs on the l-skeleton of the moment polytope A, with decorations keeping track of multiplicities and the location of the marked points. Since we consider genus-zero curves, these graphs have no loops. Each edge e of such a graph is identified by two maximal cones de = (01, a~} in C, representing the two fixed points in Xc the edge e connects. A flag is the pair of an edge and a base point (T;. For F = (e,a), de = {a, o’} we define 1 ‘dF := -w;, de
and
F(D) wtotal :-
C F=(*du))
&,
where wFCu) total ’IS defined for all vertices tl of the graph and the sum is over all flags originating at b. Now let ZI,..., Z, be the invariant divisors (codimension-one submanifolds) of Xc. These classes generate the cohomology ring H* (Xx, Z) of X C, and we will use multi-index notation to describe a cohomology class:
Ze = z;l . ..z>. The invariant divisors Zi correspond by Poincar6 duality to equivariant line bundles Li on XC via = P.D.(Zi). M ore g eneral, a class Ze corresponds to an equivariant vector bundle Le = C&L:%. Let L(D) : C( TV)-+ Xc be the inclusion of the fixed point a(b) in XC. Then we define
cl(Li)
e wdo) := e(c*)d (@)*Le) to be the equivariant Euler class of the restriction of Le to [T(D). Since the restricted bundle is topologically trivial, wzCDj is the product of the weights of the action on the fiber of Le over O(U). We are now ready to give the formula for the computation of the genus-zero Gromov-Witten invariants of XC: THEOREM 1. - Let m E N be a non-negative integer, and A E Hp(Xc, Z) be an integral degree-2 homology class. Then the genus-zero m-point Gromov-Witten invariants are given by formula (1) where the sum is over all connected one-dimensional graphs without loops on the l-skeleton of the moment polytope A, decorated with multiplicities 4 and representing the class A. LetVert = {bl,..., a,} be the set of vertices of r, and let Edge be the set of edges.The terms Sr and Tr are given by (2) and (3). The expressionAr is the automorphism group of thefied point component of I?, that is the semi-directproduct of the automorphismgroup of I? as a decorated graph and @ Zd,. &Edge
702
Gromov-Witten
invariants
of symplectic
toric manifolds
3. Examples In [ 111,using the above formula, we compute all 3-point genus-zero Gromov-Witten invariants of the and reaffirm the known formula [ 11, [5] for its quantum cohomology ring. Fano 3-foldP,,z(0(2)$1), Let us consider here the complex three-dimensional manifold Xc = PC Pi (O(3) $ 1). Its fan lives in R3 with generators given by ui = ei, ~2 = -ei, us = es, ~4 = es and us = -ez - es - 3ei. The set of primitive collections of the fan is { (01, as}, (D3, b4, 05)). Since 03 + 04 + b5 = 302 it is therefore obvious that this toric manifold is not Fano. The cohomology ring H*(Xc) and the group R(C) E Hs(Xc) are given by: H*(Xc)
= C[Zi,. . . ) z51/p3
-
25,24
-
z5,&
-
22
-
=5,-&z,,
z3z4z5)
R(C) = Al. 2 + X2.2 c Z5, where Xi = (l,l,O,O,O), X2 = (0, -3,1,1,1). The moment polytope is a frustum of a pyramid with triangles as bases.The edges of the “upper” triangle (with corners cr4 = (02,03, D4), 05 = (us, tl3, bg), 06 = (us, D4, D5)) correspond to invariant one-dimensional submanifolds with homology class X2, while the homology class of the edges of the “lower” triangle (with comers ~1 = (Ol,D3,D4), us = (Dl,D3,D5), ~3 = (Dl,b4,D5)) is 3x1 + X2. The side-edgeshave homology class Xi. We will sketch the computation of the invariants &‘z (23, 23, 2s) and @2xZ(23, 23, 2s). In both cases, graphs that enter into formula (1) live on the upper triangle of the moment polytope. The As-invariant is a sum over three graphs (each being of the edges of the triangle), the Sr term being always 1. The three Tr terms are: T
=
(-w1+
w2
+
2w5
+
w4)(-w1+
(W3
-
W5)(W3
2~5
+
w3)
(W4
-
W5)(W4
2~4
+
w3)(-Wl
(W5
-
W4)(W5
w2
+
w5
+
2W4) ,
1 T2
=
(-WI
+
w2
+
-
W4)
(-w1+
w2
+
w5
+
2W3) >
T3
=
(-w1+
w2
+
-
W3) +
w2
+
w4
+
2W3) I
-
W3)
yielding three for the invariant. A similar computation for the 2&-invariant, summing over twelve different graphs on the upper triangle, yields the second invariant. We obtain: @2(Zs, Z3, 23) = 3 and Q2xZ(Z3, Zs,Zs) = -45. Since the invariant is linear in its arguments, and because of 2 2 = 21 - 323, we obtain for the pX2-invariants wyz2, Z,,&) = - 27Qpx2(Z3, Z,, Z3). Note that we have used here that the weight of 21 on the upper triangle is zero (the comers 04, 05, (Ts do not contain bi !). Hence we obtain aq22,22,27J
=
2;
=
92&
= 9,
@z(22,
z2,
2,)
=
= -81.
-27@(23,23,23)
Now supposethe quantum cohomology ring of XC was as proposed by Batyrev 111:
QH*(Xc)
= C [Z,, . . . ,Z5,8’,
q”] I
z3-25,24-25,21-Z-2-325,
z-l.&
-
qA1,
. z3z4.z5
-
z,3qAz
>
703
H. Spielberg
The relation Z3Z4Z5 - 2,3qxQ implies on the level of Gromov-Witten
invariants that:
Taking A = $2 to be a multiple of the class XZ, and cy = P.D.(point) to be the Poincare dual of a point, this boils down to: Vp : @(P+1)xZ (23, Z’s, Z’s) = @rxZ (&,22,22) which is obviously not satisfied for p as small as p = 0 and p = 1. COROLLARY 1. - Batyrev’s definition as in [l] of the quantum cohomology ring of a symplectic toric manifold does not coincide with the one obtained from Gromov-Witten invariants as dejined in [2] for the manifold XC = PcPz(O(3) $ 1).
Remark 1. - Recently, several authors have proved the equivalence of the different approachesto construct virtual fundamental classesfor the definition of the Gromov-Witten invariants. Therefore, the Corollary applies to them as well. Acknowledgements.Most of the resultsin this Note are part of my Strasbourgth&e de doctorut [111.I want to thank Michele Audin, Olivier Debarre,EmmanuelPeyre, ClaudeSabbah,Bemd SiebertandTilmannWurzbacher. I acknowledgethe supportby the GermanAcademicExchangeOffice (DAAD) trough a graduatescholarship (Hochschulsonderprogramm III), financedby the Germanfederal stateand the GermanLiinder. Finally I want to thank the Institut de RechercheMathematiqueAvancCeat Strasbourgfor its hospitality while preparingmy thesis and this work.
’ Voir la partie principale pour plus de details, ou encore [ 1 I]. * Givental a demontre la << conjecture >>de Batyrev sur I’anneau de cohomologie quantique pour les varittes toriques de Fano. 3 These include in particular all manifolds of dimension six or less, in particular the examples we will give at the end of this Note. 4 Note that the graph l? is no longer decorated with the positions of the marked points. The sum over all these possible decorations has been integrated into the $-term, while the Tr do a priori not depend on the positions of the marked points.
References [l] Batyrev V.V., Quantum cohomology rings of toric varieties, Asterisque 218 (1993) 9-34. [2] Behrend K., Fantechi B., The intrinsic normal cone, Invent. Math. 128 (1) (1997) 45-88. [3] Delzant T., Hamiltoniens periodiques et images convexes de l’application moment, Bull. Sot. Math. France 116 (1988) 3 15-339. [4] Fukaya K., Ono K., Arnold conjecture and Gromov-Witten invariant, Topology 38 (5) (1999) 933-1048. [5] Givental A.B., A mirror theorem for toric complete intersections, in: Topological field theory, primitive forms and related topics (Kyoto, 1996). Birkhauser, Boston, 1998, pp. 141-175. [6] Graber T., Pandharipande R., Localization of virtual classes, Invent. Math. 135 (2) (1999) 487-518. [7] Li J., Tian G., Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, in: Topics in symplectic 4-manifolds (Irvine, 1996), Intemat. Press, Cambridge, 1998, pp. 47-83. [8] Li J., Tian G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Sot. I I (1) (1998) 119-174. [9] Ruan Y., Tian G., A mathematical theory of quantum cohomology, J. Differ. Geom. 42 (2) (1995) 259-367. [lo] Siebert B., Gromov-Witten invariants for general symplectic manifolds, LANL preprint alg-geom/9608005, August 1996. [I 1] Spielberg H., A formula for the Gromov-Witten invariants of toric varieties, PhD thesis, IRMA preprint 1999/l 1, 1999. [12] Witten E., Two-dimensional gravity and intersection theory on moduli spaces, Surveys Differ. Geom. 1 (1991) 243-310.
704
The Gromov-Witten manifolds, and their Holger
invariants quantum
of symplectic toric cohomology ring
SPIELBERG
Max Planck Institute for Mathematics E-mail: [email protected] (Recu le 1”’ septembre
Abstract.
1999
in the Sciences, InselstraSe
1999, accept6 le 8 septembre
22-26,
04103 Leipzig,
Allemagne
1999)
In this Note, we are interested in the Gromov-Witten invariants of symplectic toric manifolds. These are invariants of the symplectic structure of the manifold. We announce an explicit formula for their computation that uses in particular a localization argument in the corresponding equivariant theory. As an application we give an example of a (non-Fano) toric manifold for which Batyrev’s abstract computation of the quantum cohomology ring does not apply. 0 1999Academicdesscienceshditions
scientifiqueset mkdicalesElsevier SAS Les invariants symplectiques
R&urn&
de Gromov-Witten des vari6tbs toriques et leur anneau de cohomologie quantique
Nous ROUSintt+essons, dam cette Note, aux invariants de Gromov-Witten des vari&% toriques symplectiques, invariants de leurs structures symplectiques. Nous annonFons une formule explicite pour leur calcul qui utilise notamment la technique de localisation dans la thiorie Lquivariante correspondante. A l’aide de cette formule nous donnons un exemple d’une varikte’ torique (non-Fano) & laquelle le calcul abstrait de la cohomologie quantique de Batyrev ne s’applique pas. 0 1999 AcadCmiedes scienceshditions
scientifiqueset m&dicalesElsevier SAS
Version frangaise
abrbgke
Les invariants de Gromov-Witten sont des invariants de la classe de dkformation de la forme symplectique des variCtts lisses admettant une telle structure. 11sentrent Cgalementcomme fonctions de corrClation dans la definition de l’anneau de cohomologie quantique d’une telle variCt6, objet 6tudit pour la premi&re fois par Witten [12] en physique thCorique. Ruan et Tian [9] ont Ct6 les premiers ti dtfinir rigoureusementces invariants en termes mathCmatiques pour les variCt& symplectiques faiblement monotones, utilisant des structures presque complexes CC gCntriques >>.Siebert [lo], Li et Tian [7], Fukaya et Ono [4] ainsi que d’autres ont Ctendu cette dCfinition B toutes les variCt& symplectiques en construisant une classe fondamentale virtuelle dans l’espace de modules des applications stables, ainsi que Behrend et Fantechi [2] pour les va&Cs algebriques projectives. Note prBsent6e par Jean-Pierre 0764~4442/99/03290699 Tous droits r&erv&.
DEMAILLY.
0 1999 Acadkmie
des sciencesfiditions
scientifiques
et mkdicales
Elsevier
SAS.
699
H. Spielberg
Bien que les bases theoriques soient desormais disponibles, il est pourtant toujours tres difficile de calculer effectivement ces invariants, les deux constructions Ctant tres compliquees. Pour la grande classe des variCtts toriques projectives, on se trouve par contre dans une situation plus avantageuse puisque : (a) ce sont des varietes algebriques : on peut done travailler avec des courbes holomorphes que I’on connait beaucoup mieux que leurs pendants pseudo-holomorphes ; (b) on a une action d’un <
06 la sommeest sur tous graphes connexes de dimension 1 et salts cycle sur le I-squelette du polytope A, representant la classe A. Soient Vert = { nr, . . . , 0,) l’ensemble des sommetsde I?, et Edge celui des a&es. Les termes Sr et Ti- sont don&s par :
sr = fi (kwf;i;’ .W&))> i=l
Tr =
rI &Edge
(2)
j=l
(-l)dd2d (d!)2(w:;)2d
rI UZ#-P~l
ae={ol,DZ)
(3)
03 les diffe’rentsw de’crivent despoids ou des termesde poids ’ de 1‘action du tore Td sur XC. Dans (l), Ar est le groupe d’automorphismesde la composantedes points&es de r. Cette formule nous permet notamment de calculer dans [l l] tous les invariants de Gromov-Witten (a trois points) de genre 0 de la variCtC Pcp2(O(2) $ l), en accord avec les resultats de Batyrev [l] et de Givental ’ [5]. En revanche, les invariants de la varitte torique (non-Fano) Pcp2(O(3) $ l), que nous avons calcults a la fin de cette Note, nous laissent deduire que l’anneau propose par Batyrev [l] n’est pas la cohomologie quantique de cette variete (Corollary 1).
700
Gromov-Witten
invariants
of symplectic
toric
manifolds
1. Introduction Gromov-Witten invariants are invariants of the symplectic deformation class of a manifold admitting a symplectic structure. They have first been studied by Witten [12] in the context of a-models. In fact, the quantum cohomology ring of a symplectic manifold is an example of such a a-model, and Gromov-Witten invariants are basically the structure constants of this ring. Mathematically rigorous foundations have been laid by now, first by Ruan and Tian [9] for weakly monotone symplectic manifolds 3, and later by different authors for general symplectic manifolds, for example Siebert [lo], Li and Tian [7], and Fukaya and Ono [4]. Behrend and Fantechi [2], and Li and Tian [8] have also given a compatible definition for projective algebraic manifolds that works entirely in the algebraic category. However, all these definitions of the invariants use very complicated technical constructions: Ruan and Tian’s approach relies on the use of “generic” almost-complex structures J, the invariants being defined as the number of certain J-holomorphic curves; though we still understand very little about these curves in the generic case. The other approaches are based on the construction of a virtual fundamental class in the moduli space of (J)-holomorphic (stable) curves. This is necessary since the moduli space is very often “too big” for a general symplectic manifold. Moreover, the space has singularities and the codimension of the virtual class, that is a cycle representing this class, is in general not constant. In the case of symplectic toric manifold, though, we can apply a localization argument due to Graber and Pandharipande [6] to greatly simplify this construction, and actually to find an explicit formula for the genus-0 Gromov-Witten invariants of these manifolds.
2. Toric
manifolds
Toric manifolds are (compact connected) manifolds that contain an algebraic torus (C*)d as open and dense subset. The free action of the torus on itself extends to an effective action on the entire toric manifold. We find it useful to consider toric manifolds as a compactification of the big torus (C*)d by lower dimensional tori (and points which we regard as tori of dimension zero). These tori of lower dimension are in fact the submanifolds of the toric manifold that are invariant under the torus action, and therefore correspond to faces of the moment polytope of the toric manifold in case it is symplectic. In fact, Delzant [3] has proved that symplectic toric manifolds are in one-to-one correspondence with a certain class of polytopes, called Delzant polytopes. There also exists a dual combinatorial description of toric manifolds by fans (see e.g. [l] and the references therein). Let Xc be a symplectic toric manifold, A its moment polytope, and C be the corresponding complete regular fan. The combinatorial data of the fan C can in fact be used to obtain XC by symplectic reduction (or a similar construction in the algebraic category) from 15’” with the action of a subtorus (C*)caBd) c (C*)n, the latter acting on C” in the standard way. Here, 12 is the number of one-dimensional cones in C. Let WI,... , w, E (t”)* be the (constant) weights of the standard action on C”. The projection of the (C*)d-action to the quotient of (C*)n induces a map ( td)* + (t”)*; we can therefore express the weights of the (C*)d-action on XC in terms of the wi. Each maximal cone CJE C defines a chart of Xc, and on these charts the weights of the (C*)d-action are again constant. One can define coordinates indexed by neighboring cones (T’ E C, i.e. cones U’ that have a facet in common with c (we will write 0 o (T’). We will call the weights with respect to this coordinate system w;,. We will denote by wtDota, the product of these d weights for each such CT:
701
H. Spielberg
We will now describe the fixed point components of the induced action on the moduli space of stable genus-zero curves in XC, and in particular the notation in equations (l), (2), (3). First note, that fixed points in XC are given by the maximal cones in C, or dually by the comer points of the moment polytope A. Similarly, irreducible invariant one-dimensional subvarieties of Xc are in one-to-one correspondence with (d - l)-dimensional cones of C, or edges of the moment polytope A. The image of fixed points in the moduli space are invariant one-dimensional subvarieties of Xc, their irreducible components are thus sent via the moment map to comer points or edges of the polytope. Since we only consider genus-zero curves, they are characterized by the comer point (a constant map), or the edge e and the multiplicity d, of the map. Note, that special points of the stable curve have to be mapped to fixed points in XC. We can therefore describe the fixed point components in the moduli space of stable curves in XC by connected graphs on the l-skeleton of the moment polytope A, with decorations keeping track of multiplicities and the location of the marked points. Since we consider genus-zero curves, these graphs have no loops. Each edge e of such a graph is identified by two maximal cones de = (01, a~} in C, representing the two fixed points in Xc the edge e connects. A flag is the pair of an edge and a base point (T;. For F = (e,a), de = {a, o’} we define 1 ‘dF := -w;, de
and
F(D) wtotal :-
C F=(*du))
&,
where wFCu) total ’IS defined for all vertices tl of the graph and the sum is over all flags originating at b. Now let ZI,..., Z, be the invariant divisors (codimension-one submanifolds) of Xc. These classes generate the cohomology ring H* (Xx, Z) of X C, and we will use multi-index notation to describe a cohomology class:
Ze = z;l . ..z>. The invariant divisors Zi correspond by Poincar6 duality to equivariant line bundles Li on XC via = P.D.(Zi). M ore g eneral, a class Ze corresponds to an equivariant vector bundle Le = C&L:%. Let L(D) : C( TV)-+ Xc be the inclusion of the fixed point a(b) in XC. Then we define
cl(Li)
e wdo) := e(c*)d (@)*Le) to be the equivariant Euler class of the restriction of Le to [T(D). Since the restricted bundle is topologically trivial, wzCDj is the product of the weights of the action on the fiber of Le over O(U). We are now ready to give the formula for the computation of the genus-zero Gromov-Witten invariants of XC: THEOREM 1. - Let m E N be a non-negative integer, and A E Hp(Xc, Z) be an integral degree-2 homology class. Then the genus-zero m-point Gromov-Witten invariants are given by formula (1) where the sum is over all connected one-dimensional graphs without loops on the l-skeleton of the moment polytope A, decorated with multiplicities 4 and representing the class A. LetVert = {bl,..., a,} be the set of vertices of r, and let Edge be the set of edges.The terms Sr and Tr are given by (2) and (3). The expressionAr is the automorphism group of thefied point component of I?, that is the semi-directproduct of the automorphismgroup of I? as a decorated graph and @ Zd,. &Edge
702
Gromov-Witten
invariants
of symplectic
toric manifolds
3. Examples In [ 111,using the above formula, we compute all 3-point genus-zero Gromov-Witten invariants of the and reaffirm the known formula [ 11, [5] for its quantum cohomology ring. Fano 3-foldP,,z(0(2)$1), Let us consider here the complex three-dimensional manifold Xc = PC Pi (O(3) $ 1). Its fan lives in R3 with generators given by ui = ei, ~2 = -ei, us = es, ~4 = es and us = -ez - es - 3ei. The set of primitive collections of the fan is { (01, as}, (D3, b4, 05)). Since 03 + 04 + b5 = 302 it is therefore obvious that this toric manifold is not Fano. The cohomology ring H*(Xc) and the group R(C) E Hs(Xc) are given by: H*(Xc)
= C[Zi,. . . ) z51/p3
-
25,24
-
z5,&
-
22
-
=5,-&z,,
z3z4z5)
R(C) = Al. 2 + X2.2 c Z5, where Xi = (l,l,O,O,O), X2 = (0, -3,1,1,1). The moment polytope is a frustum of a pyramid with triangles as bases.The edges of the “upper” triangle (with corners cr4 = (02,03, D4), 05 = (us, tl3, bg), 06 = (us, D4, D5)) correspond to invariant one-dimensional submanifolds with homology class X2, while the homology class of the edges of the “lower” triangle (with comers ~1 = (Ol,D3,D4), us = (Dl,D3,D5), ~3 = (Dl,b4,D5)) is 3x1 + X2. The side-edgeshave homology class Xi. We will sketch the computation of the invariants &‘z (23, 23, 2s) and @2xZ(23, 23, 2s). In both cases, graphs that enter into formula (1) live on the upper triangle of the moment polytope. The As-invariant is a sum over three graphs (each being of the edges of the triangle), the Sr term being always 1. The three Tr terms are: T
=
(-w1+
w2
+
2w5
+
w4)(-w1+
(W3
-
W5)(W3
2~5
+
w3)
(W4
-
W5)(W4
2~4
+
w3)(-Wl
(W5
-
W4)(W5
w2
+
w5
+
2W4) ,
1 T2
=
(-WI
+
w2
+
-
W4)
(-w1+
w2
+
w5
+
2W3) >
T3
=
(-w1+
w2
+
-
W3) +
w2
+
w4
+
2W3) I
-
W3)
yielding three for the invariant. A similar computation for the 2&-invariant, summing over twelve different graphs on the upper triangle, yields the second invariant. We obtain: @2(Zs, Z3, 23) = 3 and Q2xZ(Z3, Zs,Zs) = -45. Since the invariant is linear in its arguments, and because of 2 2 = 21 - 323, we obtain for the pX2-invariants wyz2, Z,,&) = - 27Qpx2(Z3, Z,, Z3). Note that we have used here that the weight of 21 on the upper triangle is zero (the comers 04, 05, (Ts do not contain bi !). Hence we obtain aq22,22,27J
=
2;
=
92&
= 9,
@z(22,
z2,
2,)
=
= -81.
-27@(23,23,23)
Now supposethe quantum cohomology ring of XC was as proposed by Batyrev 111:
QH*(Xc)
= C [Z,, . . . ,Z5,8’,
q”] I
z3-25,24-25,21-Z-2-325,
z-l.&
-
qA1,
. z3z4.z5
-
z,3qAz
>
703
H. Spielberg
The relation Z3Z4Z5 - 2,3qxQ implies on the level of Gromov-Witten
invariants that:
Taking A = $2 to be a multiple of the class XZ, and cy = P.D.(point) to be the Poincare dual of a point, this boils down to: Vp : @(P+1)xZ (23, Z’s, Z’s) = @rxZ (&,22,22) which is obviously not satisfied for p as small as p = 0 and p = 1. COROLLARY 1. - Batyrev’s definition as in [l] of the quantum cohomology ring of a symplectic toric manifold does not coincide with the one obtained from Gromov-Witten invariants as dejined in [2] for the manifold XC = PcPz(O(3) $ 1).
Remark 1. - Recently, several authors have proved the equivalence of the different approachesto construct virtual fundamental classesfor the definition of the Gromov-Witten invariants. Therefore, the Corollary applies to them as well. Acknowledgements.Most of the resultsin this Note are part of my Strasbourgth&e de doctorut [111.I want to thank Michele Audin, Olivier Debarre,EmmanuelPeyre, ClaudeSabbah,Bemd SiebertandTilmannWurzbacher. I acknowledgethe supportby the GermanAcademicExchangeOffice (DAAD) trough a graduatescholarship (Hochschulsonderprogramm III), financedby the Germanfederal stateand the GermanLiinder. Finally I want to thank the Institut de RechercheMathematiqueAvancCeat Strasbourgfor its hospitality while preparingmy thesis and this work.
’ Voir la partie principale pour plus de details, ou encore [ 1 I]. * Givental a demontre la << conjecture >>de Batyrev sur I’anneau de cohomologie quantique pour les varittes toriques de Fano. 3 These include in particular all manifolds of dimension six or less, in particular the examples we will give at the end of this Note. 4 Note that the graph l? is no longer decorated with the positions of the marked points. The sum over all these possible decorations has been integrated into the $-term, while the Tr do a priori not depend on the positions of the marked points.
References [l] Batyrev V.V., Quantum cohomology rings of toric varieties, Asterisque 218 (1993) 9-34. [2] Behrend K., Fantechi B., The intrinsic normal cone, Invent. Math. 128 (1) (1997) 45-88. [3] Delzant T., Hamiltoniens periodiques et images convexes de l’application moment, Bull. Sot. Math. France 116 (1988) 3 15-339. [4] Fukaya K., Ono K., Arnold conjecture and Gromov-Witten invariant, Topology 38 (5) (1999) 933-1048. [5] Givental A.B., A mirror theorem for toric complete intersections, in: Topological field theory, primitive forms and related topics (Kyoto, 1996). Birkhauser, Boston, 1998, pp. 141-175. [6] Graber T., Pandharipande R., Localization of virtual classes, Invent. Math. 135 (2) (1999) 487-518. [7] Li J., Tian G., Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, in: Topics in symplectic 4-manifolds (Irvine, 1996), Intemat. Press, Cambridge, 1998, pp. 47-83. [8] Li J., Tian G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Sot. I I (1) (1998) 119-174. [9] Ruan Y., Tian G., A mathematical theory of quantum cohomology, J. Differ. Geom. 42 (2) (1995) 259-367. [lo] Siebert B., Gromov-Witten invariants for general symplectic manifolds, LANL preprint alg-geom/9608005, August 1996. [I 1] Spielberg H., A formula for the Gromov-Witten invariants of toric varieties, PhD thesis, IRMA preprint 1999/l 1, 1999. [12] Witten E., Two-dimensional gravity and intersection theory on moduli spaces, Surveys Differ. Geom. 1 (1991) 243-310.
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