Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. We prove the Abelian/non-Abelian correspondence in this context: this allows us to compute genus zero Gromov–Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, and disprove a conjecture of Craw. In the appendices (which can be found in the electronic supplementary material), which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way, we find at least 141 new four-dimensional Fano manifolds.

1. Introduction

Quiver flag varieties are a generalization of type A flag varieties that were introduced by Craw [1] based on work of King [2]. They are fine moduli spaces for stable representations of the associated quiver (see §§2c). Like flag varieties and toric complete intersections, quiver flag varieties can be constructed as GIT quotients of a vector space (see §2a). Unlike toric varieties, the quotienting group for a quiver flag variety is in general non-Abelian; this increases the complexity of their structure considerably: specifically, it places them largely outside of the range of known mirror symmetry constructions.

These two perspectives on quiver flag varieties—as fine moduli spaces and as GIT quotients—give two different ways to consider them as ambient spaces. From the moduli space perspective, smooth projective varieties with collections of vector bundles together with appropriate maps between them come with natural maps into the quiver flag variety. From the GIT perspective, one is led to consider subvarieties which occur as zero loci of sections of representation theoretic vector bundles. If the ambient GIT quotient is a toric variety, these subvarieties are toric complete intersections; if the ambient space is a quiver flag variety, we call these subvarieties quiver flag zero loci. While in this paper, we emphasize the GIT quotient perspective, the moduli space perspective should be kept in mind as further evidence of the fact that quiver flag varieties are natural ambient spaces. All smooth Fano varieties of dimension less than or equal to three can be constructed as either toric complete intersections or quiver flag zero loci. These constructions of the Fano threefolds were given in [3]: see theorem A.1 in [3] as well as the explicit constructions in each case. While there is an example in dimension 66 of a Fano variety which is neither a toric complete intersection nor a quiver flag zero locus, one might nevertheless hope that most four-dimensional smooth Fano varieties are either toric complete intersections or quiver flag zero loci. The classification of four-dimensional Fano varieties is open.

This paper studies quiver flag varieties with a view towards understanding them as ambient spaces of Fano fourfolds. Specifically [4] classified smooth four-dimensional Fano toric complete intersections with codimension at most four in the ambient space. This heavily computational search relied on understanding the geometry and quantum cohomology of toric varieties from their combinatorial structure. The guiding motivation of the body of the paper is to establish comparable results for quiver flag varieties to enable the same search to be carried out in this context. For example, we determine the ample cone of a quiver flag variety from the path space of the associated quiver: in this way, we are able to efficiently determine a sufficient condition for whether a quiver flag zero locus is Fano.

The main result of this paper is the proof of the Abelian/non-Abelian correspondence of Ciocan–Fontanine–Kim–Sabbah for Fano quiver flag zero loci. This allows us to compute their genus zero Gromov–Witten invariants.¹ From the perspective of the search for four-dimensional Fano quiver flag zero loci, the importance of this result is that it allows us to compute the quantum period. The quantum period (a generating function built out of certain genus 0 Gromov–Witten invariants) is the invariant that we use to distinguish deformation families of Fano fourfolds: if two quiver flag zero loci have different period sequences, they are not deformation equivalent. The appendices in the electronic supplementary, joint work with Coates and Kasprzyk, describe the search and its results.

Our primary motivation for these results is as follows. There has been much recent interest in the possibility of classifying Fano manifolds using mirror symmetry. It is conjectured that, under mirror symmetry, n-dimensional Fano manifolds should correspond to certain very special Laurent polynomials in n variables [6]. This conjecture has been established in dimensions up to three [3], where the classification of Fano manifolds is known [7,8]. Little is known about the classification of four-dimensional Fano manifolds, but there is strong evidence that the conjecture holds for four-dimensional toric complete intersections [4]. The results of the appendices will provide a first step towards establishing the conjectures for these four-dimensional Fano quiver flag zero loci.

In the appendices in the electronic supplementary material, which are joint work with Tom Coates and Alexander Kasprzyk, we use the structure theory developed here to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. One hundred and forty-one of these quantum periods were previously unknown. Thus we find at least 141 new four-dimensional Fano manifolds. This computation is described in the appendices. The quantum periods, and quiver flag zero loci that give rise to them, are also recorded there. Figure 1 shows the distribution of degree and Euler number for the four-dimensional quiver flag zero loci that we found, and for four-dimensional Fano toric complete intersections.

Figure 1. Degrees and Euler numbers for four-dimensional Fano quiver flag zero loci and toric complete intersections; cf. [4, fig. 5]. Quiver flag zero loci that are not toric complete intersections are highlighted in red. (Online version in colour.)

2. Quiver flag varieties

Quiver flag varieties are generalizations of Grassmannians and type A flag varieties [1]. Like flag varieties, they are GIT quotients and fine moduli spaces. We begin by recalling Craw's construction. A quiver flag variety M(Q, r) is determined by a quiver Q and a dimension vector r. The quiver Q is assumed to be finite and acyclic, with a unique source. Let Q₀ = {0, 1, …, ρ} denote the set of vertices of Q and let Q₁ denote the set of arrows. Without loss of generality, after reordering the vertices if necessary, we may assume that 0∈Q₀ is the unique source and that the number n_ij of arrows from vertex i to vertex j is zero unless i < j. Write s, t:Q₁ → Q₀ for the source and target maps, so that an arrow a∈Q₁ goes from s(a) to t(a). The dimension vector r = (r₀, …, r_ρ) lies in $N^{ρ + 1}$ , and we insist that r₀ = 1. M(Q, r) is defined to be the moduli space of θ-stable representations of the quiver Q with dimension vector r. Here θ is a fixed stability condition defined below, determined by the dimension vector.

(a) Quiver flag varieties as GIT quotients

Consider the vector space

Rep (Q, r) = ⨁_{a \in Q_{1}} Hom (C^{r_{s (a)}}, C^{r_{t (a)}}),

and the action of

GL (r) := \prod_{i = 0}^{ρ} GL (r_{i})

on Rep(Q, r) by change of basis. The diagonal copy of GL(1) in GL(r) acts trivially, but the quotient G: = GL(r)/GL(1) acts effectively; since r₀ = 1, we may identify G with

\prod_{i = 1}^{ρ} GL (r_{i}) .

The quiver flag variety M(Q, r) is the GIT quotient Rep(Q, r)//_θ G, where the stability condition θ is the character of G given by

θ (g) = \prod_{i = 1}^{ρ} det (g_{i}) and g = (g_{1}, \dots, g_{ρ}) \in \prod_{i = 1}^{ρ} GL (r_{i}) .

For the stability condition θ, all semistable points are stable. To identify the θ-stable points in Rep(Q, r), set

s_{i} = \sum_{a \in Q_{1}, t (a) = i} r_{s (a)}

and write

Rep (Q, r) = ⨁_{i = 1}^{ρ} Hom (C^{s_{i}}, C^{r_{i}}) .

Then w = (w_i)^ρ_i=1 is θ-stable if and only if w_i is surjective for all i.

Example 2.1

Consider the quiver Q given by

Inline Graphic

so that ρ = 1, n₀₁ = n, and the dimension vector r = (1, r). Then $Rep (Q, r) = Hom (C^{n}, C^{r})$ , and the θ-stable points are surjections $C^{n} \to C^{r}$ . The group G acts by change of basis, and therefore M(Q, r) = Gr(n, r), the Grassmannian of r-dimensional quotients of $C^{n}$ . More generally, the quiver

Inline Graphic

gives the flag of quotients Fl(n, a, b, …, c).

Quiver flag varieties are non-Abelian GIT quotients unless the dimension vector r = (1, 1, …, 1). In this case, $G ≅ \prod_{i = 1}^{ρ} {GL}_{1} (C)$ is Abelian, and M(Q;r) is a toric variety. We call such M(Q, r) toric quiver flag varieties. Not all toric varieties are toric quiver flag varieties.

(b) Quiver flag varieties as ambient spaces: Quiver flag zero loci

As mentioned in the introduction, GIT quotients have a special class of subvarieties, sometimes called representation theoretic subvarieties. In this subsection, we discuss these subvarieties in the specific case of quiver flag varieties.

We have expressed the quiver flag variety M(Q, r) as the quotient by G of the semistable locus Rep(Q, r)^ss⊂Rep(Q, r). A representation E of G, therefore, defines a vector bundle E_G → M(Q, r) with fibre E; here E_G = E × _GRep(Q, r)^ss. In the appendix in the electronic supplementary material, we will study subvarieties of quiver flag varieties cut out by regular sections of such bundles. If E_G is globally generated, a generic section cuts out a smooth subvariety. We refer to such subvarieties as quiver flag zero loci, and such bundles as representation theoretic bundles. As mentioned above, quiver flag varieties can also be considered natural ambient spaces via their moduli space construction [1,9].

The representation theory of $G = \prod_{i = 1}^{ρ} GL (r_{i})$ is well-understood, and we can use this to write down the bundles E_G explicitly. Irreducible polynomial representations of GL(r) are indexed by partitions (or Young diagrams) of length at most r. The irreducible representation corresponding to a partition α is the Schur power $S^{α} C^{r}$ of the standard representation of GL(r) [10, ch. 8]. For example, if α is the partition (k) then $S^{α} C^{r} = {Sym}^{k} C^{r}$ , the kth symmetric power, and if α is the partition (1, 1, …, 1) of length k then $S^{α} C^{r} = \overset{k}{⋀} C^{r}$ , the kth exterior power. Irreducible polynomial representations of G are therefore indexed by tuples (α₁, …, α_ρ) of partitions, where α_i has length at most r_i. The tautological bundles on a quiver flag variety are representation theoretic: if $E = C^{r_{i}}$ is the standard representation of the ith factor of G, then W_i = E_G. More generally, the representation indexed by (α₁, …, α_ρ) is $⨂_{i = 1}^{ρ} S^{α_{i}} C^{r_{i}}$ , and the corresponding vector bundle on M(Q, r) is $⨂_{i = 1}^{ρ} S^{α_{i}} W_{i}$ .

Example 2.2

Consider the vector bundle Sym²W₁ on Gr(8, 3). By duality—which sends a quotient $C^{8} \to V \to 0$ to a subspace $0 \to V^{*} \to {(C^{8})}^{*}$ —this is equivalent to considering the vector bundle Sym²S*₁ on the Grassmannian of three-dimensional subspaces of ${(C^{8})}^{*}$ , where S₁ is the tautological subbundle. A generic symmetric 2-form ω on ${(C^{8})}^{*}$ determines a regular section of Sym²S*₁, which vanishes at a point V* if and only if the restriction of ω to V* is identically zero. So the associated quiver flag zero locus is the orthogonal Grassmannian OGr(3, 8).

(c) Quiver flag varieties as moduli spaces

To give a morphism to M(Q, r) from a scheme B is the same as to give

—	globally generated vector bundles W_i → B, i∈Q₀, of rank r_i such that $W_{0} = O_{B}$ ; and
—	morphisms W_s(a) → W_t(a), a∈Q₁ satisfying the θ-stability condition

up to isomorphism. Thus M(Q, r) carries universal bundles W_i, i∈Q₀. It is also a Mori dream space (see proposition 3.1 in [1]). The GIT description gives an isomorphism between the Picard group of M(Q, r) and the character group $χ (G) ≅ Z^{ρ}$ of G. When tensored with $Q$ , the fact that this is a Mori dream space (see lemma 4.2 in [11]) implies that this isomorphism induces an isomorphism of wall and chamber structures given by the Mori structure (on the effective cone) and the GIT structure (on $χ (G) \otimes Q$ ); in particular, the GIT chamber containing θ is the ample cone of M(Q, r). The Picard group is generated by the determinant line bundles $det (W_{i})$ , i∈Q₀.

(d) Quiver flag varieties as towers of Grassmannian bundles

Generalizing example 2.1, all quiver flag varieties are towers of Grassmannian bundles [1, theorem 3.3]. For 0 ≤ i ≤ ρ, let Q(i) be the subquiver of Q obtained by removing the vertices j∈Q₀, j > i, and all arrows attached to them. Let r(i) = (1, r₁, …, r_i), and write Y_i = M(Q(i), r(i)). Denote the universal bundle W_j → Y_i by W⁽ⁱ⁾_j. Then there are maps

M (Q, r) = Y_{ρ} \to Y_{ρ - 1} \to \dots \to Y_{1} \to Y_{0} = Spec C,

induced by isomorphisms

Y_{i} ≅ Gr (F_{i}, r_{i})

, where

F_{i}

is the locally free sheaf

F_{i} = ⨁_{a \in Q_{1}, t (a) = i} W_{s (a)}^{(i - 1)}

of rank s_i on Y_i−1. This makes clear that M(Q, r) is a smooth projective variety of dimension

\sum_{i = 1}^{ρ} r_{i} (s_{i} - r_{i})

, and that W_i is the pullback to Y_ρ of the tautological quotient bundle over

Gr (F_{i}, r_{i})

. Thus W_i is globally generated, and

det (W_{i})

is nef. Furthermore, the anti-canonical line bundle of M(Q, r) is

⨂_{a \in Q_{1}} \det {(W_{t (a)})}^{r_{s (a)}} \otimes \det {(W_{s (a)})}^{- r_{t (a)}} .

2.1

In particular, M(Q, r) is Fano if

s_{i} > s_{i}^{'} := \sum_{a \in Q_{1}, s (a) = i} r_{t (a)} .

This condition is not if and only if.

(e) The Euler sequence

Quiver flag varieties, like both Grassmannians and toric varieties, have an Euler sequence.

Proposition 2.3

Let X = M(Q, r) be a quiver flag variety, and for a∈Q₁, denote W_a: = W*_s(a)⊗W_t(a). There is a short exact sequence

0 \to ⨁_{i = 1}^{ρ} W_{i}^{*} \otimes W_{i} \to ⨁_{a \in Q_{1}} W_{a} \to T_{X} \to 0.

Proof.

We proceed by induction on the Picard rank ρ of X. If ρ = 1 then this is the usual Euler sequence for the Grassmannian. Suppose that the proposition holds for quiver flag varieties of Picard rank ρ − 1, for ρ > 1. Then the fibration $π : Gr (π^{*} F_{ρ}, r_{ρ}) \to Y_{ρ - 1}$ from §2d gives a short exact sequence

0 \to W_{ρ}^{*} \otimes W_{ρ} \to π^{*} F_{ρ}^{*} \otimes W_{ρ} \to S^{*} \otimes W_{ρ} \to 0,

where S is the kernel of the projection

π^{*} F_{ρ} \to W_{ρ}

. Note that

π^{*} F_{ρ}^{*} \otimes W_{ρ} = ⨁_{a \in Q_{1}, t (a) = ρ} W_{a} andthat T_{X} = T_{Y_{ρ - 1}} \oplus S^{*} \otimes W_{ρ} .

As S*⊗W_ρ is the relative tangent bundle to π, the proposition follows by induction. ▪

If X is a quiver flag zero locus cut out of the quiver flag variety M(Q, r) by a regular section of the representation theoretic vector bundle E then there is a short exact sequence

0 \to T_{X} \to T_{M (Q, r)} |_{X} \to E \to 0.

2.2

Thus T_X is the K-theoretic difference of representation theoretic vector bundles.

3. Quiver flag varieties as subvarieties

There are three well-known constructions of flag varieties: as GIT quotients, as homogeneous spaces and as subvarieties of products of Grassmannians. Craw's construction gives quiver flag varieties as GIT quotients. General quiver flag varieties are not homogeneous spaces, so the second construction does not generalize. In this section, we generalize the third construction of flag varieties, exhibiting quiver flag varieties as subvarieties of products of Grassmannians. It is this description that will allow us to prove the Abelian/non-Abelian correspondence for quiver flag varieties.

Proposition 3.1

Let M_Q: = M(Q, r) be a quiver flag variety with ρ > 1. Then M_Q is cut out of $Y = \prod_{i = 1}^{ρ} Gr (H^{0} (M_{Q}, W_{i}), r_{i})$ by a tautological section of

E = ⨁_{a \in Q_{1}, s (a) \neq 0} S_{s (a)}^{*} \otimes Q_{t (a)},

where S_i and Q_i are the pullbacks to Y of the tautological sub-bundle and quotient bundle on the i^th factor of Y .

Proof.

As vector spaces, there is an isomorphism $H^{0} (M_{Q}, W_{i}) ≅ e_{0} C Q e_{i}$ , where $C Q$ is the path algebra over $C$ of Q (corollary 3.5, [1]). This isomorphism identifies a basis of global sections of W_i from the set of paths from vertex 0 to i in the quiver. Let $e_{a} \in C Q$ be the element associated with the arrow a∈Q₁. Thus

H^{0} (M_{Q}, W_{i}) = ⨁_{a \in Q_{1}, t (a) = i, s (a) \neq 0} H^{0} (M_{Q}, W_{s (a)}) \oplus ⨁_{a \in Q_{1}, s (a) = 0, t (a) = i} C e_{a} .

Let

F_{i} = ⨁_{t (a) = i} Q_{s (a)}

. Combining the tautological surjective morphisms

H^{0} (M_{Q}, W_{s (a)}) \otimes O_{Y} = H^{0} (Y, Q_{s (a)}) \otimes O_{Y} \to Q_{s (a)},

gives the exact sequence

0 \to ⨁_{t (a) = i, s (a) \neq 0} S_{s (a)} \to H^{0} (M_{Q}, W_{i}) \otimes O_{Y} \to F_{i} \to 0.

Thus

\frac{(H^{0} {(M_{Q}, W_{i})}^{*} \otimes O_{Y})}{F_{i}^{*}} ≅ ⨁_{t (a) = i, s (a) \neq 0} S_{s (a)}^{*},

and it follows that

E = ⨁_{i = 2}^{ρ} Hom (Q_{i}^{*}, (H^{0} {(M_{Q}, W_{i})}^{*} \otimes O_{Y}) / F_{i}^{*})

Consider the section s of E given by the compositions

Q_{i}^{*} \to H^{0} {(M_{Q}, W_{i})}^{*} \otimes O_{Y} \to \frac{(H^{0} {(M_{Q}, W_{i})}^{*} \otimes O_{Y})}{F_{i}^{*}} .

The section s vanishes at quotients (V₁, …, V_ρ) if and only if $V_{i}^{*} \subset ⨁_{t (a) = i} V_{s (a)}^{*}$ ; dually, the zero locus is where there is a surjection F_i → Q_i for each i. We now identify Z(s) with M(Q, r). Since the W_i are globally generated, there is a unique map

f : M_{Q} \to Y = \prod_{i = 1}^{ρ} Gr (H^{0} (M_{Q}, W_{i}), r_{i}),

such that Q_i → Y pulls back to W_i → M(Q, r). By construction of M_Q there are surjections

\oplus_{a \in Q_{1}, t (a) = i} Q_{s (a)} \to W_{i} \to 0,

so f(M_Q)⊂Z(s).

Any variety X with vector bundles V_i of rank r_i for i = 1, …, ρ and maps H⁰(M_Q, W_i) → V_i → 0 that factor as

H^{0} (M_{Q}, W_{i}) \to ⨁_{t (a) = i} V_{s (a)} \to V_{i}

has a unique map to M(Q, r) as the V_i form a flat family of θ-stable representations of Q of dimension r. The (Q_i|_Z(s))^ρ_i=1 on Z(s) give precisely such a set of vector bundles. The surjections H⁰(M_Q, W_i) → Q_i|_Z(s) → 0 follow from the fact that these are restrictions of the tautological bundles on a product of Grassmannians. That these maps factor as required is precisely the condition that s vanishes.

Let g:Z(s) → M_Q be the induced map. By the universal property of M(Q, r), the composition g°f:M_Q → Z(s) → M_Q must be the identity. The composition f°g:Z(s) → M(Q, r) → Y must be the inclusion Z(s) → Y by the universal property of Y . Therefore, Z(s) and M(Q, r) are canonically isomorphic. ▪

Suppose that X is a quiver flag zero locus cut out of M(Q, r) by a regular section of a representation theoretic vector bundle E_G determined by a representation E. The product of Grassmannians $Y = \prod_{i = 1}^{ρ} Gr (H^{0} (W_{i}), r_{i})$ is a GIT quotient V^ss/G for the same group G (one can see this by constructing Y as a quiver flag variety). Therefore, E also determines a vector bundle E′_G on Y :

E_{G}^{'} := E \times \frac{V^{s s}}{G} \to Y .

We see that X is deformation equivalent to the zero locus of a generic section of the vector bundle

F := E_{G}^{'} \oplus ⨁_{a \in Q_{1}, s (a) \neq 0} S_{s (a)}^{*} \otimes Q_{t (a)},

3.1

Although Y is a quiver flag variety, this is not generally an additional model of X as a quiver flag zero locus, as the summand S*_s(a)⊗Q_t(a) in F does not in general come from a representation of G. We refer to the summands of F of this form as arrow bundles.

Remark 3.2

Suppose α is a non-negative Schur partition. Then [12] shows that S^α(Q_i) is globally generated on Y (using the notation as above). This implies that S^α(W_i) is globally generated on M(Q, r).

4. Equivalences of quiver flag zero loci

The representation of a given variety X as a quiver flag zero locus, if it exists, is far from unique. In this section, we describe various methods of passing between different representations of the same quiver flag zero locus. This is important in practice, because our systematic search for four-dimensional quiver flag zero loci described in the appendices in the electronic supplementary material finds a given variety in many different representations. Furthermore, geometric invariants of a quiver flag zero locus X can be much easier to compute in some representations than in others. The observations in this section allow us to compute invariants of four-dimensional Fano quiver flag zero loci using only a few representations, where the computation is relatively cheap, rather than doing the same computation many times and using representations where the computation is expensive (see the appendices in the electronic supplementary material for more details). The results of this section are only used in the appendices in the electronic supplementary material: the rest of the paper is independent.

(a) Dualizing

As we saw in the previous section, a quiver flag zero locus X given by (M(Q, r), E) can be thought of as a zero locus in a product of Grassmannians Y . Unlike general quiver flag varieties, Grassmannians come in canonically isomorphic dual pairs:

Inline Graphic

The isomorphism interchanges the tautological quotient bundle Q with S*, where S is the tautological sub-bundle. One can then dualize some or none of the Grassmannian factors in Y , to get different models of X. Depending on the representations in E, after dualizing, E may still be a representation theoretic vector bundle, or the direct sum of a representation theoretic vector bundle with bundles of the form S*_i⊗W_j. If this is the case, one can then undo the product representation process to obtain another model (M(Q′, r′), E′_G) of X.

Example 4.1

Consider X given by the quiver

Inline Graphic

and bundle ∧²W₂; here and below the vertex numbering is indicated in blue. Then writing it as a product:

Inline Graphic

with bundle ∧²W₂⊕S*₁⊗W₂ (as in equation (3.1)) and dualizing the first factor, we get

Inline Graphic

with bundle ∧²W₂⊕W₁⊗W₂, which is a quiver flag zero locus.

(b) Removing arrows

Example 4.2

Note that Gr(n, r) is the quiver flag zero locus given by (Gr(n + 1, r), W₁). This is because the space of sections of W₁ is $C^{n + 1}$ , where the image of the section corresponding to $v \in C^{n + 1}$ at the point $ϕ : C^{n + 1} \to C^{r}$ in Gr(n + 1, r) is ϕ(v). This section vanishes precisely when $v \in \ker ϕ$ , so we can consider its zero locus to be $Gr (C^{n + 1} / ⟨ v ⟩, r) ≅ Gr (n, r)$ . The restriction of W₁ to this zero locus Gr(n, r) is W₁, and the restriction of the tautological sub-bundle S is $S \oplus O_{Gr (n, r)}$ .

This example generalizes. Let M(Q, r) be a quiver flag variety. A choice of arrow i → j in Q determines a canonical section of W*_i⊗W_j, and the zero locus of this section is M(Q′, r), where Q′ is the quiver obtained from Q by removing one arrow from i → j.

Example 4.3

Similarly, Gr(n, r) is the zero locus of a section of S*, the dual of the tautological sub-bundle, on Gr(n + 1, r + 1). The exact sequence $0 \to W_{1}^{*} \to {(C^{n + 1})}^{*} \to S^{*} \to 0$ shows that a global section of S* is given by a linear map $ψ : C^{n + 1} \to C$ . The image of the section corresponding to ψ at the point s∈S is ψ(s), where we evaluate ψ on s via the tautological inclusion $S \to C^{n + 1}$ . Splitting $C^{n + 1} = C^{n} \oplus C$ and choosing ψ to be projection to the second factor shows that ψ vanishes precisely when $S \subset C^{n}$ , that is precisely along Gr(n, r). The restriction of S to this zero locus Gr(n, r) is S, and the restriction of W₁ is $W_{1} \oplus O_{Gr (n, r)}$ .

(c) Grafting

Let Q be a quiver. We say that Q is graftable at i∈Q₀ if

—	r_i = 1 and 0 < i < ρ;
—	if we remove all of the arrows out of i we get a disconnected quiver.

Call the quiver with all arrows out of i removed Qⁱ. If i is graftable, we call the grafting set of i

{j \in Q_{0} ∣ 0 and j are in different components of Q^{i}} .

Example 4.4

In the quiver below, vertex 1 is not graftable.

Inline Graphic

If we removed the arrow from vertex 0 to vertex 2, then vertex 1 would be graftable and the grafting set would be {2}.

Proposition 4.5

Let M(Q, r) be a quiver flag variety and let i be a vertex of Q that is graftable. Let J be its grafting set. Let Q′ be the quiver obtained from Q by replacing each arrow i → j, where j∈J, by an arrow 0 → j. Then

M (Q, r) = M (Q^{'}, r) .

Proof.

Define V_j: = W*_i⊗W_j for j∈J, and V_j: = W_j otherwise.

Note that by construction of J, for j∈J, there is a surjective morphism

W_{i}^{\oplus d_{i j}} \to W_{j} \to 0.

Here, d_ij is the number of paths i → j. Tensoring this sequence with W*_i shows that V_j is globally generated.

Now we show that the V_j, j∈{0, …, ρ} are a θ-stable representation of Q′. It suffices to check that there are surjective morphisms

⨁_{a \in Q_{1}^{'}, t (a) = j} V_{s (a)} \to V_{j} .

If j∉J, this is just the same surjection given by the fact that the W_i are a θ-stable representation of Q. If j∈J, one must, as above, tensor the sequence from Q with W*_i. The V_j then give a map M(Q, r) → M(Q′, r). Reversing this procedure shows that this is a canonical isomorphism. ▪

Example 4.6

Consider the quiver flag zero locus X given by the quiver in (a) below, with bundle

W_{1} \otimes W_{3} \oplus W_{1}^{\oplus 2} \oplus det W_{1} .

Note we have chosen a different labelling of the vertices for convenience. Writing X inside a product of Grassmannians gives

W_{1} \otimes W_{3} \oplus W_{1}^{\oplus 2} \oplus det W_{1}

on the quiver in (b), with arrow bundle S*₂⊗W₁. Removing the two copies of W₁ using example 4.2 gives

W_{1} \otimes W_{3} \oplus det W_{1},

on the quiver in (c), with arrow bundle S*₂⊗W₁. We now apply example 4.3 to remove $det W_{1} = det S_{1}^{*} = S_{1}^{*}$ . As mentioned in example 4.3, W₁ on (c) becomes $W_{1} \oplus O$ after removing S*₁. The arrow bundle therefore becomes

S_{2}^{*} \otimes (W_{1} \oplus O) = S_{2}^{*} \oplus S_{2}^{*} \otimes W_{1} .

Similarly, W₁⊗W₃ becomes W₃⊕W₁⊗W₃. We can remove the new S*₂ and W₃ summands (reducing the Gr(8, 6) factor to Gr(7, 5) and the Gr(8, 2) factor to Gr(7, 2), respectively). Thus, we see that X is given by W₁⊗W₃ on the quiver in (d), with arrow bundle S*₂⊗W₁. Dualizing at vertices 1 and 2 now gives the quiver in (e), with arrow bundle S*₁⊗W₂⊕S*₁⊗W₃. Finally, undoing the product representation exhibits X as the quiver flag variety for the quiver in (f).

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5. The ample cone

We now discuss how to compute the ample cone of a quiver flag variety. This is essential if one wants to search systematically for quiver flag zero loci that are Fano. In [1], Craw gives a conjecture that would, in particular, solve this problem, by relating a quiver flag variety M(Q, r) to a toric quiver flag variety. We give a counterexample to this conjecture, and determine the ample cone of M(Q, r) in terms of the combinatorics of the quiver: this is theorem 5.13 below. Our method also involves a toric quiver flag variety: the Abelianization of M(Q, r).

(a) The multi-graded Plücker embedding

Given a quiver flag variety M(Q, r), Craw (§5 of [1], example 2.9 in [9]) defines a multi-graded analogue of the Plücker embedding:

p : M (Q, r) ↪ M (Q^{'}, 1) with 1 = (1, \dots, 1) .

Here Q′ is the quiver with the same vertices as Q but with the number of arrows i → j, i < j given by

\dim (\frac{Hom (\det (W_{i}), det (W_{j}))}{S_{i, j}}),

where S_i,j is spanned by maps which factor through maps to

det (W_{k})

with i < k < j. This induces an isomorphism

p^{*} : Pic (X) \otimes R \to Pic (X) \otimes R

that sends

det (W_{i}^{'}) \mapsto det (W_{i}) .

In [1], it is conjectured that this induces a surjection of Cox rings Cox(M(Q′, 1)) → Cox(M(Q, r)). This would give information about the Mori wall and chamber structure of M(Q, r). In particular, by the proof of theorem 2.8 of [13], a surjection of Cox rings together with an isomorphism of Picard groups (which we have here) implies an isomorphism of effective cones.

We provide a counterexample to the conjecture. To do this, we exploit the fact that quiver flag varieties are Mori dream spaces, and so the Mori wall and chamber structure on NE¹(M(Q, r))⊂Pic(M(Q, r)) coincides with the GIT wall and chamber structure. This gives GIT characterizations for effective divisors, ample divisors, nef divisors, and the walls.

Theorem 5.1 (Dolgachev & Hu [14])

Let X be a Mori dream space obtained as a GIT quotient of G acting on $V = C^{N}$ with stability condition $τ \in χ (G) = Hom (G, C^{*}) .$ Identifying Pic(X)≅χ(G), we have that:

—	v∈χ(G) is ample if V^s(v) = V^ss(v) = V^s(τ).
—	v is on a wall if V^ss(v)≠V^s(v).
—	v∈NE¹(X) if $V^{s s} \neq \emptyset .$

When combined with King's characterization [2] of the stable and semistable points for the GIT problem defining M(Q, r), this determines the ample cone of any given quiver flag variety. In theorem 5.13 below we make this effective, characterizing the ample cone in terms of the combinatorics of Q. We can also theorem 5.1 to see a counterexample to conjecture 6.4 in [1].

Example 5.2

Consider the quiver Q and dimension vector r as in (a). The target M(Q′, 1) of the multi-graded Plücker embedding has the quiver Q′ shown in (b).

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One can see this by noting that $Hom (det (W_{2}), det (W_{1})) = 0$ , and that after taking ∧³ (respectively, ∧²) the surjection $O^{\oplus 5} \to W_{1} \to 0$ (respectively, $O^{\oplus 10} \to W_{2} \to 0$ ) becomes

O^{\oplus 10} \to W_{1} \to 0 (respectively, O^{\oplus 45} \to W_{2} \to 0) .

In this case, M(Q′, 1) is a product of projective spaces and so the effective cone coincides with the nef cone, which is just the closure of the positive orthant. The ample cone of M(Q, r) is indeed the positive orthant, as we will see later. However, here we will find an effective character not in the ample cone. We will use King's characterization (definition 1.1 of [2]) of semi-stable points with respect to a character χ of

\prod_{i = 0}^{ρ} G l (r_{i})

: a representation R = (R_i)_i∈Q₀ is semi-stable with respect to χ = (χ_i)^ρ_i=0 if and only if

—	$\sum_{i = 0}^{ρ} χ_{i} \dim_{C} (R_{i}) = 0$ ; and
—	for any subrepresentation R′ of R, $\sum_{i = 0}^{ρ} χ_{i} \dim_{C} (R_{i}^{'}) \geq 0.$

Consider the character χ = ( − 1, 3) of G, which we lift to a character of $\prod_{i = 0}^{ρ} G l (r_{i})$ by taking χ = ( − 3, − 1, 3). We will show that there exists a representation R = (R₀, R₁, R₂) which is semi-stable with respect to χ. The maps in the representation are given by a triple (A, B, C)∈ Mat(3 × 5) × Mat(2 × 3) × Mat(2 × 3). Suppose that

A has full rank, B = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}] and C = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}],

and that R′ is a subrepresentation with dimensions a, b, c. We want to show that −3a − b + 3c≥0. If a = 1 then b = 3, as otherwise the image of A is not contained in R′₁. Similarly, this implies that c = 2. So suppose that a = 0. The maps B and C have no common kernel, so b > 0 implies c > 0, and −b + 3c≥0 as b ≤ 3. Therefore, R is a semi-stable point for χ, and as quiver flag varieties are Mori Dream Spaces, χ is in the effective cone.

Therefore, there cannot exist a Mori embedding of M(Q, r) into M(Q₀, 1) because it would induce an isomorphism of effective cones.

(b) Abelianization

We consider now the toric quiver flag variety associated with a given quiver flag variety M(Q, r) which arises from the corresponding Abelian quotient. Let T⊂G be the diagonal maximal torus. Then the action of G on Rep(Q, r) induces an action of T on Rep(Q, r), and the inclusion i:χ(G)↪χ(T) allows us to interpret the special character θ as a stability condition for the action of T on Rep(Q, r). The Abelian quotient is then Rep(Q, r)//_i(θ)T. Let us see that Rep(Q, r)//_θT is a toric quiver flag variety. Let λ = (λ₁, …, λ_ρ) denote an element of $T = \prod_{i = 1}^{ρ} {(C^{*})}^{r_{i}}$ , where λ_j = (λ_j1, …, λ_{jr_j}). Let (w_a)_a∈Q₁∈Rep(Q, r). Here w_a is an r_t(a) × r_s(a) matrix. The action of λ on (w_a)_a∈Q₁ is defined by

w_{a} (i, j) \mapsto λ_{s (a) i}^{- 1} w_{a} (i, j) λ_{t (a) j} .

Hence this is the same as the group action on the quiver Q^ab with vertices

Q_{0}^{ab} = {v_{i j} : 0 \leq i \leq ρ, 1 \leq j \leq r_{i}},

and the number of arrows between v_ij and v_kl is the number of arrows in the original quiver between vertices i and k. Clearly, i(θ)∈χ(T) is the character prescribed by §2a. Hence

Rep (Q, r) / /_{θ} T = M (Q^{ab}, 1) .

We call Q^ab the Abelianized quiver.

Example 5.3

Let Q be the quiver

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Then Q^ab is

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Martin [15] has studied the relationship between the cohomology of Abelian and non-Abelian quotients. We state his result specialized to quiver flag varieties, then extend this to a comparison of the ample cones. To simplify notation, denote M_Q = M(Q, r), M_Q^ab = M(Q^ab, (1, …, 1)) and V = Rep(Q, r) = Rep(Q^ab, (1, …, 1)). For v∈χ(G), let V^s_v(T) denote the T-stable points of V and V^s_v(G) denote the G-stable points, dropping the subscript if it is clear from context. It is easy to see that V^s(G)⊂V^s(T). The Weyl group W of (G, T) is $\prod_{i = 1}^{ρ} S_{r_{i}},$ where S_{r_i} is the symmetric group on r_i letters. Let π:V^s(G)/T → V^s(G)/G be the projection. The Weyl group acts on the cohomology of M(Q^ab, 1), and also on the Picard group, by permuting the W_{v_i1}, …, W_{v_{ir_i}}. It is well-known (e.g. Atiyah–Bott [16]) that

π^{*} : H^{*} {(\frac{V^{s} (G)}{T})}^{W} ≅ H^{*} (M_{Q}) .

Theorem 5.4 (Martin [15])

There is a graded surjective ring homomorphism

ϕ : H^{*} {(M_{Q^{ab}}, C)}^{W} \to H^{*} (\frac{V^{s} (G)}{T}, C) \overset{π^{*}}{\to} H^{*} (M_{Q}, C),

where the first map is given by the restriction V^s(T)/T → V^s(G)/T. The kernel is the annihilator of

e = \prod_{i = 1}^{ρ} \prod_{1 \leq j, k \leq r_{i}} c_{1} (W_{v_{i j}}^{*} \otimes W_{v_{i k}})

Remark 5.5

This means that any class σ∈H*(M_Q) can be lifted (non-uniquely) to a class $\tilde{σ} \in H^{*} (M_{Q^{ab}})$ . Moreover, $e \cap \tilde{σ}$ is uniquely determined by σ.

Corollary 5.6

Let E be a representation of G defining representation theoretic bundles E_G → M_Q and E_T → M_Q^ab. Then ϕ(c_i(E_T)) = c_i(E_G).

Proof.

Recall that

\begin{aligned} E_{G} & = \frac{(V^{s} (G) \times E)}{G} \to M_{Q} \\ and E_{T} & = \frac{(V^{s} (T) \times E)}{T} \to M_{Q^{ab}} . \end{aligned}

Define

E_{G}^{'} = \frac{(V^{s} (G) \times E)}{T} \to \frac{V^{s} (G)}{T} .

Let f be the inclusion V^s(G)/T → V^s(T)/T. Clearly, f*(E_T) = E_G′ as E_G′ is just the restriction of E_T. Considering the square

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we see that π*(E_G) = E_G′. Then we have that f*(E_T) = π*(E_G), and so in particular f*(c_i(E_T)) = π*(c_i(E_G)). The result now follows from Martin's theorem (theorem 5.4). ▪

Remark 5.7

Note that E_T always splits as a direct sum of line bundles on M(Q^ab, (1, …, 1)), as any representation of T splits into rank one representations. In particular, this means that if (Q, E_G) defines a quiver flag zero locus, (Q^ab, E_T) defines a quiver flag zero locus which is also a toric complete intersection.

The corollary shows that in degree 2, the inverse of Martin's map is

i : c_{1} (W_{i}) \mapsto \sum_{j = 1}^{r_{i}} c_{1} (W_{v_{i j}}) .

In particular, using (2.1), we have that i(ω_{M_Q}) = ω_{M_Q^ab}, where ω_X is the canonical bundle of X.

Proposition 5.8

Let Amp(Q), Amp(Q^ab) denote the ample cones of M_Q and M_Q^ab, respectively. Then

i (Amp (Q)) = Amp {(Q^{ab})}^{W} .

Proof.

Let v be a character for G, denoting its image under i:χ(G)↪χ(T) as v as well. The image of i is W-invariant, and in fact i(χ(G)) = χ(T)^W (this reflects that W-invariant lifts of divisors are unique).

Note that V^ss_v(G)⊂V^ss_v(T). To see this, suppose v∈V is semi-stable for G. Let $λ : C^{*} \to T$ be a one-parameter subgroup of T such that lim_t → 0λ(t) · v exists. By inclusion, λ is a one-parameter subgroup of G, and so 〈v, λ〉≥0 by semi-stability of v. Hence v∈V^ss_v(T). It follows that, if v∈NE¹(M_Q), then $V_{v}^{s s} (G) \neq \emptyset$ , so $V_{v}^{s s} (T) \neq \emptyset$ , and hence by theorem 5.1 v∈NE¹(M_Q^ab)^W.

Ciocan–Fontanine–Kim–Sabbah use duality to construct a projection [17]

p : {NE}_{1} (M_{Q^{ab}}) \to {NE}_{1} (M_{Q}) .

Suppose that α∈Amp(Q). Then for any C∈NE₁(M_Q^ab), i(α) · C = α · p(C) > 0. So i(α)∈Amp(Q^ab)^W.

Let Wall(G)⊂Pic(M_Q) denote the union of all GIT walls given by the G action, and similarly for Wall(T). Recall that v∈Wall(G) if and only if it has a non-empty strictly semi-stable locus. Suppose v∈Wall(G), with v in the strictly semi-stable locus. That is, there exists a non-trivial $λ : C^{*} \to G$ such that lim_t → 0λ(t) · v exists and 〈v, λ〉 = 0. Now we do not necessarily have Im(λ)⊂T, but the image is in some maximal torus, and hence there exists g∈G such that Im(λ)⊂g⁻¹Tg. Consider λ′ = gλg⁻¹. Then $λ^{'} (C^{*}) \subset T .$ Since g · v is in the orbit of v under G, it is semi-stable with respect to G, and hence with respect to T. In fact, it is strictly semi-stable with respect to T, since lim_t → 0λ′(t)g · v = lim_t → 0gλ(t) · v exists, and 〈v, λ′〉 = 〈v, λ〉 = 0. So as a character of T, v has a non-empty strictly semi-stable locus, and we have shown that

i (Wall (G)) \subset Wall {(T)}^{W} .

This means that the boundary of i(Amp(Q)) has empty intersection with Amp(Q^ab)^W. Since both are full dimensional cones in the W invariant subspace, the inclusion i(Amp(Q))⊂Amp(Q^ab)^W is in fact an equality. ▪

Example 5.9

Consider again the example

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The Abelianization of this quiver is

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Walls are generated by collections of divisors that generate cones of codimension 1. We then intersect them with the Weyl invariant subspace, generated by (1, 1, 1, 0, 0) and (0, 0, 0, 1, 1). In this subspace, the walls are generated by

(1, 1, 1, 0, 0), (0, 0, 0, 1, 1), (- 2, - 2, - 2, 3, 3) .

Combined with example 5.2, this determines the wall-and-chamber structure of the effective cone of M(Q, r). That is, it has three walls, each generated by one of v₁: = (1, 0), v₂: = ( − 2, 3) and v₃ = (0, 1). There are two cones generated by (v₁, v₃) and (v₂, v₃), respectively.

(c) The toric case

As a prelude to determining the ample cone of a general quiver flag variety, we first consider the toric case. Recall that a smooth projective toric variety (or orbifold) can be obtained as a GIT quotient of $C^{N}$ by an ρ-dimensional torus.

Definition 5.10

The GIT data for a toric variety is an ρ-dimensional torus K with cocharacter lattice $L = Hom (C^{*}, K)$ , and N characters D₁, …, D_N∈L^∨, together with a stability condition $w \in L^{\lor} \otimes R$ .

These linear data give a toric variety (or Deligne–Mumford stack) as the quotient of an open subset $U_{w} \subset C^{N}$ by K, where K acts on $C^{N}$ via the map $K \to {(C^{*})}^{N}$ defined by the D_i. U_w is defined as

{(z_{1}, \dots, z_{N}) \in C^{N} | w \in Cone (D_{i} : z_{i} \neq 0)},

that is, its elements can have zeroes at z_i, i∈I, only if w is in the cone generated by D_i, i∉I. Assume that all cones given by subsets of the divisors that contain w are full dimensional, as is the case for toric quiver flag varieties. Then the ample cone is the intersection of all of these.

In [18], the GIT data for a toric quiver flag variety is detailed; we present it slightly differently. The torus is $K = {(C^{*})}^{ρ}$ . Let e₁, …, e_ρ be standard basis of $L^{\lor} = Z^{ρ}$ and set e₀: = 0. Then each a∈Q₁ gives a weight D_a = − e_s(a) + e_t(a). The stability condition is 1 = (1, 1, …, 1). Identify L^∨≅PicM(Q, 1). Then D_a = W_a: = W*_s(a)⊗W_t(a).

A minimal generating set for a full dimensional cone for a toric quiver flag variety is given by ρ linearly independent D_{a_i}, a_i∈Q₁. Therefore, for each vertex i with 1 ≤ i ≤ ρ, we need an arrow a_i with either s(a) = i or t(a) = i, and these arrows should be distinct. For the positive span of these divisors to contain 1 requires that D_{a_i} has t(a_i) = i. Fix such a set S = {a₁, …, a_ρ}, and denote the corresponding cone by C_S. As mentioned, the ample cone is the intersection of such cones C_S. The set S determines a path from 0 to i for each i, given by concatenating (backwards) a_i with a_{s(a_i)} and so on; let us write f_ij = 1 if a_j is in the path from 0 to i, and 0 otherwise. Then

e_{i} = \sum_{j = 1}^{ρ} f_{i j} D_{a_{j}} .

This gives us a straightforward way to compute the cone C_S. Let B_S be the matrix with columns given by the D_{a_i}, and let A_S = B⁻¹_S. The columns of A_S are given by the aforementioned paths: the jth column of A_S is

\sum_{i = 1}^{ρ} f_{i j} e_{i}

. If c∈Amp(Q), then A_Sc∈A_S(Amp(Q))⊂A_S(C_S). Since A_SD_{a_i} = e_i, this means that A_Sc is in the positive orthant.

Proposition 5.11

Let M(Q, 1) be a toric quiver flag variety. Let c∈Amp(Q), c = (c₁, …, c_ρ), be an ample class, and suppose that vertex i of the quiver Q satisfies the following condition: for all j∈Q₀ such that j > i, there is a path from 0 to j not passing through i. Then c_i > 0.

Proof.

Choose a collection S of arrows a_j∈Q₁ such that the span of the associated divisors D_{a_j} contains the stability condition 1, and such that the associated path from 0 to j for any j > i does not pass through i. Then the (i, i) entry of A_S is 1 and all other entries of the i^th row are zero. As A_Sc is in the positive orthant, c_i > 0. ▪

Corollary 5.12

Let M(Q, r) be a quiver flag variety, not necessarily toric. If c = (c₁, …, c_ρ)∈Amp(Q) and r_j > 1, then c_j > 0.

Proof.

Consider the Abelianized quiver. For any vertex v∈Q^ab₀, we can always choose a path from the origin to v that does not pass through v_j1: if there is an arrow between v_j1 and v, then there is an arrow between v_j2 and v, so any path through v_j1 can be rerouted through v_j2. Then we obtain that the j1 entry of i(c) is positive-but this is just c_j. ▪

(d) The ample cone of a quiver flag variety

Let M(Q, r) be a quiver flag variety and Q′ be the associated Abelianized quiver. Here paths are defined to be directed paths consisting of at least one arrow. A path passes through a vertex i if either the source or the target of one of the arrows in the path is i. For each i∈{1, …, ρ}, define

T_{i} := {j \in Q_{0} ∣ all paths from the source to v_{j 1} pass through v_{i 1} in the abelianized quiver} .

Note that i∈T_i, as every path from 0 to v_i1 passes through v_i1 by definition. There are no paths from the source to the source, which is therefore not in T_i for any i. If r_i > 1 then T_i = {i}.

Theorem 5.13

The nef cone of M(Q, r) is given by the following inequalities. Suppose that a = (a₁, …, a_ρ)∈Pic(M_Q). Then a is nef if and only if

\sum_{j \in T_{i}} r_{j} a_{j} \geq 0 i = 1, 2, \dots, ρ .

5.1

Proof.

We have already shown that the Weyl invariant part of the nef cone of M_Q′: = M(Q′, 1) is the image of the nef cone of M_Q: = M(Q, r) under the natural map π:Pic(M_Q) → Pic(M_Q′). Label the vertices of Q′ as v_ij, i∈{0, …, ρ}, j∈{1, …, r_i}, and index elements of Pic(M_Q′) as (b_ij). The inequalities defining the ample cone of M_Q′ are given by a choice of arrow A_ij∈Q′₁, t(A_ij) = v_ij for each v_ij. This determines a path P_ij from 0 → v_ij for each vertex v_ij. For each v_ij the associated inequality is

\sum_{v_{i j} \in P_{k l}} b_{k l} \geq 0.

5.2

Suppose that a is nef. We want to show that a satisfies the inequalities (5.1). We do this by finding a collection of arrows such that the inequality (5.2) applied to π(a) is just the inequality (5.1).

It suffices to do this for i such that r_i = 1 (as we have already shown that the inequalities are the same in the r_i > 1 case). Choose a set of arrows such that the associated paths avoid v_i1 if possible: in other words, if v_i1∈P_kl, then assume k∈T_i. Notice that if v_i1∈P_kl₁, then v_i1∈P_kl₂. By assumption π(a) satisfies the i^th inequality associated with this collection of arrows, that is

\sum_{k \in T_{i}} r_{k} a_{k} = \sum_{v_{i 1} \in P_{k l}} a_{k} \geq 0.

Therefore, if C is the cone defined by (5.1), we have shown that Nef(M_Q)⊂C.

Suppose now that a∈C and take a choice of arrows A_kl. Write π(a) = (a_ij). We prove that the inequalities 5.2 are satisfied starting at v_{ρr_ρ}. For ρ, the inequality is a_{ρr_ρ}≥0, which is certainly satisfied. Suppose the (ij + 1), (ij + 2), …, (ρr_ρ) inequalities are satisfied. The inequality we want to establish for (ij) is

\sum_{v_{i 1} \in P_{k l}} a_{k l} = a_{i j} + \sum_{k \in T_{i} - {i}} \sum_{l = 1}^{r_{l}} a_{k l} + Γ = a_{i} + \sum_{k \in T_{i} - {i}} r_{k} a_{k} + Γ \geq 0,

where

Γ = \sum_{s (A_{k l}) = v_{i j}, k \notin T_{i}} (a_{k l} + \sum_{v_{k l} \in P_{s t}} a_{s t}) .

This uses the fact that for k∈T_i, v_i1∈P_kl for all l, and that if k∉T_i, and v_kl∈P_st, we also have that s∉T_i.

As a∈C it suffices to show that Γ≥0. By the induction hypothesis $a_{k l} + \sum_{v_{k l} \in P_{s t}} a_{s t} \geq 0$ , and therefore Γ≥0. This shows that π(a) satisfies (5.2). ▪

(e) Nef line bundles are globally generated

We conclude this section by proving that nef line bundles on quiver flag varieties are globally generated. Craw [1] has shown that the nef line bundles $det (W_{i})$ on M(Q, r) are globally generated; they span a top-dimensional cone contained in the nef cone (and thus all line bundles in this cone are globally generated). Nef line bundles on toric varieties are known to be globally generated. This result for quiver flag varieties will be important for us because in order to use the Abelian/non-Abelian correspondence to compute the quantum periods of quiver flag zero loci, we need to know that the bundles involved are convex. Convexity is a difficult condition to understand geometrically, but it is implied by global generation.

To prove the proposition, we will need the following lemma about the structure of the T_i.

Lemma 5.14

The set {T_i:i∈{1, …, ρ}} has a partial order given by

T_{i} \leq T_{j} \Leftrightarrow T_{j} \subset T_{i},

such that for all j, the set {T_i ≤ T_j} is a chain.

Proof.

Observe that if i∈T_j∩T_k for j < k, then T_k⊂T_j: if all paths from 0 to i1 pass through both j1 and k1, then all paths from 0 to k1 must pass through j1. So k∈T_j and hence T_k⊂T_j. Therefore, if T_j ≤ T_i and T_k ≤ T_i for j < k, then i∈T_j∩T_k, and so T_j ≤ T_k. Hence {T_k|T_k ≤ T_j} is totally ordered for all j. ▪

Proposition 5.15

Let L be a nef line bundle on M(Q, r). Then L is globally generated.

Proof.

Let M: = {T_i|T_iis minimal}. By the lemma, ${1, \dots, ρ} = ⨆_{T_{i} \in M} T_{i} .$ Suppose L is given by the character (a₁, …, a_ρ). Write L as L = ⊗_{T_i∈M}L_{T_i}, where each L_{T_i} comes from a character (b₁, …, b_ρ)∈χ(G) satisfying b_j = 0 if j∉T_i.

L is nef if and only if all the L_{T_i}, T_i∈M are nef. To see this, note that for each j the inequality

\sum_{k \in T_{j}} r_{k} a_{k} \geq 0,

involves terms from a minimal T_i if and only if j∈T_i, in which case it involves only terms from T_i. It therefore suffices to show the statement of the proposition for each L_j. Therefore suppose that {j|a_j≠0}⊂T_i for T_i minimal. If r_i > 1, then T_i = {i}, so

L = det {(W_{i})}^{\otimes a_{i}}

which is globally generated. So we further assume that r_i = 1. For k∈T_i, k > i, define h′(k) such that T_h′(k) is the maximal element such that T_i ≤ T_h′(k) < T_k. This is well defined because the set {T_j|T_j < T_k} is a chain.

A section of L is a G-equivariant section of the trivial line bundle on Rep(Q, r), where the action of G on the line bundle is given by the character $\prod χ_{i}^{a_{i}} .$ A point of Rep(Q, r) is given by ${(ϕ_{a})}_{a \in Q_{1}}, ϕ_{a} : C^{r_{s (a)}} \to C^{r_{t (a)}},$ where G acts by change of basis. A choice of path i → j on the quiver gives an equivariant map $Rep (Q, r) \to Hom (C^{r_{i}}, C^{r_{j}})$ where G acts on the image by g · ϕ = g_jϕg⁻¹_i. If r_i = r_j = 1, such maps can be composed.

For j∈T_i, define f_j as follows:

—	If j = i, let f_i be any homogeneous polynomial of degree $d_{i} = \sum_{k \in T_{i}} r_{k} a_{k} \geq 0$ in the maps given by paths 0 → i. Therefore, f_i is a section of the line bundle given by the character χ^d_i_i.
—	If j > i, r_j = 1, let f_j be any homogeneous polynomial of degree $d_{j} = \sum_{k \in T_{j}} r_{k} a_{k} \geq 0$ in the maps given by paths h′(j) → j. Note that r_h′(j) = 1 as by construction j, h′(j)∈T_h′(j). So f_j defines a section of the line bundle given by character χ^−d_j_h(j)χ^d_j_j.
—	If j > i, r_j > 1, let f_j be a homogeneous polynomial of degree a_k≥0 in the minors of the matrix whose columns are given by the paths h′(j) → j. That is, f_j is a section of the line bundle given by character χ^−r_ja_j_h′(j)χ^a_j_j.

For any x∈Rep(Q, r) which is semi-stable, and for any j∈T_i, there exists an f_j as above with f_j(x)≠0, because j∈T_h′(j). Fixing x, choose such f_j. Define $σ := \prod_{j \in T_{i}} f_{j} : Rep (Q, r) \to C$ . Then σ defines a section of the line bundle given by character

\prod_{j \in T_{i}} χ_{j}^{b_{j}} = χ_{i}^{d_{i}} \cdot \prod_{j \in T_{i}, j \neq i, r_{j} = 1} χ_{h^{'} (j)}^{- d_{j}} χ_{j}^{d_{j}} \cdot \prod_{j \in T_{i}, j \neq i, r_{j} > 1} χ_{h^{'} (j)}^{- r_{j} a_{j}} χ_{j}^{a_{j}} .

We need to check that b_l = a_l for all l. This is obvious for l∈T_i with r_l > 1. For r_l = 1,

b_{l} = \sum_{j \in T_{l}} r_{j} a_{j} - \sum_{k \in T_{l} - {l}, h^{'} (k) = l} \sum_{j \in T_{k}} r_{j} a_{j} .

This simplifies to a_l because T_l − {l} = ⊔_{j∈T_l,h′(j) = l}T_j. Therefore, σ gives a well-defined non-vanishing section of L, so L is globally generated. ▪

6. The Abelian/non-Abelian correspondence

The main theoretical result of this paper, theorem 6.4 below, proves the Abelian/non-Abelian correspondence with bundles [17, conjecture 6.1.1] for quiver flag zero loci. This determines all genus-zero Gromov–Witten invariants, and hence the quantum cohomology, of quiver flag varieties, as well as all genus-zero Gromov–Witten invariants of quiver flag zero loci involving cohomology classes that come from the ambient space. In particular, it determines the quantum period (see definition 6.1) of a quiver flag varieties or quiver flag zero locus X with c₁(T_X)≥0.

(a) A brief review of Gromov–Witten theory

We give a very brief review of Gromov–Witten theory, mainly to fix notation, see [3,17] for more details and references. Let Y be a smooth projective variety. Given $n \in Z_{\geq 0}$ and β∈H₂(Y ), let M_0,n(Y, β) be the moduli space of genus zero stable maps to Y of class β, and with n marked points [19]. While this space may be highly singular and have components of different dimensions, it has a virtual fundamental class [M_0,n(Y, β)]^virt of the expected dimension [20,21]. There are natural evaluation maps ev_i:M_0,n(Y, β) → Y taking a class of a stable map f:C → Y to f(x_i), where x_i∈C is the ith marked point. There is also a line bundle L_i → M_0,n(Y, β) whose fibre at f:C → Y is the cotangent space to C at x_i. The first Chern class of this line bundle is denoted ψ_i. Define

{⟨ τ_{a_{1}} (α_{1}), \dots, τ_{a_{n}} (α_{n}) ⟩}_{n, β} = \int_{{[M_{0, n} (Y, β)]}^{v i r t}} \prod_{i = 1}^{n} e v_{i}^{*} (α_{i}) ψ_{i}^{a_{i}},

6.1

where the integral on the right-hand side denotes cap product with the virtual fundamental class. If a_i = 0 for all i, this is called a (genus zero) Gromov–Witten invariant and the τ notation is omitted; otherwise it is called a descendant invariant. It is deformation invariant.

We consider a generating function for descendant invariants called the J-function. Write q^β for the element of $Q [H_{2} (Y)]$ representing β∈H₂(Y ). Write N(Y ) for the Novikov ring of Y :

N (Y) = {\sum_{β \in {NE}_{1} (Y)} c_{β} q^{β} | \begin{cases} c_{β} \in C, for each d \geq 0 there are only finitely many \\ β such that ω \cdot β \leq d and c_{β} \neq 0 \end{cases}} .

Here ω is the Kähler class on Y . The J-function assigns an element of H*(Y )⊗N(Y )[[z⁻¹]] to every element of H*(Y ), as follows. Let ϕ₁, …, ϕ_N be a homogeneous basis of H*(Y ), and let ϕ¹, …, ϕ^N be the Poincaré dual basis. Then the J-function is given by

J_{X} (τ, z) := 1 + τ z^{- 1} + z^{- 1} \sum_{i} ⟨ ⟨ ϕ_{i} / (z - ψ) ⟩ ⟩ ϕ^{i} .

6.2

Here, 1 is the unit class in H⁰(Y ), τ∈H*(Y ), and

⟨ ⟨ ϕ_{i} / (z - ψ) ⟩ ⟩ = \sum_{β \in {NE}_{1} (Y)} q^{β} \sum_{n = 0}^{\infty} \sum_{a = 0}^{\infty} \frac{1}{n! z^{a + 1}} {⟨ τ_{a} (ϕ_{i}), τ, \dots, τ ⟩}_{n + 1, β} .

6.3

The small J-function is the restriction of the J-function to H⁰(Y )⊕H²(Y ); closed forms for the small J-function of toric complete intersections and toric varieties are known [22].

Definition 6.1

The quantum period G_Y(t) is the component of J(0) along 1∈H^•(Y) after the substitutions z↦1, q^β↦t^{〈−K_Y,β〉}. This is a power series in t.

The quantum period satisfies an important differential equation called the quantum differential equation.

A vector bundle E → Y is defined to be convex if for every genus 0 stable map f:C → Y , H¹(C, f*E) = 0. Globally generated vector bundles are convex. Let X⊂Y be the zero locus of a generic section of a convex vector bundle E → Y and let e denote the total Chern class, which evaluates on a vector bundle F of rank r as

e (F) = λ^{r} + λ^{r - 1} c_{1} (F) + \dots + λ c_{r - 1} (F) + c_{r} (F) .

6.4

The notation here indicates that one can consider e(F) as the

C^{*}

-equivariant Euler class of F, with respect to the canonical action of

C^{*}

on F which is trivial on the base of F and scales all fibres. In this interpretation,

λ \in H_{C^{*}}^{∙} (p t)

is the equivariant parameter. The twisted J-function J_e, E is defined exactly as the J-function (6.2), but replacing the virtual fundamental class which occurs there (via equations (6.3) and (6.1)) by [M_0,n(Y, β)]^virt∩e(E_0,n,β), where E_0,n,β is π_*(ev*_n+1(E)), π:M_0,n+1(Y, β) → M_0,n(Y, β) is the universal curve, and ev_n+1:M_0,n+1(Y, β) → Y is the evaluation map. E_0,n,β is a vector bundle over M_0,n(Y, β), because E is convex. Functoriality for the virtual fundamental class [23] implies that

j^{*} J_{e, E} (τ, z) |_{λ = 0} = J_{X} (j^{*} τ, z),

where j:X → Y is the embedding [24, Theorem 1.1]. Thus one can compute the quantum period of X from the twisted J-function. We will use this to compute the quantum period of Fano fourfolds which are quiver flag zero loci.

The Abelian/non-Abelian correspondence is a conjecture [17] relating the J-functions (and more broadly, the quantum cohomology Frobenius manifolds) of GIT quotients V//G and V//T, where T⊂G is the maximal torus. It also extends to considering zero loci of representation theoretic bundles, by relating the associated twisted J-functions. As the Abelianization V//T of a quiver flag variety V//G is a toric quiver flag variety, the Abelian/non-Abelian correspondence conjectures a closed form for the J-functions of Fano quiver flag zero loci. Ciocan-Fontanine–Kim–Sabbah proved the Abelian/non-Abelian correspondence (with bundles) when V//G is a flag manifold [17]. We will build on this to prove the conjectures when V//G is a quiver flag variety.

(b) The I-function

We give the J-function in the way usual in the literature: first, by defining a cohomology-valued hypergeometric function called the I-function (which should be understood as a mirror object, but we omit this perspective here), then relating the J-function to the I-function. We follow the construction given by Ciocan-Fontanine et al. [17] in our special case. Let X be a quiver flag zero locus given by (Q, E_G) (where we assume E_G is globally generated), and write M_Q = M(Q, r) for the ambient quiver flag variety. Let (Q^ab, E_T) be the associated Abelianized quiver and bundle, M_Q^ab = M(Q^ab, (1, …, 1)). Assume, moreover, that E_T splits into nef line bundles; this implies that E_T is convex. To define the I-function, we need to relate the Novikov rings of M_Q and M_Q^ab. Let PicQ (respectively PicQ^ab) denote the Picard group of M_Q (respectively of M_Q^ab), and similarly for the cones of effective curves and effective divisors. The isomorphism PicQ → (PicQ^ab)^W gives a projection p:NE₁(M_Q^ab) → NE₁(M_Q). In the bases dual to $det (W_{1}), \dots, det (W_{ρ})$ of PicM_Q and W_ij, 1 ≤ i ≤ ρ, 1 ≤ j ≤ r_i of PicM_Q^ab, this is

p : (d_{1, 1}, \dots, d_{1, r_{1}}, d_{2, 1}, \dots, d_{ρ, r_{ρ}}) \mapsto (\sum_{i = 1}^{r_{1}} d_{1 i}, \dots, \sum_{i = 1}^{r_{ρ}} d_{ρ i}) .

For β = (d₁, …, d_ρ), define

ϵ (β) = \sum_{i = 1}^{ρ} d_{i} (r_{i} - 1) .

Then, following [17, equation 3.2.1], the induced map of Novikov rings N(M_Q^ab) → N(M_Q) sends

q^{\tilde{β}} \mapsto {(- 1)}^{ϵ (β)} q^{β},

where

β = p (\tilde{β})

. We write

\tilde{β} \to β

if and only if

\tilde{β} \in {NE}_{1} (M_{Q^{ab}})

and

p (\tilde{β}) = β

For a representation theoretic bundle E_G of rank r on M_Q, let D₁, …, D_r be the divisors on M_Q^ab giving the split bundle E_T. Given $\tilde{d} \in {NE}_{1} (M_{Q^{ab}})$ define

I_{E_{G}} (\tilde{d}) = \frac{\prod_{i = 1}^{r} \prod_{m \leq 0} (D_{i} + m z)}{\prod_{i = 1}^{r} \prod_{m \leq ⟨ \tilde{d}, D_{i} ⟩} (D_{i} + m z)} .

Note that all but finitely many factors cancel here. If E is K-theoretically a representation theoretic bundle, in the sense that there exists A_G, B_G such that

0 \to A_{G} \to B_{G} \to E \to 0,

is an exact sequence, we define

I_{E} (\tilde{d}) = \frac{I_{B_{G} (\tilde{d})}}{I_{A_{G} (\tilde{d})}} .

6.5

Example 6.2

The Euler sequence from proposition 2.3 shows that for the tangent bundle T_{M_Q}

I_{T_{M_{Q}}} (\tilde{d}) = \frac{\prod_{a \in Q_{1}^{a b}} \prod_{m \leq 0} (D_{a} + m z)}{\prod_{a \in Q_{1}^{a b}} \prod_{m \leq ⟨ \tilde{d}, D_{a} ⟩} (D_{a} + m z)} \frac{\prod_{i = 1}^{ρ} \prod_{j \neq k} \prod_{m \leq ⟨ \tilde{d}, D_{i j} - D_{i k} ⟩} (D_{i j} - D_{i k} + m z)}{\prod_{i = 1}^{ρ} \prod_{j \neq k} \prod_{m \leq 0} (D_{i j} - D_{i k} + m z)} .

Here, D_ij is the divisor corresponding to the tautological bundle W_ij for vertex ij, and D_a: = − D_s(a) + D_t(a) is the divisor on M_Q^ab corresponding to the arrow a∈Q^ab₁.

Example 6.3

If X is a quiver flag zero locus in M_Q defined by the bundle E_G, then the adjunction formula (see equation (2.2)) implies that

I_{T_{X}} (\tilde{d}) = I_{T_{M_{Q}}} (\tilde{d}) / I_{E_{G}} (\tilde{d}) .

Define the I-function of X⊂M_Q to be²

I_{X, M_{Q}} (z) = \sum_{d \in {NE}_{1} (M_{Q})} \sum_{\tilde{d} \to d} {(- 1)}^{ϵ (d)} q^{d} I_{T_{X}} (\tilde{d}) .

that $I_{T_{X}} (\tilde{d})$ is homogeneous of degree $(i (K_{X}), \tilde{d})$ , so defining the grading of q^d to be ( − K_X, d), I_{X,M_Q}(z) is homogeneous of degree 0. If X is Fano, we can write $I_{T_{X}} (\tilde{d})$ as

z^{(ω_{X}, \tilde{d})} (b_{0} + b_{1} z^{- 1} + b_{2} z^{- 2} + \dots), b_{i} \in H^{2 i} (X) .

6.6

Since I_{X,M_Q} is invariant under the action of the Weyl group on the D_ij, by viewing these as Chern roots of the tautological bundles W_i we can express it as a function in the Chern classes of the W_i. We can, therefore, regard the I-function as an element of $H^{∙} (M_{Q}, C) \otimes N (M_{Q}) \otimes C [[z^{- 1}]]$ . If X is Fano,

I_{X, M_{Q}} (z) = 1 + z^{- 1} C + O (z^{- 2}),

6.7

where O(z⁻²) denotes terms of the form αz^k with k ≤ − 2 and

C \in H^{0} (M_{Q}, C) \otimes N (M_{Q})

; furthermore (by (6.6)) C vanishes if the Fano index of X is greater than 1.

Theorem 6.4

Let X be a Fano quiver flag zero locus given by (Q, E_G), and let j : X → M_Q be the embedding of X into the ambient quiver flag variety. Then

J_{X} (0, z) = e^{- c / z} j^{*} I_{X, M_{Q}} (z)

where c = j*C.

Remark 6.5

Via the divisor equation and the string equation [25, §1.2], theorem 6.4 determines J_X(τ, z) for τ∈H⁰(X)⊕H²(X).

(c) Proof of theorem 6.4

Givental has defined [26,27] a Lagrangian cone $L_{X}$ in the symplectic vector space $H_{X} := H^{*} (X, C) \otimes N (X) \otimes C ((z^{- 1}))$ that encodes all genus-zero Gromov–Witten invariants of X. Note that J_X(τ, z)∈H_X for all τ. The J-function has the property that ( − z)J_X(τ, − z) is the unique element of $L_{X}$ of the form

- z + τ + O (z^{- 1}),

(see [26, §9]) and this, together with the expression (6.7) for the I-function and the String Equation

J_{X} (τ + c, z) = e^{c / z} J_{X} (τ, z),

shows that theorem 6.4 follows immediately from theorem 6.6 below. Theorem 6.6 is stronger: it does not require the hypothesis that the quiver flag zero locus X be Fano.

Theorem 6.6

Let X be a quiver flag zero locus given by (Q, E_G), and let j : X → M_Q be the embedding of X into the ambient quiver flag variety. Then $(- z) j^{*} I_{X, M_{Q}} (- z) \in L_{X} .$

Proof.

Let $Y = \prod_{i = 1}^{ρ} Gr (H^{0} (W_{i}), r_{i})$ . Denote by $Y^{a b} = \prod_{i = 1}^{ρ} P {(H^{0} (W_{i}))}^{\times r_{i}}$ the Abelianization of Y . In §3, we constructed a vector bundle V on Y such that M_Q is cut out of Y by a regular section of V :

V = ⨁_{i = 2}^{ρ} Q_{i} \otimes \frac{H^{0} {(W_{i})}^{*}}{F_{i}^{*}},

where

F_{i} = ⨁_{t (a) = i} Q_{s (a)}

. V is globally generated and hence convex. It is not representation theoretic, but it is K-theoretically: the sequence

0 \to F_{i}^{*} \otimes Q_{i} \to H^{0} {(W_{i})}^{*} \otimes Q_{i} \to H^{0} {(W_{i})}^{*} \otimes \frac{Q_{i}}{F_{i}^{*}} \to 0

is exact. Let i: M_Q → Y denote the inclusion.

Both Y and M_Q are GIT quotients by the same group; we can therefore canonically identify a representation theoretic vector bundle E′_G on Y such that E′_G|_{M_Q} is E_G. Our quiver flag zero locus X is cut out of Y by a regular section of V ′ = V ⊕E′_G. Note that

\frac{I_{T_{M_{Q}}} (\tilde{d})}{I_{V}} (\tilde{d}) = \frac{I_{T_{Y}} (\tilde{d})}{I_{V^{'}}} (\tilde{d}) .

The I-function I_{X,M_Q} defined by considering X as a quiver flag zero locus in M_Q with the bundle E_G then coincides with the pullback i*I_X,Y of the I-function defined by considering X as a quiver flag zero locus in Y with the bundle V ′. It therefore suffices to prove that

{(- z) (i \circ j)}^{*} I_{X, Y} (- z) \in L_{X} .

We consider a

C^{*}

-equivariant counterpart of the I-function, defined as follows. λ is the equivariant parameter given by the action on the bundle which is trivial on the base, as in (6.4). For a representation theoretic bundle W_G on Y , let D₁, …, D_r be the divisors on Y^ab giving the split bundle W_T, and for

\tilde{d} \in {NE}_{1} (Y^{ab})

set

I_{W_{G}}^{C^{*}} (\tilde{d}) = \frac{\prod_{i = 1}^{r} \prod_{m \leq 0} (λ + D_{i} + m z)}{\prod_{i = 1}^{r} \prod_{m \leq ⟨ \tilde{d}, D_{i} ⟩} (λ + D_{i} + m z)} .

We extend this definition to bundles on Y – such as V ′ – that are only K-theoretically representation theoretic in the same way as (6.5). Let

{\tilde{s}}_{i} := \dim H^{0} (W_{i}) .

Recalling that

I_{T_{Y}} (\tilde{d}) = \frac{\prod_{i = 1}^{ρ} \prod_{j \neq k} \prod_{m \leq ⟨ \tilde{d}, D_{i j} - D_{i k} ⟩} (D_{i j} - D_{i k} + m z)}{\prod_{i = 1}^{ρ} \prod_{j \neq k} \prod_{m \leq 0} (D_{i j} - D_{i k} + m z)} \frac{\prod_{i = 1}^{ρ} \prod_{j = 1}^{r_{i}} \prod_{m \leq 0} {(D_{i j} + m z)}^{{\tilde{s}}_{i}}}{\prod_{i = 1}^{ρ} \prod_{j = 1}^{r_{i}} \prod_{m \leq ⟨ \tilde{d}, D_{i j} ⟩} {(D_{i j} + m z)}^{{\tilde{s}}_{i}}},

we define

I_{X, Y}^{C^{*}} (z) = \sum_{d \in {NE}_{1} (Y)} \sum_{\tilde{d} \to d} {(- 1)}^{ϵ (d)} q^{d} I_{T_{Y}} (\tilde{d}) / I_{V^{'}}^{C^{*}} (\tilde{d}) .

The I-function I_X,Y can be obtained by setting λ = 0 in

I_{X, Y}^{C^{*}}

. In view of [24, Theorem 1.1], it therefore suffices to prove that

(- z) I_{X, Y}^{C^{*}} (- z) \in L_{e, V^{'}},

where

L_{e, V^{'}}

is the Givental cone for the Gromov–Witten theory of Y twisted by the total Chern class e and the bundle V ′.

If V ′ were a representation theoretic bundle, this would follow immediately from the work of Ciocan–Fontanine–Kim–Sabbah: see the proof of theorem 6.1.2 in [17]. In fact V ′ is only K-theoretically representation theoretic, but their argument can be adjusted almost without change to this situation. Suppose that A_G and B_G are representation theoretic vector bundles, and that

0 \to A_{G} \to B_{G} \to V \to 0,

is exact. Then we can also consider an exact sequence

0 \to A_{T} \to B_{T} \to F \to 0,

on the Abelianization, and define V_T: = F. Using the notation of the proof of [17, theorem 6.1.2], the point is that

△ (V) △ (A_{G}) = △ (B_{G}),

Here,

△ (V)

is the twisting operator that appears in the quantum Lefschetz theorem [26]. We can then follow the same argument for

\frac{△ (B_{G})}{△ (A_{G})},

After Abelianizing, we obtain

△ (B_{T}) / △ (A_{T}) = △ (F)

, and conclude that

(- z) I_{X, Y}^{C^{*}} (- z) \in L_{e, V^{'}},

as claimed. This completes the proof. ▪

Data accessibility

The electronic supplementary material contains the details of the computations finding all four-dimensional Fano quiver flag zero loci of codimension at most 4. The results of our computations are also contained there, in machine readable form. See the files called README.txt for details. The code to perform this and similar analyses, using the computational algebra system Magma [28], is available at the repository [29]. A database of Fano quiver flag varieties, which was produced as part of the calculation, is available at the repository [30]. The source code and data, but not the text of this paper, are released under a Creative Commons CC0 license: see the files called COPYING.txt for details. If you make use of the source code or data in an academic or commercial context, you should acknowledge this by including a reference or citation to this paper.

Author's contributions

E.K. is author of the main body of the paper. T.C., E.K. and A.K. are joint authors of the appendices, found in the electronic supplementary material.

Competing interests

I declare I have no competing interests.

Funding

E.K. was supported by the Natural Sciences and Engineering Research Council of Canada, and by the EPSRC Centre for Doctoral Training in Geometry and Number Theory at the Interface, grant no. EP/L015234/1. T.C. was supported by ERC Consolidator grant no. 682602 and EPSRC Programme grant no. EP/N03189X/1. A.K. was supported by EPSRC Fellowship grant no. EP/N022513/1.

Acknowledgments

The computations that underpin this work were performed on the Imperial College HPC cluster. We thank Andy Thomas, Matt Harvey and the Research Computing Service team at Imperial for invaluable technical assistance.

Footnotes

Note

1 Another proof of this, using different methods, has recently been given by Rachel Webb [5].

2 Note that usually the I-function is written as a function in (τ, z), just like the J-function. This is what you obtain if you set τ = 0 (the only case we need).

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4491959.

Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

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Volume 475Issue 2225

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DOI:https://doi.org/10.1098/rspa.2018.0791
PubMed:31236045
Published by:Royal Society
Online ISSN:1471-2946

History:

Manuscript received12/11/2018
Manuscript accepted23/04/2019
Published online15/05/2019
Published in print31/05/2019

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Four-dimensional Fano quiver flag zero loci

Abstract

1. Introduction

2. Quiver flag varieties

(a) Quiver flag varieties as GIT quotients

Example 2.1

(b) Quiver flag varieties as ambient spaces: Quiver flag zero loci

Example 2.2

(c) Quiver flag varieties as moduli spaces

(d) Quiver flag varieties as towers of Grassmannian bundles

(e) The Euler sequence

Proposition 2.3

Proof.

3. Quiver flag varieties as subvarieties

Proposition 3.1

Proof.

Remark 3.2

4. Equivalences of quiver flag zero loci

(a) Dualizing

Example 4.1

(b) Removing arrows

Example 4.2

Example 4.3

(c) Grafting

Example 4.4

Proposition 4.5

Proof.

Example 4.6

5. The ample cone

(a) The multi-graded Plücker embedding

Theorem 5.1 (Dolgachev & Hu [14])

Example 5.2

(b) Abelianization

Example 5.3

Theorem 5.4 (Martin [15])

Remark 5.5

Corollary 5.6

Proof.

Remark 5.7

Proposition 5.8

Proof.

Example 5.9

(c) The toric case

Definition 5.10

Proposition 5.11

Proof.

Corollary 5.12

Proof.

(d) The ample cone of a quiver flag variety

Theorem 5.13

Proof.

(e) Nef line bundles are globally generated

Lemma 5.14

Proof.

Proposition 5.15

Proof.

6. The Abelian/non-Abelian correspondence

(a) A brief review of Gromov–Witten theory

Definition 6.1

(b) The I-function

Example 6.2

Example 6.3

Theorem 6.4

Remark 6.5

(c) Proof of theorem 6.4

Theorem 6.6

Proof.

Data accessibility

Author's contributions

Competing interests

Funding

Acknowledgments

Footnotes

Note

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