This paper introduces a class of smooth projective varieties that generalise and share many properties with partial flag varieties of type A. The quiver flag variety M_\vartheta(Q,r) of a finite acyclic quiver Q (with a unique source) and a dimension vector r is a fine moduli space of stable representations of Q. Quiver flag varieties are Mori Dream Spaces, they are obtained via a tower of Grassmann bundles, and their bounded derived category of coherent sheaves is generated by a tilting bundle. We define the multigraded linear series of a weakly exceptional sequence of locally free sheaves E = (O_X,E_1,...,E_ρ) on a projective scheme X to be the quiver flag variety |E| = M_\vartheta(Q,r) of a pair (Q,r) encoded by E. When each E_i is globally generated, we obtain a morphism ϕ_|E| : X -> |E| realising each E_i as the pullback of a tautological bundle. As an application we introduce the multigraded Plucker embedding of a quiver flag variety
The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmanians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, …
The rim-hook rule for quantum cohomology of the Grassmannian allows one to reduce quantum calculations to classical calculations in the cohomology of the Grassmannian. We use the Abelian/non-Abelian correspondence for cohomology to prove a rim-hook removal rule for the cohomology of quiver flag varieties. Quiver flag varieties are generalisations of type A flag varieties; this result is new even in the flag case. This gives an effective way of computing products in their cohomology, reducing computations to that in the cohomology ring of the Grassmannian. We then prove a quantum rim-hook rule …
Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of homogeneous 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, disproving a conjecture of Craw. In the Appendices, 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 p…
We present a review on the recently discovered link between the Lie theory,the theory of quiver Grassmannians, and various degenerations of flag varieties. Our starting point is the induced Poincaré--Birkhoff--Witt filtration on the highest weight representations and the corresponding PBW degenerate flag varieties.
We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degener…
In a previous paper, the first two named authors established an isomorphism between the moduli space of framed flags of sheaves on the projective plane and the moduli space of stable representations of a certain quiver. In the present note, we substitute one of the claims made, namely [5, Theorem 17], for a weaker claim regarding the existence of unobstructed points in the quiver moduli space. We also extend some of the results of the cited paper, concerning the maximal stability chamber within which the isomorphism mentioned holds, and the existence of a perfect obstruction theory for the qui…
We give an explicit description of the irreducible components of two-row Springer fibers in type A as closed subvarieties in certain Nakajima quiver varieties in terms of quiver representations. By taking invariants under a variety automorphism, we obtain an explicit algebraic description of the irreducible components of two-row Springer fibers of classical type. As a consequence, we discover relations on isotropic flags that describe the irreducible components.
We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.
We relate quantum degree cones, parametrizing PBW degenerations of quantized enveloping algebras, to (negative tight monomial) cones introduced by Lusztig in the study of monomials in canonical bases, to K-theoretic cones for quiver representations, and to some maximal prime cones in tropical flag varieties.
Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of $q$-difference equations holomorphic in $\boldsymbol{z}$ with poles in $\boldsymbol{a}$ to a basis of solutions holomorphic in $\boldsymbol{a}$ with poles in $\boldsymbol{z}$. The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that for the cotangent bundle of the full flag variety, the functions obtained in this …
Several moduli spaces parametrizing linear subspaces of the projective space are cut out by linear and quadratic equations in their natural embedding: Grassmannians, Flag varieties, and Schubert varieties. The goal of this paper is to prove that a similar statement holds when one replaces linear subspaces with algebraic subschemes of the projective space. We exhibit equations of degree 1 and 2 that define schematically the Hilbert schemes $\mathbf{Hilb}^{p}_{\mathbb P^n}$ for all (possibly nonconstant) Hilbert polynomials $p$. The equations are reminiscent of the Plücker relations on the Grass…
Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kähler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same card…
