Quantum cohomology and
toric minimal model programs
We give a quantum version of the Danilov-Jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space. More precisely, we construct a canonical isomorphism from a formal version of the Batyrev ring from  to the quantum orbifold cohomology in a formal neighborhood of a canonical bulk deformation. This isomorphism generalizes results of Givental , Iritani  and Fukaya-Oh-Ohta-Ono  for toric manifolds and Coates-Lee-Corti-Tseng  for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity and an equality of dimensions deduced using a toric minimal model program (tmmp). We show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp.
According to results of Danilov and Jurkiewicz [15, 33, 34], the rational cohomology ring of a complete rationally-smooth toric variety is the quotient of a polynomial ring generated by prime invariant divisors by the Stanley-Reisner ideal. In addition to relations corresponding to linear equivalence of invariant divisors, there are higher degree relations corresponding to collections of divisors whose intersection is empty.
One can reformulate this presentation of the cohomology ring in terms of equivariant cohomology as follows. Let be a complex reductive group acting on a smooth polarized projective variety . If the action on the semistable locus is locally free then the geometric invariant theory (git) quotient , by which we mean the stack-theoretic quotient of the semistable locus by the group action, is a smooth proper Deligne-Mumford stack with projective coarse moduli space. A result of Kirwan  says that the natural map is surjective. Under suitable properness assumptions the same holds for quasi-projective .
In particular, let be a torus acting on a finite-dimensional vector space with weights contained in an open half-space. The quotient is a smooth proper Deligne-Mumford toric stack as in Borisov-Chen-Smith  and any such toric stack with projective coarse moduli space arises in this way. The equivariant cohomology may be identified with the ring of polynomial functions on , each weight maps to a divisor class in under the Kirwan map, and the Stanley-Reisner ideal is precisely the kernel of the Kirwan map. For example, if acts by scalar multiplication on , then is a polynomial ring in a single generator , the git quotient is , the intersection of the prime invariant divisors is empty, and the Stanley-Reisner ideal is the ideal generated by . This gives the standard description of the cohomology ring of projective space
In this paper we give a similar presentation of the quantum cohomology of compact toric orbifolds with projective coarse moduli spaces, via the quantum version of the Kirwan map introduced in . The results here generalize those of Batyrev , Givental , Iritani [30, 31, 32], and Fukaya-Oh-Ohta-Ono , who use results of McDuff-Tolman . In particular, Iritani  computed the quantum cohomology of toric manifolds using localization arguments for toric varieties that appear as certain complete intersections, while Fukaya et al  gave a computation using open-closed Gromov-Witten invariants defined via Kuranishi structures. The orbifold quantum cohomology of weighted projective spaces is computed in Coates-Lee-Corti-Tseng . The proof described here has several differences with the previous proofs, even in the case of Fano toric manifolds: it does not use any open Gromov-Witten invariants, proves (as the classical limit) the Danilov-Jurkewicz description, and does not use localization for the residual torus action.
A novel feature of the approach here is the appearance of minimal model programs, which are used to prove injectivity of the quantum Kirwan map modulo the quantum Stanley-Reisner ideal. The critical values of the Givental-Hori-Vafa potential acquire a natural geometric meaning in our approach: their logarithms are the transition times in the minimal model program. The dimension of the orbifold cohomology and the logarithm of the lowest eigenvalue of quantum multiplication by the first Chern class decrease under each tmmp transition. We also obtain a more conceptual understanding of the appearance of open families of non-displaceable Lagrangians in toric orbifolds, as a consequence of the existence of infinitely many minimal model programs.
We introduce the following notations.
(Novikov coefficients) Let denote the universal Novikov field of formal power series of with rational exponents
We denote by the subfield with rational coefficients, and by the subring with only non-negative powers of .
(Equivariant quantum cohomology) Let
denote the equivariant quantum cohomology of . We denote by the subspace with rational coefficients. Equivariant enumeration of stable maps to defines a family of products
forming (part of) the structure of a Frobenius manifold on  for in a formal neighborhood of a symplectic class . Explicitly the product is defined by
where the integral denotes push-forward to using the equivariant virtual fundamental class described in .
(Inertia stacks) The inertia stack of is
where in the first union, denotes representable morphisms from and the second union is over conjugacy classes of elements , with the centralizer of and the intersection of the semistable locus with the fixed point set
The rigidified inertia stack is
(Orbifold quantum cohomology of a git quotient) Let
denote the orbifold quantum cohomology of , or the version with rational coefficients. Enumeration of twisted stable maps to (representable maps from orbifold curves to ) defines a Frobenius manifold structure on ,  given by a family of products defined in a formal neighborhood of an equivariant symplectic class by
for , extended by linearity over , and the pairing on the left-hand-side is a certain re-scaled Poincaré pairing on the inertia stack , see .
(Stacky point) Let act on with weight two so that . The inertia stack is the union of two copies of corresponding to the elements of . Thus
the sum of two copies of , where is the additive generator of the twisted sector. Representable morphisms from a stacky curve to correspond to double covers of the coarse moduli space , with ramification at the stacky points. Since there is a unique double cover of the projective line with two ramification points (up to isomorphism) multiplication is given by
(Teardrop orbifold) Suppose that acts on with weights . Then is a weighted projective line, is a polynomial ring in a single generator, while
where is the point class in and is the class of the fixed point set in the twisted sector. If we identify corresponding to the dual of the Euler class of the representation with weight one then the fundamental class has image . The moduli space of twisted stable maps to of genus and class zero is then for no stacky points in the domain, or , for two stacky points in the domain. Furthermore there is a unique (up to isomorphism) homology class twisted map with two smooth marked points and one stacky marked point with automorphism group. It follows that if we take the symplectic class to have area on the fundamental class then the quantum product is defined by
Thus after inverting , the orbifold quantum cohomology is generated by with the relation .
(Alternative power series rings) Some confusion may be caused by the multitude of formal power series rings that one can work over; unfortunately almost every set of authors has a different convention.
The equivariant quantum cohomology can be defined over the larger equivariant Novikov field consisting of infinite sums with , where is the delta function at . Similarly, the quantum cohomology of the quotient can be defined over the Novikov field consisting of infinite sums with , where is the delta function at . However, these more complicated Novikov fields make some of our formulas a bit too long.
is also defined over the universal Novikov ring , and if is integral, over . Similarly, is defined over the Novikov ring , and if is integral, over for equal to the least common multiple of the orders of the automorphism groups in . However, it is convenient to work over the field . Invariance under Hamiltonian perturbation only holds for Floer/quantum cohomology over the Novikov field , and so working over is more natural for the purposes of symplectic geometry.
Unfortunately, and are non-Noetherian over and so some care is required when talking about intersection multiplicities etc. In practice, when we wish to talk about intersection multiplicities we assume that the symplectic form is integral in which case everything is defined over .
In algebraic geometry, one often uses the monoid-algebra of effective curve classes, but we prefer Novikov fields because of the better invariance properties. In fact, the cone of effective curve classes is not any more explicit than working over the Novikov field since it is the classes of connected curves that appear in the Gromov-Witten potentials, and these are rather hard to determine.
In  the second author studied the relationship between and given by virtual enumeration of affine gauged maps, called the quantum Kirwan map. An -marked affine gauged map is a representable morphism from a weighted projective line for some to the quotient stack mapping to the semistable locus . Some of the results of  are:
(Definition and properties of the quantum Kirwan map)
The stack of -marked affine gauged maps of class has a natural compactification . We denote by the evaluation maps
and their restrictions to maps of class . The moduli stack has a perfect relative obstruction theory over (the case of and trivial) where is the complexification of Stasheff’s multiplihedron.
For any , the map defined by virtual enumeration of stable -marked affine gauged maps
defines a formal map from to in a neighborhood of the symplectic class with the property that each linearization
is a -homomorphism with respect to the quantum products.
By analogy with the classical case one hopes to obtain a presentation of the quantum cohomology algebra by showing that is surjective and computing its kernel. This hope leads to the following strong and weak quantum version of Kirwan surjectivity. In the strong form, one might hope that has infinite radius of convergence, is surjective, and is surjective for any . More modestly, one might hope that is surjective for in a formal neighborhood of a rational symplectic class .
We now specialize to the toric case. Suppose that is a complex torus with Lie algebra acting on a finite-dimensional complex vector space .
(Weights) Let be the weight spaces of where and acts on with weight in the sense that for and we have . We assume that the weights are contained in an open half space, that is, for some we have . We also assume that the weights span , so that acts generically locally free on .
(Polarization and semistable locus) We assume that is equipped with a polarization, that is, an ample -line bundle , which we may allow to be rational, that is, a integer root of an honest -line bundle. Let be the vector representing the first Chern class of the polarization under the isomorphism . The point determines a rational polarization on with semistable locus
is the set of subsets so that is not in the span of the corresponding weights and is the intersection of coordinate hyperplanes
The stable=semistable condition assumption translates to the condition for each the weights span . In this case the quotient is then a smooth (possibly empty) proper Deligne-Mumford stack. We suppose that is non-empty.
(Quantum Stanley-Reisner ideal) The quantum Stanley-Reisner ideal is
If is the given symplectic class , we write . The quotient is the quantum Stanley-Reisner a.k.a Batyrev ring.
(Batyrev ring for projective space) Let act on by scalar multiplication so that all weights are equal to and the polarization vector . Then there is a unique subset in and . Thus the semistable locus is and the git quotient is . The quantum Stanley-Reisner ideal is generated by the single element . The Batyrev ring is .
Our main result says that Batyrev’s original suggestion  for the quantum cohomology is true, after passing to a suitable formal version of the equivariant cohomology and “quantizing” the divisor classes:
For a suitable formal version of the equivariant quantum cohomology (see Section 2) the linearized quantum Kirwan map induces an isomorphism
for in a formal neighborhood of any rational symplectic class .
(History) Many earlier cases of this theorem were known. Batyrev  proved a similar presentation in the case of convex toric manifolds, that is, in the case that the deformations of any stable map are un-obstructed. In the semi-Fano case (that is, is non-negative on any curve class) a presentation was given by Givental . For non-weak-Fano toric manifolds, Iritani [32, 5.11] gave an isomorphism with the Batyrev ring, see also Brown . From the symplectic point of view a presentation for the quantum cohomology of toric manifolds was given in Fukaya et al , using results of McDuff-Tolman  on the Seidel representation. The latter approach uses open-closed Gromov-Witten invariants to define a potential counting holomorphic disks whose leading order terms are the potential above. The quantum Stanley-Reisner relations were proved for toric orbifolds in Woodward , and, we understand, in unpublished work by Chung-Ciocan-Fontanine-Kim and Coates, Corti, Iritani, and Tseng. That these relations generate the ideal was expected for some time, see Iritani . Thus the main content of this paper is that these relations suffice. After the first version of this paper appeared, a quantization of the Borisov-Chen-Smith presentation of the orbifold cohomology was given in Tseng-Wang . Note that the latter is not a presentation in terms of divisor classes; for example, for weighted projective spaces the typical number of generators is much larger than one, while the Batyrev ring has a single generator.
(Necessity of the Novikov field) For the result above to hold the quantum cohomology must be defined over the Novikov field, or at least, that a suitable rational power of the formal parameter has been inverted: over a polynomial ring such as , one does not obtain an surjection because certain elements in twisted sectors are not contained in the image for . Thus one sees a Batyrev presentation of the quantum cohomology only for non-zero . The necessity of corrections to Batyrev’s original conjecture, which involved the divisor classes as generators, was noted in Cox-Katz [13, Example 22.214.171.124] for the second Hirzebruch surface and Spielberg  for a toric three-fold. The fact that the change of coordinates restores the original presentation was noted in Guest  for semi-Fano toric varieties, and Iritani [32, Section 5], for not-necessarily-Fano toric varieties in general, after passing to a formal completion. See Iritani [32, Example 5.5] and Gonzalez-Iritani [24, Example 3.5] for examples in the toric manifold case.
(Projectivity?) Note that Danilov’s results  do not require projectivity of the coarse moduli space. It seems possible that quantum cohomology might also be defined for non-projective toric varieties. Namely certain convergence conditions would remove the necessity of working over a Novikov ring, and one might have a theorem similar to 1.7, but we lack any results in this direction.
The presentation of the quantum cohomology in Theorem 1.7 can be re-phrased in terms of Landau-Ginzburg potential as follows, according to suggestions of Givental  and the physicists related to mirror symmetry. This formulation will be essential in our proof of the injectivity of the map in Theorem 1.7.
(Residual torus) Let denote the “big torus” acting on . The residual torus
has an induced action on . Let denote the unitary part of
given by exponentiating the real span of the coweights of . The Lie algebra of admits a canonical real splitting A canonical parametrization of the residual torus can be found by row-reduction on the matrix of weights, see Example 1.11 below.
(Moment polytope) The action of on is Hamiltonian, with moment map induced by the choice of moment map for the action of on . Let denote its image
the moment polytope of .
(Facets and spurious inequalities) Let be the inward normal vectors to the facets of ; these are the images of the minus the standard basis vectors of under the projection to :
The moment polytope is of the form
with positions of the facets determined by elements . We say that is a spurious inequality if it does not correspond to a facet of .
(Support constants) The support constants defining the positions of the facets of can be chosen as follows. Given an extension of to , isomorphic to , the constants are the coefficients of .
(Tropical dual torus) Our Landau-Ginzburg potential has domain a certain formal version of the dual torus to . The tropical dual torus is
where the sum over satisfies the same finiteness condition as in the definition of , namely for each the number of non-zero with is finite. Here, multiplication of elements with logarithmic coefficients and power series coefficients for is defined by addition of coefficients; that is, the product of two elements so defined is
Alternatively, the tropical dual torus is the set of exponentiation of power series . That is, the tropical dual torus is obtained by exponentiating power series with a logarithmic term in the quantum variable; we use the terminology tropical as a mellifluous synonym for logarithmic.
(Givental potential) The naive Landau-Ginzburg potential associated to the toric stack is a function on the tropical dual torus given as a sum of monomials whose exponents are the normal vectors to the facets of . (The reader may wish to compare with the treatment in Fukaya et al [19, Definition 2.1], where the potential is an element of a completed power series ring in coordinates ). It was first noticed by Givental  that this function is related to the Gromov-Witten theory of . An explanation from the point of view of mirror symmetry was given in Hori-Vafa , and a connection to Floer theory is described in Fukaya et al , where the potential appears as a count of holomorphic disks with boundary in a fiber of the moment map. In the later version, the potential receives corrections from nodal holomorphic disks, whereas in Givental  and Hori-Vafa  there are no corrections; we call the uncorrected version the Givental potential, given by
or more precisely
(Elimination of negative powers of ) As it stands, the values of have negative powers of . However, later we will always assume that is contained in the interior of the moment polytope , in which case only positive powers occur.
(Naive small potential versus corrected small potential) For the many purposes (non-displaceability, Batyrev presentation) it seems that the naive potential is “as good as” the corrected potential. A heuristic argument that the two potentials are related by a geometrically-defined change of coordinates was given in Woodward ; for semi-Fano cases it is proved in Chan et al  that this coordinate change is the mirror map from Gromov-Witten theory, while Fukaya et al [19, Theorem 11.1] show the existence of some coordinate transformation relating the two.
(Givental potential for a product of projective lines) Let and with weights with polarization vector so that . The perpendicular space to the weights is found by row-reduction to be the span of the vectors . With the corresponding parametrization of the dual torus the normal vectors to the facets are
and the moment polytope is . The potential is
(Naive potential for a projective line with extra term) The quotient of by the action of with weights and polarization vector (which projects to ) has semistable locus and git quotient . Taking the residual torus to have Lie algebra spanned by we obtain moment polytope
for which the first inequality is spurious, that is, may be removed without changing . The potential is
(Critical locus and Jacobian ring) The critical locus of is the set of points with vanishing logarithmic derivatives with respect to the coordinates on ,
The Jacobian ring of the Givental potential is the ring of functions on . Assuming that is contained in the interior of the moment polytope and is integral then the potential is defined over .
We wish to define a certain “positive part” of whose coordinate ring corresponds to the quantum cohomology of .
(Quantum embedding of the residual torus) The quantum embedding of the residual torus is the map
Note that (5) is a group homomorphism if we specialize , and in this case is dual to the projection .
(Positive part of the Jacobian ring) Let denote the ideal generated by the elements , and the completion of with respect to ,
Let denote the ring obtained from the formal completion by inverting ,
The filtered ring (resp. ) is the ring of functions on the formal scheme resp. obtained by taking a formal neighborhood of in with respect to the embedding (5) (resp. and removing the fiber over .)
(Informal interpretation of the positive part of the critical locus ) The scheme represents the locus of critical points of that have limit as with respect to the injection (5), that is, each expression has only positive powers of . Indeed, as we discuss below in Lemma 3.10, assuming is contained in the interior of the moment polytope, is finite over . After passing to a cover for some we may write each solution near as a function of a variable , that is, each component of becomes unramified over . By a simple case of the Grothendieck Existence Theorem , any point of is obtained by completion from a point of containing .
An interpretation in terms of critical points that lie over the interior of the moment polytope is given in Proposition 3.19. A reformulation of our main result is in the case the symplectic class:
For any rational symplectic class , there is a canonical isomorphism
The left-hand-side is independent of the presentation of as a git quotient of by , while the right-hand-side depends on the presentation. As we will see below, the different presentations of can lead to different splittings of the quantum cohomology.
That the rings on both sides have the same dimension follows in the Fano case from Kouchnirenko’s theorem [38, 3]. In general, we deduce the dimension equality from the toric minimal model program and an induction. A similar procedure is used by Kawamata  to show the existence of an exceptional collection in the derived category of any toric orbifold.
The Frobenius manifold structure , including the pairing, is expected to be equivalent to Saito’s Frobenius structure corresponding to the Landau-Ginzburg potential , see for example Fukaya et al . However, we do not discuss the pairings in this paper.
We end the introduction with examples of the projective plane, written in different ways as a quotient:
(Projective plane as a quotient by a circle action) The projective plane can be realized as a git quotient of affine space by a circle action as follows. Suppose that acts on by scalar multiplication. Suppose that the polarization corresponds to a trivial line bundle with a negative weight on the fiber at the origin. The semistable locus is and the git quotient is . We take the residual action of to have moment polytope in equal to
The corresponding potential is
The critical points are the solutions to
that is, . These generators and relations give a presentation of the quantum cohomology of .
(Projective plane as a quotient by a two-torus action) The projective plane can be realized as a git quotient by a two-dimensional torus as follows. Suppose that acts on with weights . The symplectic quotient is the “symplectic cut” of by the circle actions with directions in the sense of Lerman . The polytope is the intersection of a quadrant with two half-spaces with directions :
for some constants . Suppose that the polarization corresponds to the weight ; this is the right-most chamber in Figure 1.
The semistable locus consists of points with , and the git quotient is . In particular, does not define a divisor in . The potential is
The partial derivatives are
The critical values, to leading order, are the three critical points
and the two critical points
as shown in Figure 1. The first three (resp. second two) points (resp. do not) define elements of . Hence consists of three reduced points, . The other pictures in Figure 1 show the quotients for the other polarizations; the dotted line represents the span of the equivariant first Chern class , for which the quotient has a potential with all critical points located at . This ends the example.
We thank D. Cox, H. Iritani, D. McDuff, and C. Teleman for helpful comments.
2. Quantum Kirwan surjectivity for toric orbifolds
In this section we prove surjectivity for the linearization of the quantum Kirwan map on a formal completion of equivariant quantum cohomology; the surjectivity also holds for the uncompleted cohomology but does not lead to an isomorphism. Let be a smooth polarized projective -variety, or more generally, a quasiprojective smooth polarized projective -variety convex at infinity in the sense of , such as a finite-dimensional vector space with the action of a torus whose weights are contained in a half-space. The version of quantum Kirwan surjectivity we need involves a formal completion of the equivariant quantum cohomology, in which not only the powers of but also the degrees of the cohomology classes can go to infinity:
(Formal equivariant quantum cohomology ring) Let be the space of infinite sums
Equivalently, is obtained by completing with respect to the degree filtration on , and then inverting .
(Other completions) Note that there are various other natural completions. For example, completing