Abstract
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.
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 Q_{0} = {0, 1, …, ρ} denote the set of vertices of Q and let Q_{1} denote the set of arrows. Without loss of generality, after reordering the vertices if necessary, we may assume that 0∈Q_{0} 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:Q_{1} → Q_{0} for the source and target maps, so that an arrow a∈Q_{1} goes from s(a) to t(a). The dimension vector r = (r_{0}, …, r_{ρ}) lies in ${\mathbb{N}}^{\rho +1}$, and we insist that r_{0} = 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
Example 2.1
Consider the quiver Q given by
so that ρ = 1, n_{01} = n, and the dimension vector r = (1, r). Then $\mathrm{Rep}(Q,\mathbf{r})=\mathrm{Hom}({\mathbb{C}}^{n},{\mathbb{C}}^{r})$, and the θ-stable points are surjections ${\mathbb{C}}^{n}\to {\mathbb{C}}^{r}$. The group G acts by change of basis, and therefore M(Q, r) = Gr(n, r), the Grassmannian of r-dimensional quotients of ${\mathbb{C}}^{n}$. More generally, the quiver
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, $G\cong \prod _{i=1}^{\rho}{\mathrm{GL}}_{1}(\mathbb{C})$ 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 × _{G}Rep(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 $G=\prod _{i=1}^{\rho}\mathrm{GL}({r}_{i})$ 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 ${S}^{\alpha}{\mathbb{C}}^{r}$ of the standard representation of GL(r) [10, ch. 8]. For example, if α is the partition (k) then ${S}^{\alpha}{\mathbb{C}}^{r}={\mathrm{Sym}}^{k}{\mathbb{C}}^{r}$, the kth symmetric power, and if α is the partition (1, 1, …, 1) of length k then ${S}^{\alpha}{\mathbb{C}}^{r}=\stackrel{k}{\bigwedge}{\mathbb{C}}^{r}$, the kth exterior power. Irreducible polynomial representations of G are therefore indexed by tuples (α_{1}, …, 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 $E={\mathbb{C}}^{{r}_{i}}$ is the standard representation of the ith factor of G, then W_{i} = E_{G}. More generally, the representation indexed by (α_{1}, …, α_{ρ}) is $\underset{i=1}{\overset{\rho}{\u2a02}}{S}^{{\alpha}_{i}}{\mathbb{C}}^{{r}_{i}}$, and the corresponding vector bundle on M(Q, r) is $\underset{i=1}{\overset{\rho}{\u2a02}}{S}^{{\alpha}_{i}}{W}_{i}$.
Example 2.2
Consider the vector bundle Sym^{2}W_{1} on Gr(8, 3). By duality—which sends a quotient ${\mathbb{C}}^{8}\to V\to 0$ to a subspace $0\to {V}^{\ast}\to {({\mathbb{C}}^{8})}^{\ast}$—this is equivalent to considering the vector bundle Sym^{2}S*_{1} on the Grassmannian of three-dimensional subspaces of ${({\mathbb{C}}^{8})}^{\ast}$, where S_{1} is the tautological subbundle. A generic symmetric 2-form ω on ${({\mathbb{C}}^{8})}^{\ast}$ determines a regular section of Sym^{2}S*_{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∈Q_{0}, of rank r_{i} such that ${W}_{0}={\mathcal{O}}_{B}$; and | ||||
— | morphisms W_{s(a)} → W_{t(a)}, a∈Q_{1} satisfying the θ-stability condition |
up to isomorphism. Thus M(Q, r) carries universal bundles W_{i}, i∈Q_{0}. 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 $\chi (G)\cong {\mathbb{Z}}^{\rho}$ of G. When tensored with $\mathbb{Q}$, 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 $\chi (G)\otimes \mathbb{Q}$); 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∈Q_{0}.
(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∈Q_{0}, j > i, and all arrows attached to them. Let r(i) = (1, r_{1}, …, 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
(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∈Q_{1}, denote W_{a}: = W*_{s(a)}⊗W_{t(a)}. There is a short exact sequence
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 $\pi :\mathrm{Gr}({\pi}^{\ast}{\mathcal{F}}_{\rho},{r}_{\rho})\to {Y}_{\rho -1}$ from §2d gives a short exact sequence
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
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 $Y=\prod _{i=1}^{\rho}\mathrm{Gr}({H}^{0}({M}_{Q},{W}_{i}),{r}_{i})$ by a tautological section of
Proof.
As vector spaces, there is an isomorphism ${H}^{0}({M}_{Q},{W}_{i})\cong {e}_{0}\mathbb{C}Q{e}_{i}$, where $\mathbb{C}Q$ is the path algebra over $\mathbb{C}$ 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 ${e}_{a}\in \mathbb{C}Q$ be the element associated with the arrow a∈Q_{1}. Thus
Consider the section s of E given by the compositions
The section s vanishes at quotients (V_{1}, …, V_{ρ}) if and only if ${V}_{i}^{\ast}\subset \underset{t(a)=i}{\u2a01}{V}_{s(a)}^{\ast}$; 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
Any variety X with vector bundles V_{i} of rank r_{i} for i = 1, …, ρ and maps H^{0}(M_{Q}, W_{i}) → V_{i} → 0 that factor as
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 $Y=\prod _{i=1}^{\rho}\mathrm{Gr}({H}^{0}({W}_{i}),{r}_{i})$ 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 :
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:
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
and bundle ∧^{2}W_{2}; here and below the vertex numbering is indicated in blue. Then writing it as a product:
with bundle ∧^{2}W_{2}⊕S*_{1}⊗W_{2} (as in equation (3.1)) and dualizing the first factor, we get
with bundle ∧^{2}W_{2}⊕W_{1}⊗W_{2}, 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), W_{1}). This is because the space of sections of W_{1} is ${\mathbb{C}}^{n+1}$, where the image of the section corresponding to $v\in {\mathbb{C}}^{n+1}$ at the point $\varphi :{\mathbb{C}}^{n+1}\to {\mathbb{C}}^{r}$ in Gr(n + 1, r) is ϕ(v). This section vanishes precisely when $v\in \mathrm{ker}\varphi $, so we can consider its zero locus to be $\mathrm{Gr}({\mathbb{C}}^{n+1}/\u27e8v\u27e9,r)\cong \mathrm{Gr}(n,r)$. The restriction of W_{1} to this zero locus Gr(n, r) is W_{1}, and the restriction of the tautological sub-bundle S is $S\oplus {\mathcal{O}}_{\mathrm{Gr}(n,r)}$.
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 $0\to {W}_{1}^{\ast}\to {({\mathbb{C}}^{n+1})}^{\ast}\to {S}^{\ast}\to 0$ shows that a global section of S* is given by a linear map $\psi :{\mathbb{C}}^{n+1}\to \mathbb{C}$. 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 $S\to {\mathbb{C}}^{n+1}$. Splitting ${\mathbb{C}}^{n+1}={\mathbb{C}}^{n}\oplus \mathbb{C}$ and choosing ψ to be projection to the second factor shows that ψ vanishes precisely when $S\subset {\mathbb{C}}^{n}$, that is precisely along Gr(n, r). The restriction of S to this zero locus Gr(n, r) is S, and the restriction of W_{1} is ${W}_{1}\oplus {\mathcal{O}}_{\mathrm{Gr}(n,r)}$.
(c) Grafting
Let Q be a quiver. We say that Q is graftable at i∈Q_{0} 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 Q^{i}. If i is graftable, we call the grafting set of i
Example 4.4
In the quiver below, vertex 1 is not graftable.
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
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
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
Example 4.6
Consider the quiver flag zero locus X given by the quiver in (a) below, with bundle
on the quiver in (c), with arrow bundle S*_{2}⊗W_{1}. We now apply example 4.3 to remove $det{W}_{1}=det{S}_{1}^{\ast}={S}_{1}^{\ast}$. As mentioned in example 4.3, W_{1} on (c) becomes ${W}_{1}\oplus \mathcal{O}$ after removing S*_{1}. The arrow bundle therefore becomes
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:
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 NE^{1}(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 $V={\mathbb{C}}^{N}$ with stability condition $\tau \in \chi (G)=\mathrm{Hom}(G,{\mathbb{C}}^{\ast}).$ 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∈NE^{1}(X) if ${V}^{ss}\ne \varnothing .$ |
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).
One can see this by noting that $\mathrm{Hom}(det({W}_{2}),det({W}_{1}))=0$, and that after taking ∧^{3} (respectively, ∧^{2}) the surjection ${\mathcal{O}}^{\oplus 5}\to {W}_{1}\to 0$ (respectively, ${\mathcal{O}}^{\oplus 10}\to {W}_{2}\to 0$) becomes
— | $\sum _{i=0}^{\rho}{\chi}_{i}{\mathrm{dim}}_{\mathbb{C}}({R}_{i})=0$; and | ||||
— | for any subrepresentation R′ of R, $\sum _{i=0}^{\rho}{\chi}_{i}{\mathrm{dim}}_{\mathbb{C}}({R}_{i}^{\prime})\ge 0.$ |
Consider the character χ = ( − 1, 3) of G, which we lift to a character of $\prod _{i=0}^{\rho}Gl({r}_{i})$ by taking χ = ( − 3, − 1, 3). We will show that there exists a representation R = (R_{0}, R_{1}, R_{2}) 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
Therefore, there cannot exist a Mori embedding of M(Q, r) into M(Q_{0}, 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)//_{θ}T is a toric quiver flag variety. Let λ = (λ_{1}, …, λ_{ρ}) denote an element of $T=\prod _{i=1}^{\rho}{({\mathbb{C}}^{\ast})}^{{r}_{i}}$, where λ_{j} = (λ_{j1}, …, λ_{jrj}). 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
Example 5.3
Let Q be the quiver
Then Q^{ab} is
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_{Qab} = 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 $\prod _{i=1}^{\rho}{S}_{{r}_{i}},$ where S_{ri} 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_{vi1}, …, W_{viri}. It is well-known (e.g. Atiyah–Bott [16]) that
Theorem 5.4 (Martin [15])
There is a graded surjective ring homomorphism
Remark 5.5
This means that any class σ∈H*(M_{Q}) can be lifted (non-uniquely) to a class $\stackrel{~}{\sigma}\in {H}^{\ast}({M}_{{Q}^{\mathrm{ab}}})$. Moreover, $e\cap \stackrel{~}{\sigma}$ 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_{Qab}. Then ϕ(c_{i}(E_{T})) = c_{i}(E_{G}).
Proof.
Recall that
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
Proposition 5.8
Let Amp(Q), Amp(Q^{ab}) denote the ample cones of M_{Q} and M_{Qab}, respectively. Then
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 $\lambda :{\mathbb{C}}^{\ast}\to T$ be a one-parameter subgroup of T such that lim_{t → 0}λ(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∈NE^{1}(M_{Q}), then ${V}_{v}^{ss}(G)\ne \varnothing $, so ${V}_{v}^{ss}(T)\ne \varnothing $, and hence by theorem 5.1 v∈NE^{1}(M_{Qab})^{W}.
Ciocan–Fontanine–Kim–Sabbah use duality to construct a projection [17]
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 $\lambda :{\mathbb{C}}^{\ast}\to G$ such that lim_{t → 0}λ(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^{−1}Tg. Consider λ′ = gλg^{−1}. Then ${\lambda}^{\prime}({\mathbb{C}}^{\ast})\subset T.$ 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 → 0}λ′(t)g · v = lim_{t → 0}gλ(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
Example 5.9
Consider again the example
The Abelianization of this quiver is
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
(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 ${\mathbb{C}}^{N}$ by an ρ-dimensional torus.
Definition 5.10
The GIT data for a toric variety is an ρ-dimensional torus K with cocharacter lattice $L=\mathrm{Hom}({\mathbb{C}}^{\ast},K)$, and N characters D_{1}, …, D_{N}∈L^{∨}, together with a stability condition $w\in {L}^{\vee}\otimes \mathbb{R}$.
These linear data give a toric variety (or Deligne–Mumford stack) as the quotient of an open subset ${U}_{w}\subset {\mathbb{C}}^{N}$ by K, where K acts on ${\mathbb{C}}^{N}$ via the map $K\to {({\mathbb{C}}^{\ast})}^{N}$ defined by the D_{i}. U_{w} is defined as
In [18], the GIT data for a toric quiver flag variety is detailed; we present it slightly differently. The torus is $K={({\mathbb{C}}^{\ast})}^{\rho}$. Let e_{1}, …, e_{ρ} be standard basis of ${L}^{\vee}={\mathbb{Z}}^{\rho}$ and set e_{0}: = 0. Then each a∈Q_{1} 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_{ai}, a_{i}∈Q_{1}. 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_{ai} has t(a_{i}) = i. Fix such a set S = {a_{1}, …, 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(ai)} and so on; let us write f_{ij} = 1 if a_{j} is in the path from 0 to i, and 0 otherwise. Then
Proposition 5.11
Let M(Q, 1) be a toric quiver flag variety. Let c∈Amp(Q), c = (c_{1}, …, c_{ρ}), be an ample class, and suppose that vertex i of the quiver Q satisfies the following condition: for all j∈Q_{0} 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}∈Q_{1} such that the span of the associated divisors D_{aj} 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_{S}c 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 = (c_{1}, …, c_{ρ})∈Amp(Q) and r_{j} > 1, then c_{j} > 0.
Proof.
Consider the Abelianized quiver. For any vertex v∈Q^{ab}_{0}, 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
Theorem 5.13
The nef cone of M(Q, r) is given by the following inequalities. Suppose that a = (a_{1}, …, a_{ρ})∈Pic(M_{Q}). Then a is nef if and only if
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
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
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_{ρrρ}. For ρ, the inequality is a_{ρrρ}≥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
As a∈C it suffices to show that Γ≥0. By the induction hypothesis ${a}_{kl}+\sum _{{v}_{kl}\in {P}_{st}}{a}_{st}\ge 0$, 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
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_{i}is minimal}. By the lemma, $\{1,\dots ,\rho \}=\underset{{T}_{i}\in M}{\u2a06}{T}_{i}.$ Suppose L is given by the character (a_{1}, …, a_{ρ}). Write L as L = ⊗_{Ti∈M}L_{Ti}, where each L_{Ti} comes from a character (b_{1}, …, b_{ρ})∈χ(G) satisfying b_{j} = 0 if j∉T_{i}.
L is nef if and only if all the L_{Ti}, T_{i}∈M are nef. To see this, note that for each j the inequality
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 $\prod {\chi}_{i}^{{a}_{i}}.$ A point of Rep(Q, r) is given by ${({\varphi}_{a})}_{a\in {Q}_{1}},{\varphi}_{a}:{\mathbb{C}}^{{r}_{s(a)}}\to {\mathbb{C}}^{{r}_{t(a)}},$ where G acts by change of basis. A choice of path i → j on the quiver gives an equivariant map $\mathrm{Rep}(Q,\mathbf{r})\to \mathrm{Hom}({\mathbb{C}}^{{r}_{i}},{\mathbb{C}}^{{r}_{j}})$ where G acts on the image by g · ϕ = g_{j}ϕg^{−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 ${d}_{i}=\sum _{k\in {T}_{i}}{r}_{k}{a}_{k}\ge 0$ in the maps given by paths 0 → i. Therefore, f_{i} is a section of the line bundle given by the character χ^{di}_{i}. | ||||
— | If j > i, r_{j} = 1, let f_{j} be any homogeneous polynomial of degree ${d}_{j}=\sum _{k\in {T}_{j}}{r}_{k}{a}_{k}\ge 0$ 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^{−dj}_{h(j)}χ^{dj}_{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^{−rjaj}_{h′(j)}χ^{aj}_{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 $\sigma :=\prod _{j\in {T}_{i}}{f}_{j}:\mathrm{Rep}(Q,\mathbf{r})\to \mathbb{C}$. Then σ defines a section of the line bundle given by character
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,
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 c_{1}(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 $n\in {\mathbb{Z}}_{\ge 0}$ and β∈H_{2}(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
We consider a generating function for descendant invariants called the J-function. Write q^{β} for the element of $\mathbb{Q}[{H}_{2}(Y)]$ representing β∈H_{2}(Y ). Write N(Y ) for the Novikov ring of Y :
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^{〈−KY,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 , H^{1}(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
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_{Qab} = 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_{Qab}. Let PicQ (respectively PicQ^{ab}) denote the Picard group of M_{Q} (respectively of M_{Qab}), and similarly for the cones of effective curves and effective divisors. The isomorphism PicQ → (PicQ^{ab})^{W} gives a projection p:NE_{1}(M_{Qab}) → NE_{1}(M_{Q}). In the bases dual to $det({W}_{1}),\dots ,det({W}_{\rho})$ of PicM_{Q} and W_{ij}, 1 ≤ i ≤ ρ, 1 ≤ j ≤ r_{i} of PicM_{Qab}, this is
For a representation theoretic bundle E_{G} of rank r on M_{Q}, let D_{1}, …, D_{r} be the divisors on M_{Qab} giving the split bundle E_{T}. Given $\stackrel{~}{d}\in {\mathrm{NE}}_{1}({M}_{{Q}^{\mathrm{ab}}})$ define
Example 6.2
The Euler sequence from proposition 2.3 shows that for the tangent bundle T_{MQ}
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
Define the I-function of X⊂M_{Q} to be^{2}
that ${I}_{{T}_{X}}(\stackrel{~}{d})$ is homogeneous of degree $(i({K}_{X}),\stackrel{~}{d})$, so defining the grading of q^{d} to be ( − K_{X}, d), I_{X,MQ}(z) is homogeneous of degree 0. If X is Fano, we can write ${I}_{{T}_{X}}(\stackrel{~}{d})$ as
Since I_{X,MQ} 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 ${H}^{\bullet}({M}_{Q},\mathbb{C})\otimes N({M}_{Q})\otimes \mathbb{C}[[{z}^{-1}]]$. If X is Fano,
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
Remark 6.5
Via the divisor equation and the string equation [25, §1.2], theorem 6.4 determines J_{X}(τ, z) for τ∈H^{0}(X)⊕H^{2}(X).
(c) Proof of theorem 6.4
Givental has defined [26,27] a Lagrangian cone ${\mathcal{L}}_{X}$ in the symplectic vector space ${H}_{X}:={H}^{\ast}(X,\mathbb{C})\otimes N(X)\otimes \mathbb{C}(({z}^{-1}))$ 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 ${\mathcal{L}}_{X}$ of the form
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 $(-z){j}^{\ast}{I}_{X,{M}_{Q}}(-z)\in {\mathcal{L}}_{X}.$
Proof.
Let $Y=\prod _{i=1}^{\rho}\mathrm{Gr}({H}^{0}({W}_{i}),{r}_{i})$. Denote by ${Y}^{ab}=\prod _{i=1}^{\rho}\mathbb{P}{({H}^{0}({W}_{i}))}^{\times {r}_{i}}$ 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 :
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}|_{MQ} 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
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
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].
References
- 1.
Craw A . 2011 Quiver flag varieties and multigraded linear series. Duke Math. J. 156, 469–500. (doi:10.1215/00127094-2010-217) Crossref, ISI, Google Scholar - 2.
King AD . 1994 Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2) 45, 515–530. (doi:10.1093/qmath/45.4.515) Crossref, ISI, Google Scholar - 3.
Coates T, Corti A, Galkin S, Kasprzyk A . 2016 Quantum periods for 3-dimensional Fano manifolds. Geom. Topol. 20, 103–256. (doi:10.2140/gt) Crossref, ISI, Google Scholar - 4.
Coates T, Kasprzyk A, Prince T . 2015 Four-dimensional Fano toric complete intersections. Proc. A. 471,20140704 . (doi:10.1098/rspa.2014.0704) Abstract, Google Scholar - 5.
Webb R . 2018 The Abelian-nonabelian correspondence for I-functions. (http://arxiv.org/abs/1804.07786) Google Scholar - 6.
Coates T, Corti A, Galkin S, Golyshev V, Kasprzyk A . 2013 Mirror symmetry and Fano manifolds. In European congress of mathematics, pp. 285–300. Zürich, Switzerland: European Mathematical Society. Google Scholar - 7.
- 8.
Mori S, Mukai S . 1981/82 Classification of Fano 3-folds with B_{2}≥2. Manuscripta Math. 36, 147–162. (doi:10.1007/BF01170131) Crossref, ISI, Google Scholar - 9.
Craw A, Ito Y, Karmazyn J . 2018 Multigraded linear series and recollement. Math. Z. 289, 535–565. (doi:10.1007/s00209-017-1965-1) Crossref, ISI, Google Scholar - 10.
Fulton W . 1997 Young tableaux. London Mathematical Society Student Texts, vol. 35. Cambridge, UK: Cambridge University Press, Cambridge. Google Scholar - 11.
Hu Y, Keel S . 2000 Mori dream spaces and GIT. Michigan Math. J. 48, 331–348. (doi:10.1307/mmj/1030132722) Crossref, ISI, Google Scholar - 12.
Snow DM . 1986 On the ampleness of homogeneous vector bundles. Trans. Am. Math. Soc. 294, 585–594. (doi:10.1090/S0002-9947-1986-0825723-9) Crossref, ISI, Google Scholar - 13.
Levitt J . 2014 On embeddings of Mori dream spaces. Geom. Dedicata 170, 281–288. (doi:10.1007/s10711-013-9880-z) Crossref, ISI, Google Scholar - 14.
Dolgachev IV, Hu Y . 1998 Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math. 0, 5–56. (doi:10.1007/BF02698859) Crossref, Google Scholar - 15.
Martin S . 2000 Symplectic quotients by a nonabelian group and by its maximal torus. (http://arxiv.org/abs/math/0001002) Google Scholar - 16.
Atiyah MF, Bott R . 1984 The moment map and equivariant cohomology. Topology 23, 1–28. (doi:10.1016/0040-9383(84)90021-1) Crossref, Google Scholar - 17.
Ciocan-Fontanine I, Kim B, Sabbah C . 2008 The abelian/nonabelian correspondence and Frobenius manifolds. Invent. Math. 171, 301–343. (doi:10.1007/s00222-007-0082-x) Crossref, ISI, Google Scholar - 18.
Craw A, Smith GG . 2008 Projective toric varieties as fine moduli spaces of quiver representations. Am. J. Math. 130, 1509–1534. (doi:10.1353/ajm.0.0027) Crossref, ISI, Google Scholar - 19.
Kontsevich M . 1995 Enumeration of rational curves via torus actions. In The moduli space of curves (Texel Island, 1994). Progress in Mathematics, vol. 129, pp. 335–368. Boston, MA: Birkhäuser Boston. Google Scholar - 20.
Behrend K, Fantechi B . 1997 The intrinsic normal cone. Invent. Math. 128, 45–88. (doi:10.1007/s002220050136) Crossref, ISI, Google Scholar - 21.
Li J, Tian G . 1998 Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Am. Math. Soc. 11, 119–174. (doi:10.1090/S0894-0347-98-00250-1) Crossref, ISI, Google Scholar - 22.
Givental A . 1998 A mirror theorem for toric complete intersections. In Topological field theory, primitive forms and related topics (Kyoto, 1996), Progress in Mathematics, vol. 160, pp. 141–175. Boston, MA: Birkhäuser Boston. Google Scholar - 23.
Kim B, Kresch A, Pantev T . 2003 Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee. J. Pure Appl. Algebra 179, 127–136. (doi:10.1016/S0022-4049(02)00293-1) Crossref, ISI, Google Scholar - 24.
Coates T . 2014 The quantum Lefschetz principle for vector bundles as a map between Givental cones. (http://arxiv.org/abs/1405.2893) Google Scholar - 25.
Pandharipande R . 1998 Rational curves on hypersurfaces (after A. Givental). Astérisque 0, 307–340. Séminaire Bourbaki, vol. 1997/98. Google Scholar - 26.
Coates T, Givental A . 2007 Quantum Riemann-Roch, Lefschetz and Serre. Ann. Math. (2) 165, 15–53. (doi:10.4007/annals) Crossref, ISI, Google Scholar - 27.
Givental AB . 2004 Symplectic geometry of Frobenius structures. In Frobenius manifolds, Aspects Math., E36, pp. 91–112. Wiesbaden, Germany: Friedrich Vieweg. Google Scholar - 28.
Bosma W, Cannon J, Playoust C . 1997The Magma algebra system. I. The user language . J. Symbolic Comput. 24, 235–265. (doi:10.1006/jsco.1996.0125) Google Scholar - 29.
- 30.