Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Four-dimensional Fano quiver flag zero loci

Elana Kalashnikov

Elana Kalashnikov

Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK

[email protected]

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    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.

    Figure 1.

    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 nij of arrows from vertex i to vertex j is zero unless i < j. Write st:Q1 → Q0 for the source and target maps, so that an arrow aQ1 goes from s(a) to t(a). The dimension vector r = (r0, …, rρ) lies in Nρ+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)=aQ1Hom(Crs(a),Crt(a)),
    and the action of GL(r):=i=0ρGL(ri) 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 i=1ρGL(ri). The quiver flag variety M(Q, r) is the GIT quotient Rep(Q, r)//θG, where the stability condition θ is the character of G given by
    θ(g)=i=1ρdet(gi)andg=(g1,,gρ)i=1ρGL(ri).
    For the stability condition θ, all semistable points are stable. To identify the θ-stable points in Rep(Q, r), set si=aQ1,t(a)=irs(a) and write
    Rep(Q,r)=i=1ρHom(Csi,Cri).
    Then w = (wi)ρi=1 is θ-stable if and only if wi 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(Cn,Cr), and the θ-stable points are surjections CnCr. The group G acts by change of basis, and therefore M(Q, r) = Gr(n, r), the Grassmannian of r-dimensional quotients of Cn. 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, Gi=1ρGL1(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 EG → M(Q, r) with fibre E; here EG = 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 EG 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=i=1ρGL(ri) is well-understood, and we can use this to write down the bundles EG 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 SαCr of the standard representation of GL(r) [10, ch. 8]. For example, if α is the partition (k) then SαCr=SymkCr, the kth symmetric power, and if α is the partition (1, 1, …, 1) of length k then SαCr=kCr, the kth exterior power. Irreducible polynomial representations of G are therefore indexed by tuples (α1, …, αρ) of partitions, where αi has length at most ri. The tautological bundles on a quiver flag variety are representation theoretic: if E=Cri is the standard representation of the ith factor of G, then Wi = EG. More generally, the representation indexed by (α1, …, αρ) is i=1ρSαiCri, and the corresponding vector bundle on M(Q, r) is i=1ρSαiWi.

    Example 2.2

    Consider the vector bundle Sym2W1 on Gr(8, 3). By duality—which sends a quotient C8V0 to a subspace 0V(C8)—this is equivalent to considering the vector bundle Sym2S*1 on the Grassmannian of three-dimensional subspaces of (C8), where S1 is the tautological subbundle. A generic symmetric 2-form ω on (C8) 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 Wi → B, iQ0, of rank ri such that W0=OB; and

    morphisms Ws(a) → Wt(a), aQ1 satisfying the θ-stability condition

    up to isomorphism. Thus M(Q, r) carries universal bundles Wi, iQ0. 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)Zρ 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)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(Wi), iQ0.

    (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 jQ0, j > i, and all arrows attached to them. Let r(i) = (1, r1, …, ri), and write Yi = M(Q(i), r(i)). Denote the universal bundle Wj → Yi by W(i)j. Then there are maps

    M(Q,r)=YρYρ1Y1Y0=SpecC,
    induced by isomorphisms YiGr(Fi,ri), where Fi is the locally free sheaf
    Fi=aQ1,t(a)=iWs(a)(i1)
    of rank si on Yi−1. This makes clear that M(Q, r) is a smooth projective variety of dimension i=1ρri(siri), and that Wi is the pullback to Yρ of the tautological quotient bundle over Gr(Fi,ri). Thus Wi is globally generated, and det(Wi) is nef. Furthermore, the anti-canonical line bundle of M(Q, r) is
    aQ1det(Wt(a))rs(a)det(Ws(a))rt(a).2.1
    In particular, M(Q, r) is Fano if si>si:=aQ1,s(a)=irt(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 aQ1, denote Wa: = W*s(a)Wt(a). There is a short exact sequence

    0i=1ρWiWiaQ1WaTX0.

    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(πFρ,rρ)Yρ1 from §2d gives a short exact sequence

    0WρWρπFρWρSWρ0,
    where S is the kernel of the projection πFρWρ. Note that
    πFρWρ=aQ1,t(a)=ρWaandthatTX=TYρ1SWρ.
    As S*⊗Wρ 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

    0TXTM(Q,r)|XE0.2.2
    Thus TX 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 MQ: = M(Q, r) be a quiver flag variety with ρ > 1. Then MQ is cut out of Y=i=1ρGr(H0(MQ,Wi),ri) by a tautological section of

    E=aQ1,s(a)0Ss(a)Qt(a),
    where Si and Qi are the pullbacks to Y of the tautological sub-bundle and quotient bundle on the ith factor of Y .

    Proof.

    As vector spaces, there is an isomorphism H0(MQ,Wi)e0CQei, where CQ is the path algebra over C of Q (corollary 3.5, [1]). This isomorphism identifies a basis of global sections of Wi from the set of paths from vertex 0 to i in the quiver. Let eaCQ be the element associated with the arrow aQ1. Thus

    H0(MQ,Wi)=aQ1,t(a)=i,s(a)0H0(MQ,Ws(a))aQ1,s(a)=0,t(a)=iCea.
    Let Fi=t(a)=iQs(a). Combining the tautological surjective morphisms
    H0(MQ,Ws(a))OY=H0(Y,Qs(a))OYQs(a),
    gives the exact sequence
    0t(a)=i,s(a)0Ss(a)H0(MQ,Wi)OYFi0.
    Thus
    (H0(MQ,Wi)OY)Fit(a)=i,s(a)0Ss(a),
    and it follows that E=i=2ρHom(Qi,(H0(MQ,Wi)OY)/Fi).

    Consider the section s of E given by the compositions

    QiH0(MQ,Wi)OY(H0(MQ,Wi)OY)Fi.

    The section s vanishes at quotients (V1, …, Vρ) if and only if Vit(a)=iVs(a); dually, the zero locus is where there is a surjection Fi → Qi for each i. We now identify Z(s) with M(Q, r). Since the Wi are globally generated, there is a unique map

    f:MQY=i=1ρGr(H0(MQ,Wi),ri),
    such that Qi → Y pulls back to Wi → M(Q, r). By construction of MQ there are surjections
    aQ1,t(a)=iQs(a)Wi0,
    so f(MQ)⊂Z(s).

    Any variety X with vector bundles Vi of rank ri for i = 1, …, ρ and maps H0(MQ, Wi) → Vi → 0 that factor as

    H0(MQ,Wi)t(a)=iVs(a)Vi
    has a unique map to M(Q, r) as the Vi form a flat family of θ-stable representations of Q of dimension r. The (Qi|Z(s))ρi=1 on Z(s) give precisely such a set of vector bundles. The surjections H0(MQ, Wi) → Qi|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) → MQ be the induced map. By the universal property of M(Q, r), the composition g°f:MQ → Z(s) → MQ 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 EG determined by a representation E. The product of Grassmannians Y=i=1ρGr(H0(Wi),ri) is a GIT quotient Vss/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 EG on Y :

    EG:=E×VssGY.
    We see that X is deformation equivalent to the zero locus of a generic section of the vector bundle
    F:=EGaQ1,s(a)0Ss(a)Qt(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)Qt(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 Sα(Qi) is globally generated on Y (using the notation as above). This implies that Sα(Wi) 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*iWj. If this is the case, one can then undo the product representation process to obtain another model (M(Q′, r′), EG) 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 ∧2W2S*1W2 (as in equation (3.1)) and dualizing the first factor, we get

    Inline Graphic

    with bundle ∧2W2W1W2, 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 Cn+1, where the image of the section corresponding to vCn+1 at the point ϕ:Cn+1Cr in Gr(n + 1, r) is ϕ(v). This section vanishes precisely when vkerϕ, so we can consider its zero locus to be Gr(Cn+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 SOGr(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*iWj, 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 0W1(Cn+1)S0 shows that a global section of S* is given by a linear map ψ:Cn+1C. The image of the section corresponding to ψ at the point sS is ψ(s), where we evaluate ψ on s via the tautological inclusion SCn+1. Splitting Cn+1=CnC and choosing ψ to be projection to the second factor shows that ψ vanishes precisely when SCn, 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 W1OGr(n,r).

    (c) Grafting

    Let Q be a quiver. We say that Q is graftable at iQ0 if

    ri = 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

    {jQ00andjare in different components ofQi}.

    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 Qbe the quiver obtained from Q by replacing each arrow i → j, where jJ, by an arrow 0 → j. Then

    M(Q,r)=M(Q,r).

    Proof.

    Define Vj: = W*iWj for jJ, and Vj: = Wj otherwise.

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

    WidijWj0.
    Here, dij is the number of paths i → j. Tensoring this sequence with W*i shows that Vj is globally generated.

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

    aQ1,t(a)=jVs(a)Vj.
    If jJ, this is just the same surjection given by the fact that the Wi are a θ-stable representation of Q. If jJ, one must, as above, tensor the sequence from Q with W*i. The Vj 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

    W1W3W12detW1.
    Note we have chosen a different labelling of the vertices for convenience. Writing X inside a product of Grassmannians gives W1W3W12detW1 on the quiver in (b), with arrow bundle S*2W1. Removing the two copies of W1 using example 4.2 gives
    W1W3detW1,

    on the quiver in (c), with arrow bundle S*2W1. We now apply example 4.3 to remove detW1=detS1=S1. As mentioned in example 4.3, W1 on (c) becomes W1O after removing S*1. The arrow bundle therefore becomes

    S2(W1O)=S2S2W1.
    Similarly, W1W3 becomes W3W1W3. 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 W1W3 on the quiver in (d), with arrow bundle S*2W1. Dualizing at vertices 1 and 2 now gives the quiver in (e), with arrow bundle S*1W2S*1W3. Finally, undoing the product representation exhibits X as the quiver flag variety for the quiver in (f).

    Inline Graphic

    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)with1=(1,,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(Hom(det(Wi),det(Wj))Si,j),
    where Si,j is spanned by maps which factor through maps to det(Wk) with i < k < j. This induces an isomorphism p:Pic(X)RPic(X)R that sends det(Wi)det(Wi). 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=CN with stability condition τχ(G)=Hom(G,C). Identifying Pic(X)≅χ(G), we have that:

    vχ(G) is ample if Vs(v) = Vss(v) = Vs(τ).

    v is on a wall if Vss(v)≠Vs(v).

    v∈NE1(X) if Vss.

    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).

    Inline Graphic

    One can see this by noting that Hom(det(W2),det(W1))=0, and that after taking ∧3 (respectively, ∧2) the surjection O5W10 (respectively, O10W20) becomes

    O10W10(respectively,O45W20).
    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 i=0ρGl(ri): a representation R = (Ri)iQ0 is semi-stable with respect to χ = (χi)ρi=0 if and only if

    i=0ρχidimC(Ri)=0; and

    for any subrepresentation R′ of R, i=0ρχidimC(Ri)0.

    Consider the character χ = ( − 1, 3) of G, which we lift to a character of i=0ρGl(ri) 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=[100010]andC=[000001],
    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 R1. 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 TG 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, …, λρ) denote an element of T=i=1ρ(C)ri, where λj = (λj1, …, λjrj). Let (wa)aQ1∈Rep(Q, r). Here wa is an rt(a) × rs(a) matrix. The action of λ on (wa)aQ1 is defined by

    wa(i,j)λs(a)i1wa(i,j)λt(a)j.
    Hence this is the same as the group action on the quiver Qab with vertices
    Q0ab={vij:0iρ,1jri},
    and the number of arrows between vij and vkl 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)//θT=M(Qab,1).
    We call Qab the Abelianized quiver.

    Example 5.3

    Let Q be the quiver

    Inline Graphic

    Then Qab is

    Inline Graphic

    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 MQ = M(Q, r), MQab = M(Qab, (1, …, 1)) and V = Rep(Q, r) = Rep(Qab, (1, …, 1)). For vχ(G), let Vsv(T) denote the T-stable points of V and Vsv(G) denote the G-stable points, dropping the subscript if it is clear from context. It is easy to see that Vs(G)⊂Vs(T). The Weyl group W of (G, T) is i=1ρSri, where Sri is the symmetric group on ri letters. Let π:Vs(G)/T → Vs(G)/G be the projection. The Weyl group acts on the cohomology of M(Qab, 1), and also on the Picard group, by permuting the Wvi1, …, Wviri. It is well-known (e.g. Atiyah–Bott [16]) that

    π:H(Vs(G)T)WH(MQ).

    Theorem 5.4 (Martin [15])

    There is a graded surjective ring homomorphism

    ϕ:H(MQab,C)WH(Vs(G)T,C)πH(MQ,C),
    where the first map is given by the restriction Vs(T)/T → Vs(G)/T. The kernel is the annihilator of e=i=1ρ1j,kric1(WvijWvik).

    Remark 5.5

    This means that any class σH*(MQ) can be lifted (non-uniquely) to a class σ~H(MQab). Moreover, eσ~ is uniquely determined by σ.

    Corollary 5.6

    Let E be a representation of G defining representation theoretic bundles EG → MQ and ET → MQab. Then ϕ(ci(ET)) = ci(EG).

    Proof.

    Recall that

    EG=(Vs(G)×E)GMQandET=(Vs(T)×E)TMQab.
    Define
    EG=(Vs(G)×E)TVs(G)T.
    Let f be the inclusion Vs(G)/T → Vs(T)/T. Clearly, f*(ET) = EG′ as EG′ is just the restriction of ET. Considering the square

    Inline Graphic

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

    Remark 5.7

    Note that ET always splits as a direct sum of line bundles on M(Qab, (1, …, 1)), as any representation of T splits into rank one representations. In particular, this means that if (Q, EG) defines a quiver flag zero locus, (Qab, ET) 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:c1(Wi)j=1ric1(Wvij).
    In particular, using (2.1), we have that i(ωMQ) = ωMQab, where ωX is the canonical bundle of X.

    Proposition 5.8

    Let Amp(Q), Amp(Qab) denote the ample cones of MQ and MQab, respectively. Then

    i(Amp(Q))=Amp(Qab)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 Vssv(G)⊂Vssv(T). To see this, suppose vV is semi-stable for G. Let λ:CT be a one-parameter subgroup of T such that limt → 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 vVssv(T). It follows that, if v∈NE1(MQ), then Vvss(G), so Vvss(T), and hence by theorem 5.1 v∈NE1(MQab)W.

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

    p:NE1(MQab)NE1(MQ).
    Suppose that α∈Amp(Q). Then for any C∈NE1(MQab), i(α) · C = α · p(C) > 0. So i(α)∈Amp(Qab)W.

    Let Wall(G)⊂Pic(MQ) 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:CG such that limt → 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 gG such that Im(λ)⊂g−1Tg. Consider λ′ = gλg−1. Then λ(C)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 limt → 0λ′(t)g · v = limt → 0(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))Wall(T)W.
    This means that the boundary of i(Amp(Q)) has empty intersection with Amp(Qab)W. Since both are full dimensional cones in the W invariant subspace, the inclusion i(Amp(Q))⊂Amp(Qab)W is in fact an equality. ▪

    Example 5.9

    Consider again the example

    Inline Graphic

    The Abelianization of this quiver is

    Inline Graphic

    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 CN 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, …, DNL, together with a stability condition wLR.

    These linear data give a toric variety (or Deligne–Mumford stack) as the quotient of an open subset UwCN by K, where K acts on CN via the map K(C)N defined by the Di. Uw is defined as

    {(z1,,zN)CN|wCone(Di:zi0)},
    that is, its elements can have zeroes at zi, iI, only if w is in the cone generated by Di, iI. 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, …, eρ be standard basis of L=Zρ and set e0: = 0. Then each aQ1 gives a weight Da = − es(a) + et(a). The stability condition is 1 = (1, 1, …, 1). Identify L≅PicM(Q, 1). Then Da = Wa: = W*s(a)Wt(a).

    A minimal generating set for a full dimensional cone for a toric quiver flag variety is given by ρ linearly independent DaiaiQ1. Therefore, for each vertex i with 1 ≤ i ≤ ρ, we need an arrow ai 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 Dai has t(ai) = i. Fix such a set S = {a1, …, aρ}, and denote the corresponding cone by CS. As mentioned, the ample cone is the intersection of such cones CS. The set S determines a path from 0 to i for each i, given by concatenating (backwards) ai with as(ai) and so on; let us write fij = 1 if aj is in the path from 0 to i, and 0 otherwise. Then

    ei=j=1ρfijDaj.
    This gives us a straightforward way to compute the cone CS. Let BS be the matrix with columns given by the Dai, and let AS = B−1S. The columns of AS are given by the aforementioned paths: the jth column of AS is i=1ρfijei. If c∈Amp(Q), then AScAS(Amp(Q))⊂AS(CS). Since ASDai = ei, this means that ASc is in the positive orthant.

    Proposition 5.11

    Let M(Q, 1) be a toric quiver flag variety. Let c∈Amp(Q), c = (c1, …, cρ), be an ample class, and suppose that vertex i of the quiver Q satisfies the following condition: for all jQ0 such that j > i, there is a path from 0 to j not passing through i. Then ci > 0.

    Proof.

    Choose a collection S of arrows ajQ1 such that the span of the associated divisors Daj 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 AS is 1 and all other entries of the ith row are zero. As ASc is in the positive orthant, ci > 0. ▪

    Corollary 5.12

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

    Proof.

    Consider the Abelianized quiver. For any vertex vQab0, we can always choose a path from the origin to v that does not pass through vj1: if there is an arrow between vj1 and v, then there is an arrow between vj2 and v, so any path through vj1 can be rerouted through vj2. Then we obtain that the j1 entry of i(c) is positive-but this is just cj. ▪

    (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

    Ti:={jQ0allpathsfromthesourcetovj1passthroughvi1intheabelianizedquiver}.
    Note that iTi, as every path from 0 to vi1 passes through vi1 by definition. There are no paths from the source to the source, which is therefore not in Ti for any i. If ri > 1 then Ti = {i}.

    Theorem 5.13

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

    jTirjaj0i=1,2,,ρ.5.1

    Proof.

    We have already shown that the Weyl invariant part of the nef cone of MQ: = M(Q′, 1) is the image of the nef cone of MQ: = M(Q, r) under the natural map π:Pic(MQ) → Pic(MQ). Label the vertices of Q′ as vij, i∈{0, …, ρ}, j∈{1, …, ri}, and index elements of Pic(MQ) as (bij). The inequalities defining the ample cone of MQ are given by a choice of arrow AijQ1, t(Aij) = vij for each vij. This determines a path Pij from 0 → vij for each vertex vij. For each vij the associated inequality is

    vijPklbkl0.5.2

    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 ri = 1 (as we have already shown that the inequalities are the same in the ri > 1 case). Choose a set of arrows such that the associated paths avoid vi1 if possible: in other words, if vi1Pkl, then assume kTi. Notice that if vi1Pkl1, then vi1Pkl2. By assumption π(a) satisfies the ith inequality associated with this collection of arrows, that is

    kTirkak=vi1Pklak0.
    Therefore, if C is the cone defined by (5.1), we have shown that Nef(MQ)⊂C.

    Suppose now that aC and take a choice of arrows Akl. Write π(a) = (aij). We prove that the inequalities 5.2 are satisfied starting at vρrρ. For ρ, the inequality is aρrρ≥0, which is certainly satisfied. Suppose the (ij + 1), (ij + 2), …, (ρrρ) inequalities are satisfied. The inequality we want to establish for (ij) is

    vi1Pklakl=aij+kTi{i}l=1rlakl+Γ=ai+kTi{i}rkak+Γ0,
    where
    Γ=s(Akl)=vij,kTi(akl+vklPstast).
    This uses the fact that for kTi, vi1Pkl for all l, and that if kTi, and vklPst, we also have that sTi.

    As aC it suffices to show that Γ≥0. By the induction hypothesis akl+vklPstast0, 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(Wi) 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 Ti.

    Lemma 5.14

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

    TiTjTjTi,
    such that for all j, the set {Ti ≤ Tj} is a chain.

    Proof.

    Observe that if iTjTk for j < k, then TkTj: 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 kTj and hence TkTj. Therefore, if Tj ≤ Ti and Tk ≤ Ti for j < k, then iTjTk, and so Tj ≤ Tk. Hence {Tk|Tk ≤ Tj} 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: = {Ti|Tiis minimal}. By the lemma, {1,,ρ}=TiMTi. Suppose L is given by the character (a1, …, aρ). Write L as L = ⊗TiMLTi, where each LTi comes from a character (b1, …, bρ)∈χ(G) satisfying bj = 0 if jTi.

    L is nef if and only if all the LTi, TiM are nef. To see this, note that for each j the inequality

    kTjrkak0,
    involves terms from a minimal Ti if and only if jTi, in which case it involves only terms from Ti. It therefore suffices to show the statement of the proposition for each Lj. Therefore suppose that {j|aj≠0}⊂Ti for Ti minimal. If ri > 1, then Ti = {i}, so L=det(Wi)ai which is globally generated. So we further assume that ri = 1. For kTi, k > i, define h′(k) such that Th′(k) is the maximal element such that Ti ≤ Th′(k) < Tk. This is well defined because the set {Tj|Tj < Tk} 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 χiai. A point of Rep(Q, r) is given by (ϕa)aQ1,ϕa:Crs(a)Crt(a), where G acts by change of basis. A choice of path i → j on the quiver gives an equivariant map Rep(Q,r)Hom(Cri,Crj) where G acts on the image by g · ϕ = gjϕg−1i. If ri = rj = 1, such maps can be composed.

    For jTi, define fj as follows:

    If j = i, let fi be any homogeneous polynomial of degree di=kTirkak0 in the maps given by paths 0 → i. Therefore, fi is a section of the line bundle given by the character χdii.

    If j > i, rj = 1, let fj be any homogeneous polynomial of degree dj=kTjrkak0 in the maps given by paths h′(j) → j. Note that rh′(j) = 1 as by construction j, h′(j)∈Th′(j). So fj defines a section of the line bundle given by character χdjh(j)χdjj.

    If j > i, rj > 1, let fj be a homogeneous polynomial of degree ak≥0 in the minors of the matrix whose columns are given by the paths h′(j) → j. That is, fj is a section of the line bundle given by character χrjajh′(j)χajj.

    For any x∈Rep(Q, r) which is semi-stable, and for any jTi, there exists an fj as above with fj(x)≠0, because jTh′(j). Fixing x, choose such fj. Define σ:=jTifj:Rep(Q,r)C. Then σ defines a section of the line bundle given by character

    jTiχjbj=χidijTi,ji,rj=1χh(j)djχjdjjTi,ji,rj>1χh(j)rjajχjaj.

    We need to check that bl = al for all l. This is obvious for lTi with rl > 1. For rl = 1,

    bl=jTlrjajkTl{l},h(k)=ljTkrjaj.
    This simplifies to al because Tl − {l} = ⊔jTl,h′(j) = lTj. 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(TX)≥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 nZ0 and βH2(Y ), let M0,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 [M0,n(Y, β)]virt of the expected dimension [20,21]. There are natural evaluation maps evi:M0,n(Y, β) → Y taking a class of a stable map f:C → Y to f(xi), where xiC is the ith marked point. There is also a line bundle Li → M0,n(Y, β) whose fibre at f:C → Y is the cotangent space to C at xi. The first Chern class of this line bundle is denoted ψi. Define

    τa1(α1),,τan(αn)n,β=[M0,n(Y,β)]virti=1nevi(αi)ψiai,6.1
    where the integral on the right-hand side denotes cap product with the virtual fundamental class. If ai = 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 qβ for the element of Q[H2(Y)] representing βH2(Y ). Write N(Y ) for the Novikov ring of Y :

    N(Y)={βNE1(Y)cβqβ|cβC,foreachd0thereareonlyfinitelymanyβsuch thatωβdandcβ0}.
    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, …, ϕN be a homogeneous basis of H*(Y ), and let ϕ1, …, ϕN be the Poincaré dual basis. Then the J-function is given by
    JX(τ,z):=1+τz1+z1iϕi/(zψ)ϕi.6.2
    Here, 1 is the unit class in H0(Y ), τH*(Y ), and
    ϕi/(zψ)=βNE1(Y)qβn=0a=01n!za+1τa(ϕi),τ,,τn+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 GY(t) is the component of J(0) along 1∈H(Y) after the substitutions z↦1, qβt〈−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 , H1(C, f*E) = 0. Globally generated vector bundles are convex. Let XY 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)=λr+λr1c1(F)++λcr1(F)+cr(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, λHC(pt) is the equivariant parameter. The twisted J-function Je, 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 [M0,n(Y, β)]virte(E0,n,β), where E0,n,β is π*(ev*n+1(E)), π:M0,n+1(Y, β) → M0,n(Y, β) is the universal curve, and evn+1:M0,n+1(Y, β) → Y is the evaluation map. E0,n,β is a vector bundle over M0,n(Y, β), because E is convex. Functoriality for the virtual fundamental class [23] implies that
    jJe,E(τ,z)|λ=0=JX(jτ,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 TG 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, EG) (where we assume EG is globally generated), and write MQ = M(Q, r) for the ambient quiver flag variety. Let (Qab, ET) be the associated Abelianized quiver and bundle, MQab = M(Qab, (1, …, 1)). Assume, moreover, that ET splits into nef line bundles; this implies that ET is convex. To define the I-function, we need to relate the Novikov rings of MQ and MQab. Let PicQ (respectively PicQab) denote the Picard group of MQ (respectively of MQab), and similarly for the cones of effective curves and effective divisors. The isomorphism PicQ → (PicQab)W gives a projection p:NE1(MQab) → NE1(MQ). In the bases dual to det(W1),,det(Wρ) of PicMQ and Wij, 1 ≤ i ≤ ρ, 1 ≤ j ≤ ri of PicMQab, this is

    p:(d1,1,,d1,r1,d2,1,,dρ,rρ)(i=1r1d1i,,i=1rρdρi).
    For β = (d1, …, dρ), define
    ϵ(β)=i=1ρdi(ri1).
    Then, following [17, equation 3.2.1], the induced map of Novikov rings N(MQab) → N(MQ) sends
    qβ~(1)ϵ(β)qβ,
    where β=p(β~). We write β~β if and only if β~NE1(MQab) and p(β~)=β.

    For a representation theoretic bundle EG of rank r on MQ, let D1, …, Dr be the divisors on MQab giving the split bundle ET. Given d~NE1(MQab) define

    IEG(d~)=i=1rm0(Di+mz)i=1rmd~,Di(Di+mz).
    Note that all but finitely many factors cancel here. If E is K-theoretically a representation theoretic bundle, in the sense that there exists AG, BG such that
    0AGBGE0,
    is an exact sequence, we define
    IE(d~)=IBG(d~)IAG(d~).6.5

    Example 6.2

    The Euler sequence from proposition 2.3 shows that for the tangent bundle TMQ

    ITMQ(d~)=aQ1abm0(Da+mz)aQ1abmd~,Da(Da+mz)i=1ρjkmd~,DijDik(DijDik+mz)i=1ρjkm0(DijDik+mz).
    Here, Dij is the divisor corresponding to the tautological bundle Wij for vertex ij, and Da: = − Ds(a) + Dt(a) is the divisor on MQab corresponding to the arrow aQab1.

    Example 6.3

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

    ITX(d~)=ITMQ(d~)/IEG(d~).

    Define the I-function of XMQ to be2

    IX,MQ(z)=dNE1(MQ)d~d(1)ϵ(d)qdITX(d~).

    that ITX(d~) is homogeneous of degree (i(KX),d~), so defining the grading of qd to be ( − KX, d), IX,MQ(z) is homogeneous of degree 0. If X is Fano, we can write ITX(d~) as

    z(ωX,d~)(b0+b1z1+b2z2+),biH2i(X).6.6

    Since IX,MQ is invariant under the action of the Weyl group on the Dij, by viewing these as Chern roots of the tautological bundles Wi we can express it as a function in the Chern classes of the Wi. We can, therefore, regard the I-function as an element of H(MQ,C)N(MQ)C[[z1]]. If X is Fano,

    IX,MQ(z)=1+z1C+O(z2),6.7
    where O(z−2) denotes terms of the form αzk with k ≤  − 2 and CH0(MQ,C)N(MQ); 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, EG), and let j : X → MQ be the embedding of X into the ambient quiver flag variety. Then

    JX(0,z)=ec/zjIX,MQ(z)
    where c = j*C.

    Remark 6.5

    Via the divisor equation and the string equation [25, §1.2], theorem 6.4 determines JX(τ, z) for τH0(X)⊕H2(X).

    (c) Proof of theorem 6.4

    Givental has defined [26,27] a Lagrangian cone LX in the symplectic vector space HX:=H(X,C)N(X)C((z1)) that encodes all genus-zero Gromov–Witten invariants of X. Note that JX(τ, z)∈HX for all τ. The J-function has the property that ( − z)JX(τ, − z) is the unique element of LX of the form

    z+τ+O(z1),
    (see [26, §9]) and this, together with the expression (6.7) for the I-function and the String Equation
    JX(τ+c,z)=ec/zJX(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, EG), and let j : X → MQ be the embedding of X into the ambient quiver flag variety. Then (z)jIX,MQ(z)LX.

    Proof.

    Let Y=i=1ρGr(H0(Wi),ri). Denote by Yab=i=1ρP(H0(Wi))×ri the Abelianization of Y . In §3, we constructed a vector bundle V on Y such that MQ is cut out of Y by a regular section of V :

    V=i=2ρQiH0(Wi)Fi,
    where Fi=t(a)=iQs(a). V is globally generated and hence convex. It is not representation theoretic, but it is K-theoretically: the sequence
    0FiQiH0(Wi)QiH0(Wi)QiFi0
    is exact. Let i: MQ → Y denote the inclusion.

    Both Y and MQ are GIT quotients by the same group; we can therefore canonically identify a representation theoretic vector bundle EG on Y such that EG|MQ is EG. Our quiver flag zero locus X is cut out of Y by a regular section of V ′ = VEG. Note that

    ITMQ(d~)IV(d~)=ITY(d~)IV(d~).
    The I-function IX,MQ defined by considering X as a quiver flag zero locus in MQ with the bundle EG then coincides with the pullback i*IX,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)(ij)IX,Y(z)LX.
    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 WG on Y , let D1, …, Dr be the divisors on Yab giving the split bundle WT, and for d~NE1(Yab) set
    IWGC(d~)=i=1rm0(λ+Di+mz)i=1rmd~,Di(λ+Di+mz).
    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 s~i:=dimH0(Wi). Recalling that
    ITY(d~)=i=1ρjkmd~,DijDik(DijDik+mz)i=1ρjkm0(DijDik+mz)i=1ρj=1rim0(Dij+mz)s~ii=1ρj=1rimd~,Dij(Dij+mz)s~i,
    we define
    IX,YC(z)=dNE1(Y)d~d(1)ϵ(d)qdITY(d~)/IVC(d~).
    The I-function IX,Y can be obtained by setting λ = 0 in IX,YC. In view of [24, Theorem 1.1], it therefore suffices to prove that
    (z)IX,YC(z)Le,V,
    where Le,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 AG and BG are representation theoretic vector bundles, and that

    0AGBGV0,
    is exact. Then we can also consider an exact sequence
    0ATBTF0,
    on the Abelianization, and define VT: = F. Using the notation of the proof of [17, theorem 6.1.2], the point is that
    (V)(AG)=(BG),
    Here, (V) is the twisting operator that appears in the quantum Lefschetz theorem [26]. We can then follow the same argument for
    (BG)(AG),
    After Abelianizing, we obtain (BT)/(AT)=(F), and conclude that
    (z)IX,YC(z)Le,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.

    References