Quantum K-theory of Incidence Varieties

We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1,n-1;n) and determine the structure of its (T-equivariant) quantum K-theory ring. Our results are an interplay between geometry and combinatorics. The geometric side concerns Gromov-Witten varieties of 3-pointed genus 0 stable maps to X with markings sent to Schubert varieties, while on the combinatorial side are formulas for the (equivariant) quantum K-theory ring of X. We prove that the Gromov-Witten variety is rationally connected when one of the defining Schubert varieties is a divisor and another is a point. This implies that the (equivariant) K-theoretic Gromov-Witten invariants defined by two Schubert classes and a Schubert divisor class can be computed in the ordinary (equivariant) K-theory ring of X. We derive a positive Chevalley formula for the equivariant quantum K-theory ring of X and a positive Littlewood-Richardson rule for the non-equivariant quantum K-theory ring of X. The Littlewood-Richardson rule in turn implies that non-empty Gromov-Witten varieties given by Schubert varieties in general position have arithmetic genus 0.


Introduction
Let X be a homogeneous space G/P defined by a complex semisimple linear algebraic group G and a parabolic subgroup P .Let d ∈ H 2 (X, Z) be an effective degree and M d := M 0,3 (X, d) be the Kontsevich moduli space parametrizing 3-pointed, genus 0, degree d stable maps to X. Let X u and X v be opposite Schubert varieties in X.Let M d (X u , X v ) := ev −1 1 (X u ) ∩ ev −1 2 (X v ) ⊆ M d be the Gromov-Witten variety of stable maps with one marking sent to X u and another to X v , and Γ d (X u , X v ) := ev 3 (M d (X u , X v )) be the closure of the union of degree d rational curves in X connecting X u and X v .
Buch, Chaput, Mihalcea, and Perrin [16] proved that when X is a cominuscule variety (certain homogeneous spaces with Picard rank one), (1) is cohomologically trivial.This means the pushforward of the structure sheaf of M d (X u , X v ) is the structure sheaf of Γ d (X u , X v ) and the higher direct images of the first sheaf are zero.As a consequence, the (T -equivariant) K-theoretic (3-pointed, genus 0) Gromov-Witten invariants of X satisfy the identity (2) , where χ T X denotes the T -equivariant pushforward along the structure morphism of X. Identity (2) says that these invariants defined on M d can be computed in the equivariant ordinary K-theory ring of X.
When X is not cominuscule, (2) doesn't always hold (see [6, remark 2] for a counterexample).However, Buch and Mihalcea conjecture the following [14].Conjecture 1.1.When X is of type A and X v is a divisor D, the map (1) is cohomologically trivial.
In this paper, we consider the incidence variety X = Fl(1, n − 1; n), which is a special two-step flag variety of type A and one of the simplest homogeneous spaces that are not cominuscule.
Our main geometric result is the following.
Theorem 1.2 ( .= Theorem 3.7).The general fibre of is rationally connected, where D is an opposite Schubert divisor.
Using a result of Kollár [34] (see [7,13]), Theorem 1.2 implies Conjecture 1.1 and the identity (3) The right-hand side of (3) is easily computable using Lenart and Postnikov's Chevalley formula for the equivariant ordinary K-theory ring of G/P [41], because Γ d (X u , D) is either a Schubert variety or the intersection of a Schubert variety with D (see Section 2.2.2 for details).
Our main application is in studying the (T -equivariant) quantum K-theory ring of X.Quantum K-theory was introduced by Givental and Lee [28,38] as a K-theoretic analogue of quantum cohomology.The latter has been widely studied in theoretical physics and various branches of mathematics following the pioneering works of Witten [49] and Kontsevich [35].Rationality properties of Gromov-Witten varieties often have implications in quantum K-theory, see for example [7,8,13,18,39].
Before stating our next main results, we shall introduce a bit more notations.(Opposite) Schubert varieties in X are indexed by W P := {[i, j] : i and j are unequal integers between 1 and n}.
Here [i, j] corresponds to the permutation w ∈ S n such that w(1) = i, w(n) = j and w(2) < • • • < w(n − 1).In order to state our formulas in a simple form, we define W P := {[i, j] ∈ Z × Z : i ≡ j mod n}, and for w = [i, j] ∈ W P we define where w := [i, j] ∈ W P is defined by i ≡ i and j ≡ j (mod n) being the unique such integers between 1 and n, and QK T (X) q := QK T (X) ⊗ Z[q] Z[q, q −1 ] denotes the T -equivariant quantum K-theory ring of X with the deformation parameters q inverted.We let O [1] and O [2] be the classes of the structure sheaves of the two opposite Schubert divisors.Identity (3) allows us to compute K-theoretic Gromov-Witten invariants and derive the following Chevalley formula for the equivariant quantum K-theory ring of X.By [9,Section 5.3], this formula uniquely determines the ring struture.Setting q = 0 recovers Lenart and Postnikov's Chevalley formula for equivariant ordinary K-theory [41] (see also [33] for a Littlewood-Richardson rule for the equivariant ordinary K-theory ring of two-step flag varieties).
A Chevalley formula for the equivariant quantum K-theory ring of the complete flag variety G/B has been proved by Lenart, Naito, and Sagaki [40].Kato has established a ring homomorphism from the equivariant quantum K-theory ring of G/B to that of G/P [31].Theorem 1.3 can also be proved by combining these results and working out the cancellations combinatorially.This approach is taken by Kouno, Lenart, Naito, and Sagaki in [36], where they prove Chevalley formulas for the equivariant quantum K-theory ring of Grassmannians of type A and C. 1We also obtain the following closed formula for Littlewood-Richardson coefficients for the nonequivariant quantum K-theory ring of X.
Theorem 1.4 (= Theorem 5.1).In QK(X), for , As an application, we prove that for general g ∈ G and u, v, w d) has arithmetic genus 0 whenever it is non-empty (see Corollary 5.6).
Positivity of Schubert structure constants has been a topic of interest in the study of (quantum) Schubert calculus [1, 2,5,10,30,42].The above-mentioned formulas that we derive are positive in the appropriate sense (see Corollary 4.20 and Corollary 5.8).
There have also been studies on the powers of the quantum parameters q that appear with nonzero coefficients in a product of Schubert classes in the quantum cohomology and quantum K-theory rings of homogeneous spaces [10,12,27,40,44,47].Our Littlewood-Richardson rule for Fl(1, n − 1; n) shows that the powers of q that appear with nonzero coefficients in O u ⋆ O v form an interval between two degrees (see Corollary 5.9).
Using the ring homomorphism from the quantum K-theory ring of X to that of P n−1 supplied by the aforementioned result of Kato [31], we recover known formulas for projective spaces [9,13,21,36,37] (see Corollary 4.23 and Corollary 5.10).In a different direction, from our formulas one can easily derive analogous formulas for the (equivariant) (quantum) cohomology ring of X. Chevalley formulas are known for the (equivariant) (quantum) cohomology rings of all G/P [43] (see also [27] for the non-equivariant case and [22] for the non-equivariant case in type A), but Littlewood-Richardson rules for two-step flag varieties are only known in the classical case [15,17,23,33].
Our approach for proving Theorem 1.2 is as follows.It suffices to show that when p is a general point in Γ d (X u , D), the Gromov-Witten variety M d (X u , D, p) is rationally connected.By a result of Behrend and Manin [3], the projection p 1 : X → P n−1 induces a morphism We restrict ρ to a morphism from M d (X u , D, p) to the corresponding Gromov-Witten variety for P n−1 .While the restriction is not surjective in general, we use an involution of X to reduce to a situation where it is surjective (see Corollary 3.3).Together with the irreducibility of M d (X u , D, p) (see Proposition 3.6) and parametrizations of the target and the general fibre, a result of Graber, Harris, and Starr [29] implies that M d (X u , D, p) is rationally connected.
This paper is organized as follows.In Section 2, we set up notations for working with the Kontsevich moduli space and the incidence variety and review some known geometric results that will be used in later sections.In Section 3, we study the geometry of Gromov-Witten varieties and prove Theorem 1.2.In Section 4 we apply (3) to compute K-theoretic Gromov-Witten invariants, derive the Chevalley formula for the equivariant quantum K-theory ring of X and discuss some immediate consequences.Section 5 is devoted to the quantum K-theoretic Littlewood-Richardson rule and its implications.
While writing this paper, the author learned through private communications that Sybille Rosset proved independently in her doctoral thesis [46] Theorem 1.2, the nonequivariant case of Theorem 1.3, and the positivity of all structure constants of QK(X).

Preliminaries
Varieties are not assumed to be irreducible, but rational, unirational, and rationally connected varieties are.Let f : X → Y be a morphism of varieties, then the general fibre is a fibre f −1 (u), where u is a closed point chosen from a suitable dense open subset U ⊆ Y .

Moduli Space of Stable Maps.
2.1.1.Overview.Given a smooth complex projective variety X and an effective degree let M g,m (X, d) be the moduli space of maps f from a connected projective nonsingular curve C of genus g with m distinct marked points to X such that f * whose points correspond to isomorphism classes of m-pointed stable maps f : C → X representing the class d, i.e., f * [C] = d, where C is allowed to be an m-pointed connected nodal curve of arithmetic genus g.Evaluating each f at the i-th marked point gives a morphism We write ev for ev The contravariant functor M g,m (X, d) from the category of complex schemes to sets is defined as follows: given a complex scheme S, M g,m (X, d)(S) is the set of isomorphism classes of stable families over S of maps from m-pointed curves of genus g to X representing the class d.M g,m (X, d) is a projective coarse moduli space for M g,m (X, d).
given by composing stable families with ξ and collapsing components of source curves in families as necessary to make the compositions stable.
Proof.We adopt the definitions in [3].
Let S be a complex scheme.A stable map (C, x, f ) over S from an m-pointed curve (of genus g) to Y (of class d) is the same as a stable (Y, τ, α)-map over S.
By [3,Theorem 3.6], there exists a pushforward (D, y, h) of (C, x, f ) of stable maps under (ξ, id, τ, id).Since (D, y, h) is a stable map over S from an m-pointed curve of genus g to Z of class ξ * d, this pushforward defines a map ρ S : M g,m (Y, d)(S) → M g,m (Z, ξ * d)(S).Moreover, this pushforward commutes with base change, making ρ a natural transformation.The proof of [3,Theorem 3.6] shows that the pushforward is constructed by composing stable families with ξ and collapsing components of the source curves in families as necessary to make the compositions stable.

Corollary 2.2. Given a morphism of smooth projective varieties
given by composing stable maps with ξ and collapsing components of source curves as necessary to make the compositions stable.
2.1.3.Stable Maps to Flag Varieties.We are mainly interested in the case where g = 0 and X is a flag variety G/P defined by a complex connected semisimple linear algebraic group G and a parabolic subgroup P .Schubert varieties in X are orbit closures for the action of a Borel subgroup B ⊆ G.We write B − for the Borel subgroup conjugate to B by the longest Weyl group element.
Let X 1 , . . ., X l be Schubert varieties in X with l ≤ 3 (not necessarily in general position).We will be concerned with i.e., the closure of the union of degree d rational curves (images of degree d maps from P 1 ) in X that meet X 1 , and )), which is the closure of the union of degree d rational curves in X connecting X 1 and X 2 .
Using automorphisms of P 1 , we can think of closed points in M 0,3 (X, d) as maps P 1 → X of degree d with marked points at 0, 1, ∞ ∈ P 1 .2.2.Incidence Varieties.

Definition and Basic Properties.
From now on n ≥ 3.
be the projections.Schubert varieties in X are indexed by pairs of unequal integers [i, j] (we use square brackets to distinguish them from degrees), where i and j are each between 1 and n.We denote this indexing set by W P .X embedds in P(C n ) × P(C n * ) with image given by where x 1 , . . ., x n are the standard coordinates on C n and y 1 , . . ., y n are the dual coordinates on C n * .Under this identification, the Schubert variety X [i,j] is the subvariety given by and the opposite Schubert variety X [i,j] is the subvariety given by , where The two Schubert divisors D [1] := X [2,n] and D [2] := X [1,n−1] are cut out by x 1 = 0 and y n = 0, respectively.The involution of P(C n )×P(C n * ) given by swapping x i and y n+1−i induces an involution ϕ of X. ϕ swaps X [i,j] and X ι [i,j] , where In particular, ϕ swap D [1] and D [2] .An effective degree of X is a pair of non-negative integers , T ⊂ G be the maximal torus of diagonal matrices and B ⊂ G be the Borel subgroup of upper triangular matrices.For i = 1, . . ., n, let ε i : T → C * be the character that sends a diagonal matrix to its i-th diagonal entry.Then the root system of (G, B, T ) is and the subset of simple roots is The Weyl group W consists of permutations of {ε 1 , . . ., ε n }, and can be identified with S n .Let ∆ P determines a parabolic subgroup P , which consists of block upper triangular matrices with 3 blocks sized 1, n − 2, 1. W P is the subgroup of W consisting of permutations of {ε 2 , . . ., ε n−1 }.We identify W P with the set of coset representatives of W/W P of minimal length by sending [i, j] to the unique permutation w such that w(1) = i, w(n) = j, and w(2) < • • • < w(n − 1).For u ∈ W , write ℓ(u) for the length of u.The Bruhat order on W gives W P a poset structure.The set of closed points in X can be identified with the quotient set G/P by the map (4) g.(E 1 , E n−1 ) → gP, where E i = Span{e 1 , . . ., e i } is the span of the first i standard basis vectors in C n .In P(C n ) × P(C n * ), gP has coordinates where C(g) is the cofactor matrix of g.Let φ : G → G be defined as Then φ is an involution in the category of algebraic groups.Moreover, φ restricts to an involution of P and an involution of T which sends diag(t 1 , . . ., t n ) to diag(t n −1 , . . ., t 1 −1 ).In particular, φ induces an involution ϕ of G/P .Under our identifications, this is exactly the ϕ defined previously.
When d 1 > 0 and d 2 = 0, since the image of every stable map of degree d is contained in a fibre of p 2 , and every two points in a fibre of p 2 are connected by a line and therefore the image of a stable map of degree d, Similarly, when d 1 = 0 and Then f has degree (1, 1), and ) is the closure of the union of degree d rational curves in X meeting X [i,j] and D [k] .
When d k > 0, every degree d rational curve in X meets D [k] , so Γ d (X [i,j] , D [k] ) is the closure of the union of degree d rational curves in X meeting X When ) is a Richardson variety, i.e., an intersection of two opposite Schubert varieties.Schubert varieties are known to have rational singularities [45].It then follows from [5,Lemma 2] that Richardson varieties have rational singularities.

Helpful Lemmas.
The following lemmas will be used in the proofs of Proposition 3.6 and Theorem 3.7.

Geometry of Gromov-Witten Varieties
By Corollary 2.2, there is a map ρ : M 0,3 (X, d) → M 0,3 (P n−1 , d 1 ) given by composing with p 1 and collapsing components of the source curve as necessary to make the composition stable.In particular, the diagram )) be restrictions of ρ, where g ∈ G is not assumed to be general.Using the automorphism ϕ, we can often reduce to the case where (5) either can be extended to all of P 1 .The extended map f has degree (d Therefore, d ′ 2 ≤ d 1 by the definition of d 1 .By attaching additional components to the domain as necessary using [26, Lemma 12], we get a stable map f : C → X of degree d with image f 1 .Note that the added components collapse after composing with p 1 . )) is non-empty if and only if i = n and in this case it is a point.
, where we think of B 1 , B 2 as We write D + (s) and V + (s) for the non-vanishing and vanishing sets in P(C n ) for the function s, respectively.
If i < n, let )) (when i = n this is guaranteed by a j = 0).Therefore, there exists ) and the conclusion follows.
Corollary 3.3.Assume ( 5) is satisfied, and Let a ∈ C be the 3rd marked point and A = (A 1 , A 2 ) = f (a).If A ∈ g.D [2] , by [26,Lemma 12] we can attach an additional component to C to obtain f ∈ M d (X [i,j] , X [n,1] , g.D [2] ).Otherwise, pick A ′ 2 ∈ g.p 2 (D [2] ) such that Attach P 1 to C by identifying (1 : 0) with a and map this copy of P 1 to X by Replace the 3rd marked point with (0 : 1) on the attached Note that the added component, among others, collapses after composing with p 1 , so This parametrizes ρ−1 (f 1 ) as an open subset of the intersection of some hyperplanes in [32, Theorem 2 and Remark 7], it is reduced and every irreducible component meets M 0,3 (P n , e).On the other hand, We thank an anonymous referee for pointing out the following Proposition was proved independently in [46].Proposition 3.6.For general g, M d (X [i,j] , X [n,1] , g.D [k] ) is irreducible whenever non-empty.
) is equidimensional and every component meets M 0,3 (X, d) and hence Md (X [i,j] , X [n,1] , g.D [k] ).Consider the morphism By Lemma 3.4, the general fibre intersected with Md (X [i,j] , X [n,1] , g.D [k] ) is connected, so at most one component of M d (X [i,j] , X [n,1] , g.D [k] ) maps dominantly.On the other hand, since ρ g,k 3 is closed and surjective by Corollary 3.3 and )) is irreducible by Lemma 3.5, at least one component maps surjectively to it.Therefore, exactly one component Y maps surjectively to By Lemma 2.5 and Lemma 3.2, every irreducible component of the general fibre meets Md (X [i,j] , X [n,1] ), and it follows from Lemma 3.4 that the general fibre is irreducible.Let U be an open dense subset of )) over which fibres of ρ 2 are irreducible and of expected dimension By [32,Theorem 2], every component of is also non-empty and therefore dense in ρ 2 (Z).
By generic flatness, the general fibre of Z → ρ 2 (Z) has dimension )), which is a contradiction.When k = 2, we have Since Z doesn't map surjectively, ( 7) must be an equality, which is equivalent to m ′ = m.This implies Z contains an irreducible fibre of ρ 2 , but then Z meets Y , which is a contradiction.Therefore, M d (X [i,j] , X [n,1] , g.D [k] ) is connected for g general in G.
Theorem 3.7.The general fibre of is rationally connected.
Proof.When d = 0, the domain and target are both isomorphic to X [i,j] ∩ D [k] and ev 3 is an isomorphism.When d > 0, without loss of generality, we assume (5).Case 1: ) is non-empty.By Lemma 2.5 and Lemma 2.6, there is a dense open subset U ⊆ Γ d (X [i,j] , D [k] ) such that for all u ∈ U , ev −1 3 (u) is reduced and every irreducible component meets Md (X [i,j] , D [k] ).On the other hand, since M0 (p 1 (X ) for general g.By Corollary 3.3, Lemma 3.5 and Proposition 3.6, the restriction ρ : )) is a surjective morphism between complex irreducible varieties and the target is rational.By Lemma 2.5 and Lemma 2.6, the general fibre is reduced and every irreducible component meets Md (X [i,j] , D [k] , g.X [n,1] ).It then follows from Lemma 3.4 that the general fibre is rational.[29, We shall construct a unique preimage µ : ), so L intersects p 1 (D [1] ) at one point B 1 .Let {v} be a basis for B 1 .Since are unique and nonzero.Let µ : P 1 → X be given by Then µ is the unique degree (1, 1) morphism µ : Proof.This follows from Theorem 3.7 and [13, Theorem 3.1] (see also [7,Proposition 5.2]).

Chevalley Formula for T -equivariant Quantum K-theory
See [20] for an introduction to equivariant K-theory and [9, Section 2] for a summary of the (equivariant) (quantum) K-theory of flag varieties.We will use the same set-up and notations as in [9].4.1.Pullbacks along Morphisms of Algebraic Groups.Let G, H be algebraic groups and f : G → H a morphism.Let Z be an H-variety.Then Z is also a G-variety with the action given by g.z := f (g).z for all g ∈ G, z ∈ Z, and an H-equivariant O Z -module F is naturally also a G-equivariant O Z -module, which we denote by is a module over the Grothendieck ring K H (Z) of H-equivariant vector bundles over Z, and f * is compatible with the module structures in the sense that the diagram Lemma 4.1.Let G, H be algebraic groups and f : G → H a morphism.
(1) Let λ : H → GL(V ) be a representation, and [V λ ] the corresponding class in K H (pt). Then (2) Let Z be an H-variety and Ω ⊆ Z an H-stable closed subvariety, then (3) Given φ : Y → Z a flat H-equivariant morphism of H-varieties, the diagram Proof.This is clear.
Notation 4.4.From now on, when we write "≡" or " ≡", it shall be understood that the equivalence is taken modulo n.
For w = [i, j] ∈ W P , we define where w := [i, j] ∈ W P is defined by i ≡ i and j ≡ j, and the subscript q stands for localization with respect to {q d : d ∈ Z 2 ≥0 }.Recall that ε i : T → C * is the character that sends a diagonal matrix to its i-th diagonal entry.We let Recall that K T (X) is a free module over K T (pt).The classes O w , w ∈ W P form a basis.For where χ X is the sheaf Euler characteristic map.Let I ∂Xw ⊆ O Xw be the ideal sheaf of ∂X w := X w \ BwP .Then O ∨ w = [I ∂Xw ] (see [5]).We extend the definition to allow for w ∈ W P by letting O ∨ w := q d(w) O ∨ w ∈ QK T (X) q .For i = 1, 2, we will write O [i] for the class of The goal of this subsection is to prove the following T -equivariant Chevalley formula and derive some immediate consequences.Theorem 4.5.In QK T (X) q , for [i, j] ∈ W P , the quantum multiplication by divisor classes is given by The Chevalley formula for the equivariant ordinary K-theory of X can be recovered from Theorem 4.5 by interpreting O w as 0 unless w ∈ W P .We rely on [41] for this special case.The formula is derived from [41, Corollary 7.1 and Corollary 8.2] using that (α 1,2 , . . ., α 1,n ) is a reduced ε 1 -chain in the sense of [41,Definition 5.4].
Note that Corollary 3.9 gives us the following "quantum-equals-classical" type result, which allows us to compute some equivariant K-theoretic Gromov-Witten invariants using the Chevalley formula for equivariant ordinary K-theory.Corollary 4.6.For σ ∈ K T (X), [i, j] ∈ W P , and k = 1, 2, .
For the second equality, see [13,Section 4.1] and the references therein.The third equality follows from the projection formula.The last equality follows from Corollary 3.9.
The rest follows from Lemma 2.3.
Notation 4.7.For σ 1 , σ 2 ∈ K T (X) and w ∈ W P , we write I T (σ 1 , σ 2 ; O w ) for , with the convention that they are 0 when d(w) ≥ 0. We will only allow the third argument to be outside K T (X), and the semicolon is there to remind the reader of it.We do the same in the non-equivariant case.
When specialized to the non-equivariant case, the following proposition says that for general g and k = 1, 2, ( 11) In other words, when M d (g.X u , D [k] , X v ) is non-empty, its arithmetic genus is 0. See Corollary 5.6 for a generalization.Gromov-Witten varieties associated to Schubert varieties in general position are known to have rational singularities [7,Corollary 3.1].By [24, Corollary 4.18(a)], (11) will follow if M d (g.X u , D [k] , X v ) is rationally connected whenever non-empty.We have proved this when X u is a point (see the proof of Theorem 3.7).
By the Chevalley formula for the equivariant ordinary K-theory of X, where The conditions (15) implies a = n > b when i + 1 = j = n and either a > i + 1 or b < j when i + 1 = j < n.
) = 1 and the right hand side of ( 14 where If l < 1, then X [a,b] = X and the right hand side of ( 17) equals 1. Otherwise By Section 2.2.2, a ≥ i and b = l ≤ j.
Proposition 4.10 (Buch-Chaput-Mihalcea-Perrin).For σ 1 , σ 2 ∈ K T (X), their product Proof.By linearity, it suffices to prove that for u, v ∈ W P , ( 19) It suffices to show that O w has the same coefficients on both sides of (19), which means (20)

By definition, ((O
where Taking out the coefficients of q d(w) , we have By [7, Proposition 3.2], Equation ( 20) follows. Since by Proposition 4.10, the verification of Theorem 4.5 requires computing I T (O u , O [i] ; O ∨ w ) for all u ∈ W P , w ∈ W P ; i = 1, 2. In Proposition 4.8, we have computed I T (O u , O [i] ; O w ).We now express each O ∨ w as a linear combination of the classes O w .
Recall from Section 2.2.1 that we identify W P with the set of minimal coset representatives of W/W P .Definition 4.11.For v ∈ W P , we define I(v) to be the set of u ∈ W P such that every element in W between u and v is contained in W P , i.e., The next lemma is a special case of [25, Theorem 1.2 and Section 3].Lemma 4.12.Let f be any function on W P with values in an abelian group and define function g on W P by Then for all v ∈ W P , using that the Richardson variety X w ∩ X u is non-empty if and only if w ≤ u, and in this case it is rational with rational singularities.The result follows.

Now we give explicit computation of I(v)
. Some examples are illustrated in Figure 2.
Proof.This is clear.
Proof.By reasoning as follows, the conclusion can be read from the poset W P .When n = 5, the poset structure of W P is illustrated in Figure 1.
, and that at least one of them is in , and that at least one of them is in W P ∩ [u, v] except when we are in either of the following cases: (1) Definition 4.17.For u = [x, y] ∈ W P , we now let where T X is the tangent bundle of X, and d(u) c 1 (T X ) is the degree of q d(u) in the (graded) quantum cohomology ring of X.
For u, v ∈ W P , we write u ∈ I(v) if Proof of Theorem 4.5.Without loss of generality, we may assume [i, j] ∈ W P .We will prove ( 21) or X [2,1]  or X [2,1] (h) 1 when we are in one of the following situations: (h1 The coefficient of O w on the left-hand side of ( 21) is ), which we will now compute and compare with (a) through (i).By Corollary 4.14, By Proposition 4.8, ) is nonzero only when l ≤ j, and in this case, the value of S k is easily computable using Lemma 4.16.For example, when 1 For another example, when a ≡ 1 and b = 0, We shall consider the three cases in Lemma 4.16 separately.Table 6, Table 7, and Table 8 list values of S k in each case, respectively.When the conditions in the tables are not satisfied, S k = 0. 6 says that out of all S k , only S a and S a−1 can be nonzero when a ≡ 1, and only S a can be nonzero when a ≡ 1.Therefore, we can reduce (22) to a finite sum.When a ≡ 1, equation ( 22) becomes By Table 6, S a and S a−1 are nonzero only when either b = j or b = n − d 2 n for some d 2 > 0, and in either case they are 1 and −1, respectively.Therefore, I T (O [i,j] , O [1] ; (5) 0 otherwise.When a − b = 1, 2 and a ≡ 1, the situations (a1), (a2), (b1), (b2), (d), (e1), (e2), (f3), and (i) are realizable while the other ones are not.In all realizable situations, the value of I T (O [i,j] , O [1] ; O ∨ [a,b] ) agrees with the coefficient of O w on the right hand side of (21).
In this case, the situations (a1), (b1), (d), (e1), (g), and (i) are realizable while the other ones are not.In all realizable situations, the value of I T (O [i,j] , O [1] ; O ∨ [a,b] ) agrees with the coefficient of O w on the right-hand side of (21).
As a consequence, we have the following algorithm that recursively expresses O [i,j] as a polynomial in O [1] , O [2] , q 1 , q 2 with coefficients in K T (pt) for all [i, j] ∈ W P .Algorithm 4.18.Case 1: Assume i < j = n or i > j + 1.
There is a conjecture (see [9] for example) about the positivity of structure constants in QK T (G/P ).In the case of incidence varieties, [9, Conjecture 2.2] specializes to the following statement.
It follows from Theorem 4.5 that the conjecture holds when O v is (a power of q times) the class of a Schubert divisor.Equivalently: Corollary 4.20.For u, w ∈ W P , k = 1, 2, let N w u,k be the coefficient of O w in the product Proof.Without loss of generality, we may assume u = [i, j] ∈ W P .One checks that when i + 1 ≡ j, as polynomials with positive coefficients in the classes [C ε r+1 −εr ] − 1, ( 9) and ( 10) are positive.
Let f : {id} → T be the trivial group homomorphism.By Section 4.1, f * : K T (X) → K(X) has the property that for σ 1 , . . ., σ m ∈ K T (X), ).This implies that the Chevalley formula for non-equivariant quantum K-theory of X can be obtained by replacing all the [C ε r+1 −εr ] with 1 in Theorem 4.5.
where x 1 , . . ., x n are the homogeneous coordinates of P n−1 .

Quantum K-theoretic Littlewood-Richardson Rule
The goal of this section is to prove the following closed formula for quantum K-theoretic Littlewood-Richardson coefficients as well as some corollaries. , Remark 5.2.When i = j + 1, we have where ℓ([j + k, j + l − n]) is computed by Definition 4.17.These are exactly the Seidel products and the same formula holds in quantum cohomology, see [4,11,19].
Lemma 5.4.Let R be an associative ring with unit 1.Let S ⊂ R be a subset that generates R as a Z-algebra.Let M be a left R-module.Let µ : R × M → M be any Z-bilinear map.Assume that for all r ∈ R, s ∈ S, and m ∈ M we have: (1) µ(1, m) = m; (2) µ(rs, m) = µ(r, sm).Then µ(r, m) = rm for all r ∈ R and m ∈ M .
The following was proven independently in [46]: for d ∈ H 2 (X) + , a general g ∈ G, and u, v, w ∈ W P , the variety M d (g.X u , X v , X w ) is rationally connected.
To prove Corollary 5.6, we shall need the following simple lemma.Recall from Definition 4.9 that for [x, y] ∈ W P and d ≥ d([x, y]), we define Γ d (X [x,y]   In view of (37), the only remaining possibility is (42) x = a + d 1 n, y = b − d 2 n.
Assuming (42), M d (g.X u , X v , X w ) = ∅ implies by dimension count that Therefore, nχ(a > b) ≤ a − b, which is a contradiction.
In the non-equivariant case, positivity is verified by Theorem 5.1 as follows.First, recall from Definition 4.17 The proof of this is completely analogous and is therefore omitted.
For the quantum cohomology and quantum K-theory of all homogeneous spaces G/P , it is known that there is a unique minimal power of the quantum parameters q that appear with nonzero coefficient in the product of any two given Schubert classes.This minimal power is the same in quantum cohomology and quantum K-theory, and corresponds to the minimal degree of a stable curve connecting opposite Schubert varieties [12].For the quantum cohomology of cominuscule flag varieties, it is known that the powers of q that appear with nonzero coefficients in the product of any two given Schubert classes form an interval between two degrees [10,44].However, this is not true for the quantum cohomology of complete flag varieties.A counterexample is the product of [Y 164532 ] with itself in the quantum cohomology ring of Y = Fl (6), where the maximal powers of q with nonzero coefficients are not unique and the powers of q with nonzero coefficients do not form an interval [10].For the quantum K-theory of type A complete flag varieties, it is known that there is a unique maximal power of q that appear with nonzero coefficient in the product of a Schubert divisor class and a Schubert class; however, the powers of q that appear with nonzero coefficients don't always form an interval [40].It remains an open problem whether there is always a unique maximal power of q that appear with nonzero coefficient in the product of two given Schubert classes in the quantum K-theory ring of a homogeneous space.In the case of incidence varieties, it is immediate from Theorem 5.1 that: Corollary 5.9.In QK(X), for u, v ∈ W P , the powers of q that appear with nonzero coefficients in O u ⋆ O v form an interval between two degrees.
Using [31,Theorem 2.18] with the projection p 1 : X → P n−1 , we get the following closed formula for quantum K-theoretic Littlewood-Richardson coefficients for projective spaces, stated in terms of Notation 4.22.This is a special case of [13,Theorem 5.4].In particular, the quantum K-theory ring and quantum cohomology rings are isomorphic for projective spaces.

Lemma 2 . 5 .
Let φ : Y → Z be a dominant morphism between irreducible varieties and U be a non-empty open subset of Y .Then every irreducible component of the general fibre meets U .Proof.Let W be an irreducible component of Y \ U .If φ(W ) is not dense in Z, then the general fibre avoids W .If φ(W ) is dense in Z, then by generic flatness, the irreducible components of the intersection of the general fibre with W have strictly smaller dimension than do irreducible components of the general fibre.Therefore, no irreducible component of the general fibre is contained in W . Lemma 2.6.Let φ : Y → Z be a dominant morphism between irreducible varieties over a field of characteristic 0. Then the general fibre is reduced.Proof.Without loss of generality, assume Z = SpecA, Y = SpecB.Then the generic fibre is Spec(B ⊗ A A 0 ).φ being dominant implies A ⊆ B. So B ⊗ A A 0 ⊆ B 0 is a domain.Since the characteristic is 0, the generic fibre is geometrically reduced by [48, Lemma 020I], and the conclusion follows from [48, Lemma 0578].

X
and write C η for the one-dimensional T -representation corresponding to the character η, and [C η ] for the corresponding class in K T (pt).

Conjecture 4 . 19 .
For u, v, w ∈ W P , let N w u,v be the coefficient of O w in the product
Proposition 2.1.Given a morphism of smooth projective varieties ξ : Y → Z and d ∈ H 2 (Y ) + , there exists a natural transformation