Our main results are formulated in terms of the formal series
called the genus-0 descendant potential of M, where M is respectively (as in the Abstract), the base of the toric fibration E, or a suitable divisor of the base. The ingredients of the series are defined in the generality of almost-Kähler manifolds M, as follows. The spaces are moduli spaces of (equivalence classes of) degree-D stable maps into M of genus-0 (possibly nodal) compact connected holomorphic curves with n marked points. Two such stable maps and are equivalent if there is a holomorphic automorphism mapping marked points to marked points and preserving the ordering, such that . For a stable map , the degree-D condition reads .
Then, denotes the virtual fundamental class of . The ingredient is the 1st Chern class of the universal cotangent line bundle over whose fiber at a stable map is the cotangent line along the stable map at the a-th marked point. The maps evaluate the stable maps at the a-th marked points.
The Mori cone MC of M is the semigroup in generated by classes representable by compact holomorphic curves. Then is the element in the Novikov ring (the power-series completion of the semigroup algebra of the Mori cone) representing the degree . Lastly, are arbitrary cohomology classes of M with coefficients in a suitable ground ring (for now, the Novikov ring with rational coefficients ).
It is convenient for this purpose to choose a basis of , and extend to a basis of . Then, the dual basis can be thought of in terms of a corresponding basis of curve classes, and its extension to . Define Novikov’s variables ; these record, for the exponent, the pairing of on a curve class . Equivalently, the variable records the coefficients of the curve classes along the dual basis vector to , in the dual basis expansion of .
1.2. Toric Fibrations
Let be an integer matrix, and consider the action of on the Hermitian space that, for each , multiplies the coordinate by . Let be the map given by
If the moment map has a regular value , then is a symplectic manifold. This construction is called symplectic reduction. The space is also denoted by , and is equipped with a canonical symplectic form, call it , induced by the standard symplectic form on . All complex line bundles over B may be assumed to have the unitary circle as structure group, as they are induced by pullback from the tautological line bundle over . Given complex line bundles over B, it follows that is the structure group of the vector bundle . Thus, the fiberwise symplectic reduction of is well-defined giving the toric fibration . The ith coordinate on the torus defines a circle bundle over E for which the expression defines connection 1-forms in the bundle. Denote by the first Chern class of the ith circle bundle over E, and its restriction to a fiber. The classes are of Hodge -type by the Fubini-Study construction, though they need not be Kähler classes1. Let be any -fixed point of , and representing the -equivalence class . The orbits and are then identical. It follows that there is some coordinate subspace with coordinates , containing p, such that none of the coordinates vanishes. It will be convenient to think of the -fixed strata of E in terms of the corresponding indices . For each , the restriction of to a fiber is Poincaré dual to the jth coordinate divisor . Define for . The expressions for the pullbacks in terms of may be summarized by the equations .
Set . All bundles introduced thus far are T-equivariant, so their Chern classes may be assumed to take values in the T-equivariant cohomology group , or , with coefficient ring .
1.3. The Cone
Associated to the genus-0 Gromov-Witten theory of M is a Lagrangian cone in a symplectic loop space   . The space is a module over the ground ring . Pending further completions, consists of Laurent series in 1/z with coefficients in , completed so that consists of elements of at each order in Novikov’s variables, and . Identify each with the domain variables of by the dilaton shift convention . Take the ring of coefficients for Novikov’s variables to be the (super-commutative) power series ring (with coefficients in the field of fractions , in all of our applications) in the formal coordinates along , and require the variables to vanish when Novikov’s variables and formal coordinates along are all set to zero. This gives a Novikov ring that is consistent with the formula for in our Main Theorem.
Let be a basis of and the Poincaré-dual basis. Consider the symplectic manifold with standard symplectic form . It is symplectomorphic to with symplectic form
where is the Poincaré pairing.
Let us implement this symplectomorphism via the map
Consider the graph of the differential of , which is a Lagrangian submanifold in . From there, we arrive at
by rigid translation in the direction of the dilaton shift. Thus is also a Lagrangian submanifold. Henceforth we consider as a submanifold of . The work of Coates-Givental  , establishes that is a (Lagrangian) cone as a formal Lagrangian section of near ; that is,
In particular, each tangent space is preserved by multiplication by z.
It may be that contains (as a limit point) the -coordinate origin (0, 0), as a special case of Getzler’s  , Givental’s  solution, and its geometric formulation  , of Eguchi-Xiong’s, Dubrovin’s -jet conjecture, as follows.
The shift of the formal variable in the z-(or -) direction appears to be well-understood, so perhaps formality of the geometry (to guarantee convergence of ) in the z-direction need not be assumed. This existence (via convergence) of the “vertex” or the “limiting vertex” of the cone gives an intuitive way to think about the introductory material; however, the author has not studied this convergence sufficiently. In our main theorem, the domain variable is consistent with the setting of formal geometry.
The Lagrangian cone of the T-equivariant genus-0 Gromov-Witten theory of lies in the corresponding symplectic loop space as above. A point in the cone can be written as
where denotes the virtual push-forward by the evaluation map , and is an arbitrary element of with coefficients . Define the J-function to be the restriction of to values and to for all . For each there is a unique such that
This property of the set of all tangent spaces2 of to be in 1-1 correspondence with the set H, which is a finite-dimensional -module, is called overruled. For each and for each open set , the J-function generates a module over the algebra of differential operators as follows,
is the unital, associative, (super-)commutative quantum cup product. Additionally, the J-function satisfies the string and divisor equations:
respectively. The graded homogeneity, defined by degrees of formal variables, makes the quantum cup product a degree 0 operation, the J-function graded homogeneous of degree 1, and z of degree 1.
1.4. Twisted Lagrangian Cones
The forgetful maps induce the K-theoretic push-forward maps . Let be a complex vector bundle over M. The evaluation maps induce the (virtual-) bundles , in terms of which the (virtual-) virtual bundles
are defined. The fiber of over a stable map is
Given a characteristic class , define the twisted Poincaré pairing .
A point in the -twisted cone can be written as
The overruled Lagrangian cone in the -twisted genus-0 Gromov-Witten theory of M lies in the symplectic loop space , where is an element of with arbitrary coefficients . The examples we will consider are:
Example 1.4.1. , and is a convex line bundle; i.e., ; or equivalently, for all genus-0 stable maps to M.
Example 1.4.2. , and is a complex vector bundle with a hamiltonian T-action that decomposes into a direct sum of complex line bundles, each of which carries a non-trivial T-action.
1.5. Torus Action on
The manifold may be described as the result of surgery on E, along the divisor of the T-fixed section , as we now recall. The notation , and recall the detailed local geometry near the exceptional divisor . Define a map from a tubular neighborhood of the bundle over the projective space bundle over to a tubular neighborhood of over as follows. Fiberwise, it is described by the projection map
This construction holds in the generality3 of complex manifolds and submanifolds, respectively, where is replaced by (normal bundle to the submanifold within the ambient manifold). The map collapses the projective space fibers fiberwise over . The map is the identity map away from the points above, and thus extends over the entire gluing space . This map identifies T-equivariantly the complements of the 0-sections of the total spaces of the preceding two vector bundles. Remove a tubular neighborhood of from E, and replace it by a tubular neighborhood of the bundle over .
We will call the resulting manifold the projective-space (surgery, gluing, quotient) of E along , the quotient-space of E along , the quotient-space of E along (Section 2.1), the surgery-space of E along (or normal to) . Henceforth, we denote this by , for simplicity of notation.
1.6. Simplification: Toric Manifold
Let X be a compact symplectic toric manifold and let be the maximal unitary torus, and let Y be a T-invariant submanifold of X. Then is again a toric manifold. As explained in Section 1.5, though not in the generality needed here, the action of T on X induces an action of T on . Thus, we may study T-equivariant genus-0 Gromov-Witten invariants of , the -quotient of X along (or normal to) Y directly, using fixed-point localization. All faces of the moment polytope of Y are faces of the moment polytope of X. The moment polytope of admits a canonical inclusion into the moment polytope of X, for which all faces of the moment polytope of are contained in faces of the corresponding same dimension of the moment polytope of X. Let be the primitive integer normal vectors to the codimension one faces of the moment polytope of a toric manifold. Let be a basis of the -vector space
consisting of primitive integer vectors. The toric manifold is then recovered from symplectic reduction referred to the matrix , whose row vectors are . By a mirror theorem of Givental  and its extensions  , a particular family of points on the Lagrangian cone of the genus-0 Gromov-Witten theory of a toric manifold is given by an explicit formula4 in terms of ,
This project has its roots in the following instructive example. Let E be the total space of the projective bundle
described by symplectic reduction with respect to the matrix
Let be the section of E that maps each point to the point in the fiber over x. When X is the toric bundle E and Y is then a calculation gives
This example provides a reference point for navigating the project. The matrix may be computed using Appendix A in  , which is itself a summary of literature     on moment maps and aspects of toric manifolds. Namely, in the momentum polyhedron of a toric manifold, the 1-dim edge vectors leaving a vertex at a T-fixed point are positive multiples of the elements of the set . These latter are the weights of the T-action on the normal bundle to in the toric manifold.
Apply this first to the original projective bundle . Then compute the weights of the T-action on the normal bundles in to the T-fixed points of the exceptional divisor. Finally, compute the normals to the codimension one faces of the momentum polyhedron of . A basis of linear relations among them is given by the rows of the matrix.
However, in fact, our main theorem arises as a generalization of this example. Here we are using the toric mirror theorems    as a guide to the structure of genus-0 Gromov-Witten invariants more generally (following the initial proposals of A. Elezi and A. Givental). Elezi’s work focused on projective bundles . In  , Givental proposed a toric bundles generalization of Elezi’s approach using toric mirror integral representations  . This is an ingredient in  and in the present work.
1.7. Organization of the Text
We recall in Section 2.2 the Atiyah-Bott fixed-point localization Theorem which implies, in particular, that any element of is uniquely determined by its restrictions to the T-fixed strata of . Points on the overruled Lagrangian cone of the genus-0 T-equivariant Gromov-Witten theory of are certain H-valued formal functions, which we study in terms of their restrictions . As we recall  in Section 5, the projection of each of the restrictions consists of two types of terms. Namely, there are terms ii) that form simple poles expanded as series about non-zero -values of z. The remaining terms i) are polynomial in at any given order in formal variables . The organising principle of the text, formulated as Theorem 2, characterizes the Lagrangian cone of the genus-0 T-equivariant Gromov-Witten theory of in terms of two conditions i) ii) on . The condition ii) says that the residues of at its simple poles at non-zero values of z are governed recursively with respect to . The condition i) describes the remaining poles at in terms of a certain twisted Lagrangian cone of the stratum . The Main Theorem gives a family of points whose restrictions satisfy the conditions of Theorem 2.
In Section 6 we verify condition ii) for the restrictions directly, using their defining formulae. In Section 7, we verify condition i) using transformation laws  of Lagrangian cones with respect to the twisting construction from Section 1.4 and example 1.8 (expanding simple poles at non-zero values of z in non-negative powers of z). A new aspect of the present work relative to toric bundles is that ii) relates the series that, according to condition i), lie in Lagrangian cones derived from genus-0 Gromov-Witten invariants of B and of , respectively. The Quantum Lefschetz Theorem relates the Lagrangian cone associated to the genus-0 Gromov-Witten theory of A with that of B. If the push-forward does not identify the Mori cone of with that of B, the opposite relation describing the Lagrangian cone of B in terms of that of is realised algebraically by the Birkhoff factorization procedure and dividing by powers of z. Division by z does not preserve the Lagrangian cone, so we must then clear denominators on both sides. For each , denote the greatest power of z that we divide by up to order in this process by . We work out an example where A is a smooth quintic 3-fold.
It suffices without loss of generality to assume that is a single connected manifold A, as regards most aspects of the project. In case there is a subtlety, we address it as it arises.
A key result to keep in mind while reading the paper is the Proposition in Section 2.1, describing the T-equivariant normal bundles to the T-fixed sections of the exceptional divisor. The Proposition is used for both the Atiyah-Bott fixed-point localization theorem for in Section 2.2, and for stating the twisting construction in genus-0 Gromov-Witten theory in Section 7.1 for .
1.8. T-Fixed Strata of
Recall that L gives rise to as the zero locus of a generic section. The tautological line bundle with fiber , i.e. the bundle, over the exceptional divisor is central to the results.
The T-action on induces a T-action on the moduli spaces of stable maps to , which in turn induces a T-action on the universal cotangent line bundles at each of the marked points. For a given T-fixed stratum of and a line bundle with a fiberwise T-action, we refer to the class as the T-weight of at . The T-fixed strata of are in comparison with those of E as follows. The stratum of E is replaced by , which is canonically diffeomorphic to . Let take on the values as a substratum of , as well as .
Example 1.8. If is a T-fixed stratum in the complement of the exceptional divisor, then take in Example 1.4.2. If is a T-fixed stratum in the exceptional divisor, then take in Example 1.4.2. In either case, set in Example 1.4.2 and also define .
Finally, set .
For each T-fixed section of E, the strata of E is canonically a stratum of that we also denote by . Lastly, there are T-fixed strata of that have no counterpart in E. Namely, each T-equivariant line bundle summand of gives rise to a T-fixed section over A in the exceptional divisor.
In the case , will denote (Section 2.1 for the definition) rather than the pullback to (which is modded out as in Section 2.1). Thus, is given a new definition in this case. Let also take the value .
In particular (Section 2.1), the summand gives rise to a section
over A in the total space of the exceptional divisor. The T-fixed set is only a proper subset of the T-fixed stratum . Thus we must check the conditions 1.a) and 2.aa) for the projections to (see Section 7.3 for the integral asymptotics of ), and not for , for the series .
From now on let the symbol stand for the T-fixed strata denoted above, or for the “substrata” of . Let us denote the situation of a torus fixed point connected to by a 1-dimensional edge of the momentum polyhedron of a fiber of E, by . In this case , , and . Let be the coordinate from and the coordinate from . Similarly, we have the notation and . In the next section, we enhance this description of the T-fixed points of E to a description of the T-fixed points of .
2. Geometry of
2.1. Geometric Preliminaries and Decomposition of Cohomology
The action of T on E decomposes into a direct sum of 1-dimensional eigenspaces,
Let be an ordering index of these eigenspaces, where the index value corresponds to the bundle , and indexes the summand of with T-weight . Denote the T-fixed section of corresponding to the index by . In the case, we need to include the index a, for the divisors of B along which we replace the geometry of E by the geometry. The strata is connected to the strata by the T-invariant edges . Denote by the T-weight of .
Denote by the composition of with the projection to the base B. It is now mandatory that we introduce the diagram
Let be the normal bundle within to the T-fixed section over A with index in .
Proposition. The action of T on decomposes into a direct sum of T-equivariant line bundles, whose T-equivariant Euler classes are the elements of the set
Let us now turn attention to the restriction map . Denote the T-equivariant Euler class of the bundle on the exceptional divisor. By the Lerray-Serre theorem,
In the following, we extend the definition of to the entire . With this interpretation of , recall the isomorphism of vector spaces 
where the quotient is an additive quotient and . On the other hand, . The restriction of to the exceptional divisor is , which restricts to to .
Let us assume that , so that
This holds true in the examples of quintic 3-folds for which the base is projective space. More generally, examples follow from the Lefschetz hyperplane Theorem and the Hard Lefschetz Theorem.
The restriction map and the Poincaré pairing give the orthogonal projection :
The short exact sequence
gives a direct sum decomposition with respect to the Poincaré-pairing on .
The result of “division by ” is only defined at the level of coset representatives of . The choice of a basis of coset representatives from suffices for integration over weighted by , which represents integration over the fundamental class of . Thus, the subspace represents the span of an arbitrary basis of coset representatives from , and is not uniquely defined. The space can be thought of as .
For the purpose of integrating over fundamental cycles, the pullback , , of b to can be described with respect to by a multiplicative factor of ,
Let us now establish that
Both are subsets of . In general, the subspaces can differ only on , about which the Hodge diamond is symmetric w.r.t. the Lefschetz theorems. The inclusion is clearly an isomorphism when the base is projective space.
Thus, . This gives the inclusion. Finally, taking the quotients of gives
Let us assume the map on the LHS is an isomorphism (an equality). This is also assumed as hypotheses for the main theorem (Section 4) and Theorem 2.
The RHS is used in the comparison of projection maps. Then, and extend to by
2.2. Fixed-Point Localization
For each , the action of T on decomposes into a direct sum of 1-dimensional eigenspaces. Define as in Example 1.8. Let be the classes that restrict to the T-equivariant Poincaré duals of the torus-invariant divisors in the fibers of the exceptional divisor :
The Atiyah-Bott Theorem says that the pairing of a class against the fundamental class of is given by
Namely, we sum over each of the T-fixed strata the pairing of the class
against the fundamental class of .
Thus, denote the class in that restricts to the T-equivariant Euler class of the bundle on the exceptional divisor, and restricts to zero at all T-fixed strata in the complement of the exceptional divisor.
Define a T-equivariant line bundle over the union of torus-invariant edges of as follows. It restricts to the bundle over the edges of the exceptional divisor, restricts to the trivial bundle over the edges and whose T-equivariant Euler class restricts to over the edges .
Proposition. The pairings on elements of take values in .
Proof. The restriction of to the union of torus invariant edges coincides with the class . Apply the Atiyah-Bott fixed-point localization Theorem to the restriction of to the union of torus-invariant edges of ,
and for all . Thus, induces an element of .
3. The htA Function
Let be the coordinate along . Let be a basis of , and a basis of , with dual bases and . Let be coordinates on . Define
Quantum Lefschetz Theorem   . Suppose , or more generally that L is convex. Then for each and for each smooth family , the series modification
lies in the image by of the Lagrangian cone associated to the genus-0 Gromov-Witten theory of A with domain inputs encoded by coefficients of by the dilaton shift.
Let us assume that has the property (Div + Str primary) that its dependence on is of the form
where do not depend on , are Laurent polynomials in z valued in , and . Then, both series and have the property Div+Str primary.
In the case that define by . Let us define a partial order on by if . In the case that the inclusion is only proper, our goal is to prove well-definedness of the least positive integer function such that, for each , the truncation of to order on both of and in the Novikov’s variables of the base is a formal linear combination of vectors in the linear space
(both sides truncated to order ≤ D on both of and ),
The need for this is as follows. Condition 1.a of Theorem 2 refers to twisted Lagrangian cones of the Gromov-Witten theory of A. The Quantum Lefschetz Theorem also refers to the (image by of the) Lagrangian cone of A, but does so in terms of a family of points of the (image by of the) Lagrangian cone of the Gromov-Witten theory of B. The difficulty is that the Quantum Lefschetz Theorem only uses certain terms of the series-those that lie in the Novikov ring associated with . The input for the Main Theorem is a family of points on the Lagrangian cone of B, which uses the Novikov ring of B.
The difficulty with this is that the Mori cone (resp Novikov ring) of A is only a subcone (resp. subring) of the Mori cone (resp. Novikov ring) of B. The natural algebraic tool for working with Langragian cones in genus-0 Gromov-Witten theory is the Birkhoff factorization technique. We will do this using the divisor equations. Thus, assume is generated by , in which case is generated multiplicatively by .
We now prove well-definedness of by giving a combinatorial algorithm for computing it. We observe the following (Divisor-, String-) differential equations
For any polynomial in variables with coefficients in , it follows that
Now replace the series by a differential operator series. Let be the (maximal) pole order of at . Then define through the formula
Namely, expand the RHS (right-hand side) at order D,
to get the formula for in terms of , inductively. Define
Let be the unique5 family of points of whose truncation to order on both of and in the Novikov’s variables of the base satisfies
Example. Let , and a smooth quintic 3-fold.
Let be the Kähler generator of , and take to be the J-function of at the point ,
Thus, we deduce the relation . The A series has been reindexed relative to the original A series.
The coefficient of in is
This is a polynomial in powers of the nilpotent of maximal non-vanishing degree 3 variable , with coefficients in .
A quick check by induction shows that when n is a positive multiple of 5, in which case . Also by induction, for each for which n is not a multiple of 5, is the maximal power of in the series; i.e., . The preceding discussion allows us to deduce the following.
Proposition. Suppose that is not surjective, so that is not identically zero. If B is , if A is the zero locus of a generic section of a convex line bundle L over B, if is the J-function of B, and if the class of the base is nonnegative as a functional on , then .
Proof. Group each numerator factor with a denominator factor and expand analogously to the above. Each factor in the denominator that is not grouped with a factor in the numerator gives a power of beyond those that come with powers of .
4. Main Results
4.1. The I-Function
Upon extension of scalars of homology groups, the Mori cone of includes into . Given or , define , , , , and these values uniquely determine .
Henceforth we use the gamma function convention:
Let us assume the conditions in Section 3 hold true. Our main theorem assumes the hypotheses of Section 2.1; however, the latter hypotheses may not be necessary (as noted in Section 5.3). Then,
Main Theorem. Let E be a toric fibration over base B, whose fibers are not copies of the point, and let be a T-fixed section. Let be convex line bundles over B, and smooth divisors of B arising as the zero loci of generic sections of . Further assume the to be mutually disjoint. In the Case 1 below, assume is generated by , so that each is generated multiplicatively by .
Case 1: If the push-forwards do not identify the Mori cone of with that of B, then for each , for each , for each and for each smooth family with the property Div + Str primary, the version of the series (a completion6 of) defined by
lies in the truncation to order on both of and (in the Novikov’s variables of the base) of the Lagrangian cone associated to the genus-0 Gromov-Witten theory of . Case 2: If the push-forwards identify the Mori cone of with that of B, then , and the preceding series lies in the preceding cone without any truncation condition on either, while still assuming the property Div + Str primary for the smooth family .
Since the genus-0 generating functions of Gromov-Witten theory of E and are described in  , we may think of the main result as a gluing result or a gluing formula. Similarly, the integral (Section 7.3) is defined in terms of the integrals for and .
Remark. When the fibers are copies of the point then we omit the sum over and we set to zero, since the projective fibers are also copies of the point. Keeping these interpretations in mind, the theorem remains true when the fiber of the toric fibration is the point. The theorem reduces to the statement .
Remark. The natural generalization of the Main Theorem to the case of several T-fixed sections of E coincides, at the first level of analysis, with the natural generalization of the mirror theory of Section 7.
Remark. The analogue of the proof of Theorem 2 in  indicates the dependence of points of upon domain variables from .
Conjecture. The dependence on domain variables
may be incorporated into the Main Theorem by replacing in the argument of and in the argument of , for some function , . The latter shift of the argument of by u is free, and then the shift of is determined.
Some examples of the main Theorem.
1) Let B be a smooth toric variety obtained by -symplectic reduction of and A a (nef) coordinate hyperplane divisor of B. An instance of in this case is the example in Section 1.6. The series , constructed from , is not supported in the Mori cone of . See the inequality conditions on the support of the series, in the Remark in (2.bb) of Section 6. However, if we construct the series from then the latter conditions at the fixed point are updated by the additional condition . The class is, apriori, an element of . If the bundle L is considered as -equivariant, then is -equivariant. The class is not the same equivariantly as , but they define the same functionals on the Mori cone of . The above inequality reads . This inequality rules out “the class of a in a fiber of the exceptional divisor”, as well as the curve classes , from the solutions to the original set of inequalities in the Remark.
Thus, the series of the Main Theorem is an extension outside the Novikov ring of the series of the toric mirror theorems, in example 1.5 and more generally for symplectic toric manifolds   .
2) Let B be , and A the manifold of complete flags in .
Corollary. Let E be a toric fibration over base B, whose fibers are not copies of the point, and let be a T-fixed section. Then for each , for each and for each smooth family with the property Div + Str primary, the version of the series
lies in the Lagrangian cone associated to the genus-0 Gromov-Witten theory of .
Application to codimension > 1 subvarieties . Let be a symplectic reduction of a direct sum of line bundles pulled back from A, and a symplectic reduction of a direct sum of line bundles also pulled back from A. T-fixed sections and may be considered as index subsets, respectively. The disjoint union of index subsets defines a T-fixed section . Then Corollary applies to , where the matrix used for the symplectic reduction is block diagonal with a block for each of the fibers.
4.2. Graded Homogeneity
Let be a basis of extending a basis of . Define for all Novikov’s variables . This determines the degrees of Novikov’s variables as follows: , let denote the coefficent of a along the basis vector . Thus, and . Let us refer to Sections 1.8, 2.1 and 2.2 for the definition of classes and for the projection maps and onto subspaces of . The first Chern class of away from the exceptional divisor is the restriction of the first Chern class of TE. The first Chern class of TE is .
Let be any T-fixed stratum in the complement of the exceptional divisor. The tangent space to the fibers of E at decomposes as the direct sum of the line bundles with the equivariant first Chern classes . Since the classes all vanish, the above formula for the first Chern class accounts correctly for the normal bundle to in E (Section 1.2). On the exceptional divisor, the tangent bundle of restricts to . The fiber line summand, along with the and maps, will give the difference between the tangent bundle to the projective bundle itself, and the pullback of from the ambient space.
At each T-fixed section on the projective space bundle over A, refer to Section 2.1 for the first Chern classes of the normal line bundle summands. Then, the first Chern class of the preceding is , as follows. The dimension of the fiber of E is , and there are T-fixed point sections of the projective bundle fibers of the exceptional fibers. At each , the first Chern class of one of the line bundle summands, , vanishes. Thus, we get contributions to the first Chern class of the tangent bundle to the projective space fibers at each such T-fixed section. Thus, we arrive at for measuring the degree of , . The restriction of the class to fibers of is the dual vector to the fiber curve classes.
The class vanishes away from the exceptional divisor. Now compare the preceding formulae for to the universal formula
The latter restricts correctly to the exceptional divisor and to the standard locus.
Let us now check the degree of the total series is . The degree of is 1. Then, for each , the term is of degree . Thus, if we simply define , then the degree of the latter becomes 1. When the factor is included we thus arrive at degree , which is independent of . Let us note geometry of the latter definition of , as follows. The summand from
contributes to the pairings . The data beyond to determine a class is the pairing , realized as
This is the latter degree of , for all .
Then compare the degrees of the terms , where is defined by and , with the degrees of the hypergeometric factors. Then, the remaining terms of the main series are of total degree 0, as follows. If and , all factors are denominator series with the total degree
In the case , the “denominator” series with upper limit is actually a numerator series. The index goes from 0 to , giving factors in the numerator rather than the denominator; thus the preceding counting of is correct in this case too. Let us simply note that the degree counting is the same in all cases. Thus, the mechanism that establishes the degree formula is the ratio of infinite products, from Section 4.
The degree of Novikov variables is . Thus, this need only be compared to the hypergeometric factors of the series. In view of the above remarks, we compare the Novikov variables degrees to the upper limit indices on the product series. The Novikov variable degrees and the product series degrees should be equal, so that they cancel out to 0. The comparison is immediately verified.
5. The T-Equivariant Cone
5.1. Localization of Stable Maps
The work of Graber-Pandharipande  justifies the fixed-point localization technique for computing integrals of T-equivariant cohomology classes over virtual fundamental cycles in the moduli spaces of stable maps to . Here the T-equivariant normal “bundle” to a T-fixed stable map is actually a virtual (orbi-) virtual bundle in T-equivariant K-theory. The description of T-fixed stable maps is then analogous to the description in . Namely, the connected components of the T-fixed loci in the moduli spaces of genus-0 stable maps are fiber products of moduli spaces of genus-0 stable maps into the T-fixed strata of . Let C be a leg of ; i.e., an irreducible component of that maps surjectively to a T-invariant edge of (a fiber of of) . The fiber product is defined by reference to the curves from , from and toric edges . The image points and coincide with the images of marked points of stable maps from and from in their roles as nodal points of . There is also the case that either or may be a marked or unmarked point of , not connecting C to any other curve component of .
There are three disjoint cases to consider, depending upon how the 1-dimensional -orbit (i.e., edge) intersects the exceptional divisor. Equivalently, these cases are distinguished by the projection image of the point set . Firstly (2.bb), the projection of the toric edge along the projection map is again a toric edge at each point of the given fiber product. Suppose that the two strata connecting a toric edge map via the projection to the T-fixed sections and . The fibre products involving factors of genus-0 stable maps into can be non-compact, as follows. Given a toric edge connecting to , the nodal point in is unable to enter the exceptional divisor. The Atiyah-Bott formula7 implies that the correct cohomology group to use for the non-compact space is the subset . A similar case to consider is when the toric edge connects to , where .
Secondly (2.ab), the T-fixed points of the toric edge connect to the rest of the T-fixed stable map at and at ; . In this case too, the projection image of the toric edge is also a toric edge.
Third (2.aa), the toric edge is contracted by the projection map at each point of the given fiber product. The T-fixed points of the toric edge connect to the rest of the T-fixed stable map at and at , .
There are three types of terms that contribute to the series . Namely, the polynomial term , and then two types of contributions to the projection of the series . Given a T-fixed stable map to , which we denote by , now let C be the smooth irreducible component of that contains the first marked point of the source of the stable map. In order for the stable map to contribute to , f must map the first marked point into the stratum . The latter two types of contributions are determined by whether
i) All points of C are mapped by f into the T-fixed stratum . In this case, let be the maximal connected subset of containing C that maps to , and let .
ii) C maps to a T-invariant in connecting T-fixed strata and . Let us assume that, in the normalization of , C is a with two marked points—which we may take to be 0 and via the action of on —, that there is a marked point of at , and that the marked point at corresponds to a node of . Thus the stable map takes C to a , maps the first marked point of at to and maps to a nodal point of the stable map at , and as it follows from the work of Kontsevich  , is given by .
Each point of lies in either the (normal bundle to the) exceptional divisor, or its complement—this is close enough to the toric bundles case for the following decomposition in  to hold, since the details are local.
Let us recall (Section 1.8) the definition of . The fiber of the virtual normal bundle to the T-fixed strata of stable maps to the T-fixed strata at the T-fixed stable map , as in case i) above, is given by
The virtual normal bundle
to the T-fixed stable map with source , deforming the map to a non T-fixed stable map, decomposes into the direct sum of:
i) The virtual normal bundle over the stable map with source , and
ii) A virtual vector space
over the point . This virtual vector space is the fiber of a virtual bundle. Let us use the same notation for the bundle and for the fiber.
This is along the same lines as in  , with the only new subtlety coming from the case when . Namely, the deformation of a stable map to along a section of the bundle is a stable map into and is still T-fixed. If there are no componenents of of type ii) connecting to the domain curves of the latter maps of type i) then, the line bundle does not contribute to the virtual normal bundle to the fiber product factors of stable maps to (or ) in the T-fixed loci in the moduli spaces of stable maps to . For more details of this subtlety, see the decomposition of the map near the end of 5.2 (with slightly expanded definition of C.).
A second technical issue regarding the T-equivariant deformation theory of f, comes from the case that the bundle contributes to the T-equivariant normal deformation theory of , but not to the restriction of f to the component (of ) of that connects to C in . This mismatch can occur at (or ), but does not occur in the toric bundles case. This case requires modifying the -term of the deformation bundle from i) to , where maps all points of to the image of the nodal point .
This bundle is not quite what is needed for geometrical deformation theory. For that, we might take the bundle of sections of that restrict to over A. However, that will not suffice for reasons that follow. In any case, we need some bundle that contains to use in the role of the third non-zero term in the short exact sequences defining the gluing maps of the deformation theory.
There is the complication here that we don’t want the bundle to contribute, via the Quantum Riemann Roch theorem, to the twisted cone . Thus, the present solution to the deformation problem would not be consistent with the twisted cones conditions.
Let be an index value for local cooordinates with non-trivial T-action, as in Section 6. Let X be either the fiber of the toric bundle E or of the total space of the normal line bundle to the projective space bundle of the exceptional divisor.
In Appendix 1 of  , we described T-equivariant line bundles defined as the normal line bundles to the local coordinate hyperplane divisors on the fibers of X; . For = “else” (Section 6), the associated T-equivariant line bundle is ; the corresponding divisor is , by definition. Let us recall that T acts trivially on the pullback
The normal line bundle along the exceptional divisor extends to the T-invariant edges of as (see Section 2.2). The line bundle is in the role of , for the index value = “else”.
Let be an equivalent notation for , etc. There is an ambient set on which the ingredients
are defined by subsets, as the direct sum over in such subsets. In the case of toric bundles, the ambient set is . In the present geometry, in the case that , the ambient set is .
Define (resp., ) to be the direct sum over T-equivariant line bundles with non-trivial torus action for which is negative (resp., non-negative). Similarly, define (resp., ) by replacing C by in the above definitions.
Let us consider the identity
In the toric bundles case, the equation holds only in (T-equivariant) K-theory. Namely, the LHS is missing the direct summands for all values of (which are T-equivariantly trivial), when .
The first set in parentheses on the RHS is interpreted as “an element of the ambient set is considered as ; i.e., as an element of one of , ”. The remaining three direct summands (counting as well) are interpreted similarly by a Ven diagram.
The LHS in the present case, in analogy with the Appendix in  , is dependent upon , so we need to update the LHS by , which we define as follows.
Let us now refer to Appendix A. 7-9 to elaborate.
In the first case, the summand with index contributes to the T-equivariant deformation theory, since the pairing of the bundle on C is .
Let be the connected component of , connecting to C, that maps to (or ). The direct summand L of the coefficient sheaf of is for the gluing construction defined by short exact sequences that glue the separate deformation bundles from the fiber product of stable map moduli spaces. Namely, the direct summand L provides constant deformations (within the T-fixed stratum ) of i), to coincide with the given deformation of at ii). The direct summand from ii) is deduced as a result in  ; it is not a definition. By analogy with the derivation there, in the case that , define
In the case that ,
In the case that ,
which can be understood (from eigenspaces, though not established further here), in terms of . In the formula for the recursion coefficient in terms of and the deformation bundles, the term contributes the factor of .
In the case , ; else . See Section 5.3.
The formula for is given in terms of , , , , and thus can be expressed independently of , as in Corollary A.4 in  ; i.e.,
In particular, depends only on , and not on .
The numerator factor does not contain the terms in the product formula, while the denominator includes the terms. The numerator term then contributes the terms to the class, and cancels the terms from the class. This gives the product formulae in Section 5.4, defined by the analytic continuation in Section 4.1.
The factor of must also be verified by the fixed-point localization formula for gluing nodal curves, in the moduli spaces of stable maps. This gluing was worked out for the toric bundles case in Appendix 3 in . The factor of , from one of the numerator8 factors, in the formulae for (Section 5.4) from fixed-point localization, is used as the Poincaré-dual of A as a submanifold of B, as follows. Consider the terms in the formula for . The leg with the first marked point is mapped to the toric edge in the recursion relation, and the cohomology class representing the toric edge is restricted to A by Poincaré-duality by the factor of as follows ,
The description here counts deformations along on C at , and along on at respectively, glueing them at for a global deformation, when they can be identified for glueing. The overcounting is then removed by subtracting at , by including it in the overall subtraction of in the formula for . This is along the same lines as for toric bundles themselves.
An equivalent description for the deformation theory, would be to keep the deformation on C, remove it on and remove it on . Keeping it on C has the affect in geometry of restricting to . This description gives the correct twisted cones condition for . Thus, we update the preceding deformation theory description accordingly, which only modifies and omits the summand .
5.2. A Key Ingredient of Theorem 2
Let have the same meaning as in Section 5.1 case i), and reserve the notation C for case ii) except that the first marked point will also be allowed the role of nodal point of attaching C to . As in 5.1 the connected component of in the space of T-fixed stable maps into is a fiber product of stable maps into the T-fixed strata of . A tree with root C may be attached, via a nodal point, to stable maps carrying the first marked point of . The smoothing of such a node deforms away from the locus of T-fixed stable maps into . The inverse T-equivariant Euler class of the latter smoothing mode is given by where is the smooth point of in the normalization of that corresponds to the latter nodal point of . Its presence is required by the fixed-point localization technique. The tree with root C yields a cohomology class of B that is proportional to in contribution to the terms of type ii) in . Let us observe that if we substitute , then we get the inverse T-equivariant Euler class of the latter smoothing mode. Let us integrate last over the moduli of where is defined as in i). The precedingly described nodal attachments to , with effectively yield new descendant input terms to the integrals over moduli of type i) in .
If the tree with root C is rooted at there are two possible ways can intersect with the stratum at , according to the decomposition . Namely, the image by constrains to lie in , while may be interpreted as constraining to lie in .
Define “the sum of all contributions to ii) where the first marked point of is contained in C ”.
Let be the completion of (Example 1.8) by allowing additional additive terms that are infinite z series at each order in Novikov’s variables, of the form where and . Denote by restrictions of where is expanded in non-negative powers of z.
When the image of the first marked point lies in (resp., ), the series is constructed as a power series in z, from the cohomology (resp., ) with coefficients in . The source component C from ii) maps to different cases of T-invariant edge curves, depending on the image of the marked point . The trees can be described by the data of Theorem 2, in Section 5.4. The series of our main theorem, which is verified by the techniques of Section 5.4, thus gives a special case of the trees (by Section 5.4). Intuitively, the trees should be described by formulas with some degree of similarity, by reference to the main series.
In the following let us simply note how the numerator twisting factors in the Quantum Lefschetz theorem cancel with some of the factors from the denominator series. This is interpreted in terms of twisted lagrangian cones (an ingredient in Section 5.4) by the Lemmas in Section 7.3.
Begin by writing the main Theorem in terms of , rather than in terms of , using the quantum Lefschetz Theorem. The twisting factors cancel with a denominator series. The particular denominator series depends upon the direct summand
for the restriction of the series . In the first case, the denominator series
is partially cancelled by the preceding numerator series. In the second case, the denominator series
is partially cancelled by the preceding numerator series. The is from the shift of the summation index for , in Section 6.
The preceding observations establish that is the point of the -twisted Lagrangian cone of with input . Let us denote this Lagrangian cone contained in by .
Finally, apply the discussion in 2.2 and 5.1 combined with the general computational details given in  to compute9
Given two of the T-fixed strata and connected by an edge, define submanifolds of each where the strata intersect with edges connecting the two strata. The two submanifolds are diffeomorphic, call it , by the connecting edges.
The discussion in 5.1 and in 5.2 gives the recursion relation along the same line of argument as in Appendix 2 of  :
Let us note the orthogonal10 direct sum decomposition . Both direct summands are --modules. The role of , is understood by observing that are valued in . Recall from Section 2.1 (up to isomorphisms) the inclusion . This gives a way to interpret the -module structure. The map in the recursion relation is applied to a multiple of from the recursion coefficient .
Perhaps the operators in the recursion relation can be composed with suitable projection maps, defined w.r.t. the Lefschetz decomposition so that both sides of the recursion refer to the same ambient vector space, while still sufficing for Theorem 2 (Section 5.4). The author has not worked in this generality.
5.4. Theorem 2
Theorem 2. Points of the overruled Lagrangian cone of the T-equivariant genus-0 Gromov-Witten theory of are characterized by the conditions:
In the case , we are nearly in the case (2.aa) as far as considering the LHS of the recursion relation as nearly a point of , as follows. The normal bundle to A in B doesn’t deform curves into A out of the T-fixed loci in the moduli spaces of stable maps to . Recall that the line bundle is the restriction of the tautological line bundle to the T-fixed section . The normal bundle to A in B thus extends over the T-fixed curves . Then, local sections of the extended normal bundle do deform multiple covers of the latter curve out of the T-fixed stratum in the moduli spaces of stable maps to . Hence, the inverse T-equivariant Euler class of the associated deformation bundle contributes to the fixed-point localization formula in the moduli spaces of stable maps.
Aside from the many cases to consider, the proof is identical to the proof of the corresponding Theorem 2 in .
To prove the equivariant version of the Main Theorem, it suffices to show that satisfies conditions (1.a), (1.b) and (2) of Theorem 2.
The hypergeometric modifications are -series whose coefficients have simple poles at when such values are non-zero, finite order poles at , and essential singularities at .
Thus, we need to show that: (1.a) , (1.b) , and (2) residues at the simple poles satisfy the recursion relations of Theorem 2. We check conditions (2) here by direct calculation of the residues. We check conditions (1.a), (1.b) in Section 7.
Our first goal is to argue that the series is supported in the Mori cone of , . The mechanism that insures
this is to look at the support of the factors of
for which .
Proposition. Any element of may be represented as the sum of a curve whose irreducible components are preserved by the action of T on , and an integer multiple of “the class of a T-invariant in a fiber of the exceptional divisor”.
Proof. Let be a curve class in MC. The action of T on is induced by that on E. We would like to take a lift of the projection , , which we may assume  to be preserved by the action of T on E; i.e., that is represented by such a curve. Apriori, there may be any number of toric edge component curves among the curves representing . These may intersect with a curve component of in . These toric edges each lift to the in such a way that one of their T-fixed points intersects the exceptional divisor at a T-fixed point of a projective space fiber. The preceding irreducible component curves in may be lifted to arbitrary T-fixed sections of the exceptional divisor. This may result in a disconnected lifted curve.
The curve classes are determined by their pairings with elements of . Thus, add the multiple of “the class of a T-invariant in a fiber of the exceptional divisor” to the lift of .
With the Proposition in place, let us now compare and . This will allow us to interpret the support conditions along of the series , in terms of .
A first source of difference between the two comes from the inclusion . Another difference is that the T-invariant curves in do not have any geometric analogues in . However, the latter curve may be represented as the sum of the class of a and the class of a in a fiber of the exceptional divisor. Thus all elements of have geometric analogues in .
Remark. Any curve from a fiber of has a geometric analogue in . The ’s are determined by the geometry of E, and thus have the same meaning whether pulled back to or to . The class is determined11 by the local geometry of the exceptional divisor and thus has the same meaning whether referred to or to .
(2.aa) Residue of at . Given
, rename and , and then redefine and ,
Then, the pairings translate into .
The classes , contribute in terms of (resp. ) to d (resp. ) from the definition in Section 4. Such contributions from are accounted for already, by redefining the summation indices d and as above. Then, the remaining contributions to d (resp. ) are from the Mori cones of the fibers.
Proposition. The series is supported in the Mori cone of .
Proof. For , the support of the series is
characterised by the inequality . For each , the support of
the series is characterised by the inequality . Let us now
argue that the set of solutions to the same inequalities is contained in . By the comparison of with , and by the Remark, it suffices to establish the analogous result for . This follows from the Corollary and the same (strictly speaking, analogous) inequalities that arise there, as a special case of a general result in toric geometry describing the Mori cone in terms of inequalities.
In the following recursion verification, let .
For the recursion relation, the series takes values in the image of . We noted the role of in the recursion coefficient for this purpose, at the end of Section 5.3.
Residue of at . Given
, rename and , and then redefine and ,
Then, the pairings translate into .
Proposition. The series is supported in the Mori cone of .
Proof. For , the inequalities describing the support of the series are and , whose solution set is “a subset of ” “the class of a in a fiber of the exceptional divisor”.
(2.bb) Residue of at . Let be the delta-function
. Given , rename and , and then redefine and ,
The pullbacks vanish. In particular , and
Then, the pairings translate into .
Proposition. If then the series is supported in the Mori cone of .
Proof. The support of the series is characterised by the
inequality . The terms of the series that determine the remaining support conditions are those with ; i.e., . The set coincides with the set . For each , the
support of the series is characterised by the inequality
. For each , the support of the series
is characterised by the inequality . The proof proceeds as
in the case of 2.aa .
Remark. For , the inequalities describing the support of the series are and , whose solution set is “a
subset of ” “the
class of a in a fiber of the exceptional divisor” at each order .
Since for all it follows that . Hence the “index” does not transform presently. Thus, the asymmetry between the factors indexed by and is removed for the purposes of the present recursion process. It follows that the present recursion process is identical to the toric bundles case  ,
In the case then is replaced by . Then reverse the change in the summation index. This gives the recursion relation, as in all other cases. For use “ ” in the transformation of the summation index, following the case.
(2.ab) Residue of at . Given
, rename and , and then redefine and ,
Then, the pairings translate into .
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 Kontsevich, M. (1995) Enumeration of Rational Curves via Torus Actions. In: Dijkgraaf, R.H., Faber, C.F. and van der Geer, G.B.M., Eds., The Moduli Space of Curves. Progress in Mathematics, Birkhäuser Boston, Boston, MA, 335-368.