Markus Reineke
A factorization formula for certain automorphisms of a Poisson algebra associated to a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing formulas for Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman.
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Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers.
We compute the Brauer groups of several moduli spaces of stable quiver representations.
Fano quiver moduli
本文通过研究箭图模空间上的线丛及其整体截面，证明了一大类箭图模空间是Fano簇，并给出了点构型、Kronecker和环面箭图模空间等具体例子。
The fixed point set under a natural torus action on projectivized moduli spaces of simple representations of quivers is described. As an application, the Euler characteristic of these moduli is computed.
Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth.
It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra of a quiver.
These lecture notes consist of an introduction to moduli spaces in algebraic geometry, with a strong emphasis placed on examples related to the theory of quiver representations. The goal is to provide the background necessary to understand the construction of the moduli spaces of stable and semistable quiver representations due to King, Nakajima's quiver varieties, as well as some 'quiver versions' of existing moduli spaces of algebra-geometric objects.
Moduli spaces of representations of quivers with multiplicities via non-reductive GIT
该研究利用非约化群的相对仿射GIT与新稳定性条件，构造了截断幂级数环上带重数的箭图表示模空间（含带标架模空间及Nakajima箭图簇类似物），并证明部分光滑模空间上同调纯净。
We construct moduli spaces of representations of quivers over arbitrary schemes and show how moduli spaces of pointed curves of genus zero like the Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n}$ and the Losev-Manin moduli spaces $\overline{L}_n$ can be interpreted as inverse limits of moduli spaces of representations of certain bipartite quivers. We also investigate the case of more general Hassett moduli spaces $\overline{M}_{0,a}$ of weighted pointed stable curves of genus zero.
The Mukai conjecture for Fano quiver moduli
本文验证了与基本域内部维数向量相关联的 Fano quiver 模空间的 Mukai 猜想。
A system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert scheme-type) framed versions of quiver moduli is derived. This is applied to wall-crossing formulas for the Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman, in particular confirming their integrality.
We show that a semi-stable moduli space of representations of an acyclic quiver can be identified with two GIT quotients by reductive groups. One of a quiver Grassmannian of a projective representation, the other of a quiver Grassmannian of an injective representation. This recovers as special cases the classical Gelfand-MacPherson correspondence and its generalization by Hu and Kim to bipartite quivers, as well as the Zelevinsky map for a quiver of Dynkin type A with the linear orientation.
Finding the walls for quiver moduli
本文提出了与箭图和维向量相关的几何不变量理论问题中壁变化的有效刻画方法。
A result of A. King and C. Walter asserts that the Chow ring of a fine quiver moduli space is generated by the Chern classes of universal bundles if the quiver is acyclic. We will show that defining relations between these Chern classes arise geometrically as degeneracy loci associated to the universal representation.
Rigidity and Schofield's partial tilting conjecture for quiver moduli
本文利用Teleman量子化计算箭图表示模空间上万有丛自同态丛的高阶上同调，由此证明Schofield部分倾斜猜想，并证明箭图表示模空间作为簇具有无穷小刚性。
We compute rational equivariant Chow rings with respect to a torus of quiver moduli spaces. We derive a presentation in terms of generators and relations, use torus localization to identify it as a subring of the Chow ring of the fixed point locus, and we compare the two descriptions.
Torus fixed points of quiver moduli spaces are given by stable representations of the universal (abelian) covering quiver. As far as the Kronecker quiver is concerned they can be described by stable representations of certain bipartite quivers coming along with a stable colouring. By use of the glueing method it is possible to construct a huge class of such quivers implying a lower bound for the Euler characteristic. For certain roots it is even possible to construct all torus fixed points.
We study torus actions on moduli spaces of quivers. First we give a description of the weight spaces of the induced action of the tangent space to a torus-fixed point. Then we focus on actions of tori of rank one and derive an explicit form for the attractors in the Białynicki-Birula decomposition.
We give a moduli-theoretic treatment of the existence and properties of moduli spaces of semistable quiver representations, avoiding methods from geometric invariant theory. Using the existence criteria of Alper--Halpern-Leistner--Heinloth, we show that for many stability functions, the stack of semistable representations admits an adequate moduli space, and prove that this moduli space is proper over the moduli space of semisimple representations. We construct a natural determinantal line bundle that descends to a semiample line bundle on the moduli space and provide new effective bounds for …
Motivated by string-theoretic arguments Manschot, Pioline and Sen discovered a new remarkable formula for the Poincare polynomial of a smooth compact moduli space of stable quiver representations which effectively reduces to the abelian case (i.e. thin dimension vectors). We first prove a motivic generalization of this formula, valid for arbitrary quivers, dimension vectors and stabilities. In the case of complete bipartite quivers we use the refined GW/Kronecker correspondence between Euler characteristics of quiver moduli and Gromov-Witten invariants to identify the MPS formula for Euler cha…
We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.
Framed quiver moduli parametrize stable pairs consisting of a quiver representation and a map to a fixed graded vector space. Geometric properties and explicit realizations of framed quiver moduli for quivers without oriented cycles are derived, with emphasis on their cohomology. Their use for quantum group constructions is discussed.
Brauer groups of resolved quiver moduli via gerbes
研究半稳定箭图表示模空间奇点解消的Brauer群，通过扩展箭图 - 曲线字典，将关于曲线上向量丛模的证明转换到箭图表示模情形，给出平凡性新证明及相关消失结果的新代数几何证明。
We study the convolution algebra $H_{*}(Z^θ_{W})$ of homology on diagram automorphism fixed point quiver variety and prove that there exists an algebra homomorphism from the universal enveloping algebra of the diagram automorphism fixed algebra of the split quiver to $H_{*}(Z^θ_{W})$.
It is a well established fact, that any projective algebraic variety is a moduli space of representations over some finite dimensional algebra. This algebra can be chosen in several ways. The counterpart in algebraic geometry is tautological: every variety is its own Hilber scheme of sheaves of length one. This holds even scheme theoretic. We use Beilinson's equivalence to get similar results for finite dimensional algebras, including moduli spaces and quiver grassmannians. Moreover, we show that several already known results can be traced back to the Hilbert scheme construction and Beilinson'…
Virasoro constraints and representations for quiver moduli spaces
本文利用Joyce顶点代数研究带关系的箭图表示模空间的Virasoro约束，在光滑假设下构造了Virasoro代数的表示，并通过例外簇将其应用于del Pezzo曲面层模空间及Grassmannian。
We construct Mori Dream Spaces as fine moduli spaces of representations of bound quivers, thereby extending results of Craw--Smith \cite{CrawSmith} beyond the toric case. Any collection of effective line bundles $\mathscr{L}=(\mathscr{O}_X, L_1,..., L_r)$ on a Mori Dream Space $X$ defines a bound quiver of sections and a map from $X$ to a toric quiver variety $|\mathscr{L}|$ called the multigraded linear series. We provide necessary and sufficient conditions for this map to be a closed immersion and, under additional assumptions on $\mathscr{L}$, the image realises $X$ as the fine moduli space…
Series of Fano quiver moduli from extended Dynkin quivers
该研究针对小型扩展 Dynkin 箭图，构造了皮卡秩≤5的任意偶维 Fano 簇系列，给出其到射影空间的平坦态射及贝蒂数闭式公式，还阐释了上同调霍尔代数对其的作用。
Ample stability implies rigidity for quiver moduli
本文证明丰富稳定性足以将Teleman量子化应用于箭图模空间的普遍表示自同态丛，从而推出其无穷小刚性，并回答了先前工作中的开放问题，且该条件对互素维数向量保证Picard秩最大。
On Chow rings of quiver moduli
本文借助 Ellingsrud-Strømme 结果与周环表示研究箭袋模空间的积分计算，构造泛表示典范态射，并计算小型 Kronecker 模空间不变量及刻画首个非平凡6维情形。
Moduli of super-representations of quivers
该研究将King的几何不变量理论框架扩展到超情形，构造箭图超表示的模空间，还实现了超格拉斯曼流形等作为几何商，为超表示研究提供了模空间框架。
Fano 4-fold quiver moduli from subspace quivers
研究在特定维数向量假设下子空间箭图表示的模空间，通过 GIT 商分类出四个法诺四维流形，用箭图模技术描述其几何，其中两个已知，两个新，为大皮卡秩法诺四维流形分类提供有趣例子。
This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver $Q$ with relations $R$ corresponding to the finite-dimensional algebra $\bigl(\bigoplus_{i=0}^{r} L_i \bigr)$ where $\mathcal{L} := (\mathscr{O}_X,L_1, ...c, L_r)$ is a list of line bundles on a projective toric variety $X$. The quiver $Q$ defines a smooth projective toric variety, called the multilinear series $|\mathcal{L}|$, and a map $X \to |\mathcal{L}|$. We provide necessary and sufficient conditions for the indu…
The main result of this paper is the statement that the Hodge theoretic Donaldson-Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson-Thomas "function" to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson-Thomas theory. This will be the topic of the s…
We consider quiver representations respecting a quiver automorphism and show that the dimension vectors of the indecomposables are precisely the positive roots of an associated symmetrisable Kac-Moody Lie algebra. Moreover, every such Lie algebra arises in this way. We also discuss the relationship with species of valued quivers over finite fields.
Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a network category whose isomorphism classes of objects correspond to the orbits of quiver representations, in which neural networks map input data. We then prove that the output of a neural network depends only on the corresponding point in the moduli space. Finally, we present a different perspective on mapping neural networks with a specific activation funct…
In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Marcus Reineke. Obtained is a generalization of this construction for finite dimensional associative algebras and for quivers with oriented cycles. As an application we get an explicit realization of fibers for the moduli space bundle over the categorical quotient for the quiver $A_{n-1}^{(1)}$.
For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are constructed.
Quiver moduli spaces of a given dimension
本文通过箭图-维数向量对上的变换诱导模空间同构，并结合组合分析，证明了每个维数下满足简单约束的箭图表示模空间仅有有限多个射影代数簇。
Let $Q$ be a connected quiver with no oriented cycles, $k$ the field of complex numbers and $P$ a projective representation of $Q$. We study the adjoint action of the automorphism group $\Aut_{kQ} P$ on the space of radical endomorphisms $\radE_{kQ}P$. Using generic equivalence, we show that the quiver $Q$ has the property that there exists a dense open $\Aut_{kQ} P$-orbit in $\radE_{kQ} P$, for all projective representations $P$, if and only if $Q$ is a Dynkin quiver. This gives a new characterisation of Dynkin quivers.
This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. In the study of the symmetries, a realization of the moduli spaces as quiver varieties is given and plays an essential role.
Vector fields and admissible embeddings for quiver moduli
本文提出一种用于箭图表示模空间的双重标架构造，将涉及泛表示的层上同调计算简化为线丛计算，并证明模空间向量场同构于路代数的第一 Hochschild 上同调，同时给出导出范畴的容许嵌入。
Degrees of Fano Quiver Moduli
本文提出计算Fano箭图模空间度数的方法，给出两类模空间的显式公式，其中一类涉及二项式系数和，特例与中心MacMahon数一致，另一类为有理数公式，效率优于直接计算。
We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the numb…
Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver representations are more general than G-valued representations of finitely generated groups, we show by collapsing arrows that their quotient spaces are equivalent. We then establish a general criterion for the moduli of G-valued quiver representations with relations to admit a strong deformation retraction to a compact quotient by pinching vertices on the quiver. Thi…
We use categorical method and birational geometry to study moduli spaces of quiver representations. From certain "representable" functor, we construct a birational transformation from the moduli space of representations of one quiver to another new quiver with one vertex less. The dimension vector and the stability for the new moduli are determined functorially. We introduce several relative notions of stability to study such a birational transformation. The essential case is proven to be the usual blow-ups. Moreover, we compare the induced ample divisors for the two moduli. We illustrate this…
For a classical group $G$ over a field $F$ together with a finite-order automorphism $θ$ that acts compatibly on $F$, we describe the fixed point subgroup of $θ$ on $G$ and the eigenspaces of $θ$ on the Lie algebra $\mathfrak{g}$ in terms of cyclic quivers with involution. More precise classification is given when $\mathfrak{g}$ is a loop Lie algebra, i.e., when $F=\mathbb{C}((t))$.
An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed.
Maximal Rigid Representations of Continuous Quivers of Type A with Automorphism
本文利用Buan和Krause对循环箭图最大刚性表示的分类结果，推导出了带自同构的A型连续箭图一类最大刚性表示的同构类数量公式。
We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the `split-quotient quiver' introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quive…
Quiver moduli and quantum loop algebras
本文通过临界K理论将量子环形代数中所有简单模的q-特征与有向图模空间的欧拉特征联系起来。
Cluster scattering diagrams via quiver moduli and tight gradings
该研究结合箭图表示模空间与紧分次组合框架，研究秩2丛散射图，证明并推广了Elgin等人关于壁函数系数的相关猜想，还给出了散射图Weyl群对称性的新证明。
Rational singularities for moment maps of totally negative quivers
本文证明全负箭图矩映射的零纤维具有有理奇点，通过推广Budur的射流空间维数界方法，并将该性质传递到2-Calabi-Yau范畴的模空间，进而推广Wyss的有限域射流计数渐近结果并给出p进体积解释。
We construct a special embedding of the translation quiver $\mathbb{Z}Q$ in the three-dimensional affine space $\mathbb{R}^{3}$ where $Q$ is a finite connected acyclic quiver and $\mathbb{Z}Q$ contains a local slice whose quiver is isomorphic to the opposite quiver $Q^{op}$ of $Q.$ Via this embedding, we show that there exists an involutive anti-automorphism of the translation quiver $\mathbb{Z}Q.$ As an immediate consequence, we characterize explicitly the group of cluster automorphisms of the cluster algebras of seed $(X,Q)$, where $Q$ and $Q^{op}$ are mutation equivalent.
This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of moduli spaces of quiver representations and derived categories arising in toric geometry. The first main goal is to present the noncommutative geometric approach to semiprojective toric varieties via quivers. To achieve this, we use geometric invariant theory to construct both semiprojective toric varieties and moduli spaces of quiver representations. The second main goal builds on the first by present…
Motivated by the counting of BPS states in string theory with orientifolds, we study moduli spaces of self-dual representations of a quiver with contravariant involution. We develop Hall module techniques to compute the number of points over finite fields of moduli stacks of semistable self-dual representations. Wall-crossing formulas relating these counts for different choices of stability parameters recover the wall-crossing of orientifold BPS/Donaldson-Thomas invariants predicted in the physics literature. In finite type examples the wall-crossing formulas can be reformulated in terms of id…
We study moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations. This class of moduli spaces unifies Grassmannians of subrepresentations of rigid representations and moduli spaces of representations of generalized Kronecker quivers. With homological methods, we find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel'fand MacPherson correspondence, and derive a formula for the Poincare polynomial in singular cohomology of these moduli spaces.
Combining the MPS degeneration formula for the Poincaré polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.
We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the non-trivial Dynkin diagram automorphism. As applications of the study, we prove that folded AR-quivers encode crucial information on the representation theory of quantum affine algebra $U_q'(B^{(1)}_{n+1})$ such as Dorey's rule and denominator formulas.
Tensor stable moduli stacks and refined representations of quivers
本文引入张量稳定模叠以推广Serre-稳定模叠，将凝聚层范畴的幺半结构纳入模问题，用于解决Deligne-Mumford叠的恢复与导出等价的模实现问题，并研究精细表示的模及其与倾斜丛情形的联系。
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