William Crawley-Boevey, Jan Schröer
We prove some basic results about irreducible components of varieties of modules for an arbitrary finitely generated associative algebra. Our work generalizes results of Kac and Schofield on representations of quivers, but our methods are quite different, being based on deformation theory.
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We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are reinterpreted using perverse sheaves and reducible projective varieties with dependent Lyubeznik numbers are constructed.
We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.
We study the global geometry of the ten dimensional O'Grady irreducible symplectic variety. We determine its second Betti number, its Beauville form and its Fujiki constant.
Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
Irreducible symplectic varieties via relative Prym varieties
本文利用带反辛对合的 K3 曲面及商曲面线性系构造相对 Prym 簇，给出其为本原或不可约辛簇的判据，并在无穷多维度中产生不可约辛簇序列。
We show that the Andre motive of an irreducible, symplectic variety, which is deformation equivalent to the Hilbert scheme of points on a K3 surface, is an object the category generated by its motive truncated in degree two.
We show the termination of any log-minimal model program for a pair $(X,Δ)$ of a symplectic manifold $X$ and an effective $\mathbb R$-divisor $Δ$.
We study the $J_b(F)$-action on the set of top-dimensional irreducible components of affine Deligne--Lusztig varieties in the affine Grassmannian. We show that the stabilizer of any such component is a parahoric subgroup of $J_b(F)$ of maximal volume, verifying a conjecture of X.~Zhu. As an application, we give a description of the set of top-dimensional irreducible components in the basic locus of Shimura varieties.
We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.
Fujiki relations and fibrations of irreducible symplectic varieties
本文针对推广不可约辛流形的奇异复射影簇，证明广义Beauville-Bogomolov形式满足Fujiki关系且秩与光滑情形一致，并用新定义验证其纤维化表现与不可约辛流形相似。
We determine the regular irreducible components of the variety mod(A,d), where A=kQ/I is a string algebra and I is generated by a set of paths of length two. Our case is among the first examples of descriptions of irreducible components, aside from hereditary, tubular and Gelfand-Ponomarev algebras.
Let A be a tame quasi-tilted algebra and d the dimension vector of an indecomposable A-module. In the paper we prove that each irreducible component of the variety of A-modules of dimension vector d is regular in codimension one.
Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.
We examine the set of $J_b(F)$-orbits in the set of irreducible components of affine Deligne-Lusztig varieties for a hyperspecial subgroup and minuscule coweight $μ$. Our description implies in particular that its number of elements is bounded by the dimension of a suitable weight space in the Weyl module associated with $μ$ of the dual group.
We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $Γ$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_Γ$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_Γ$ to be irreducible and exhibit several examples where $\mathrm M_Γ$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_Γ$.
Solvable and Nilpotent Matroids: Realizability and Irreducible Decomposition of Their Associated Varieties
该研究引入可解与幂零拟阵族，探究其可实现性及相关簇的不可约分解，给出对应理想的有限生成集，证明了幂零拟阵相关拟阵簇的可实现性与不可约性等结论，并借助Grassmann-Cayley代数等技术分析簇的定义方程，还给出森林构型拟阵理想的完整生成集。
Let $G$ be the Weil restriction of a general linear group. By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we obtain a stratification of the affine Deligne-Lusztig varieties for $G$ (in the affine Grassmannian) attached to a minuscule coweight and a basic element. As an application, we verify a conjecture by Chen and Zhu on irreducible components of affine Deligne-Lusztig varieties for $G$.
Irreducible symplectic varieties via K3-del Pezzo double covers
对于2≤n≤10，构造第二Betti数为16≤b₂≤24的2n维不可约辛簇，作为一般K3-del Pezzo双重覆盖上Beauville-Mukai系统的有限辛商群的非平凡终端化。
Although regular semisimple Hessenberg varieties are smooth and irreducible, semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of semisimple Hessenberg varieties corresponding to the standard Hessenberg space. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of thi…
We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.
Minimal matroids in dependency posets: algorithms and applications to computing irreducible decompositions of circuit varieties
本文研究点线构型及其极小拟阵与环路簇，提出一种在依赖序下识别极小拟阵的算法，用于计算环路簇的不可约分解，并应用于 Fano 等经典构型及探讨极小拟阵唯一性猜想。
On the finiteness of twists of irreducible symplectic varieties
本文利用不可约辛簇的锥猜想，证明了在特征0的固定有限域扩张下不可约辛簇扭曲的有限性，并讨论了非闭域锥猜想、导出等价扭曲有限性及正特征情形。
We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.
In this paper we study the geometric structure of affine Deligne-Lusztig varieties for $GL_3$ and $b$ basic. We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.
Irreducible symplectic varieties with a large second Betti number
本文证明了非紧Lagrangian fibration存在不可约辛紧化，并应用于证明包含固定cubic fourfold的cubic fivefolds的relative Jacobian fibration可被第二Betti数至少为24的不可约辛簇紧化，属于新形变类型。
By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we introduce a stratification for the affne Deligne-Lusztig variety (in the affne Grassmannian) attached to attached to a minuscule cocharacter and a basic element. As an application, we complete the proof of a conjecture on the irreducible components of affne Deligne-Lusztig varieties, due to Miaofen Chen and Xinwen Zhu.
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.
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
Supersingular Tate conjecture for irreducible symplectic varieties of known types: I
本文证明多种已知类型的超奇异不可约辛簇在特定Artin不变量条件下具有Tate或超奇异阿贝尔Chow motive，进而其乘积满足超奇异Tate猜想。
We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and several mobile points -- are irreducible.
This work is a PhD thesis. First we provide some general context on wonderful varieties and moduli spaces of rational curves. Working over complex numbers we prove that the moduli space of rational curves with no marked points on the wonderful compactification of a symmetric space is not irreducible in general. Lastly we show that in the case of wonderful group compactifications the set of rational curves with no marked points and irreducible source is irreducible.
This is a quick survey on the characteristic varieties associated to rank one local systems on a smooth, irreducible, quasi-projective complex variety $M$. A key new result is Proposition 1.8, giving additional information on the constructible sheaf $\F=R^0f_*(\LL)$, where $\LL$ is a rank one local system on $M$ and $f:M \to S$ is a surjective morphism with $S$ a smooth curve and the generic fiber $F$ of $f$ connected. Corollary 1.9 says that for $S$ compact, the singular support of $\F$ cannot be a singleton.
We expand the existing arsenal of methods for exploring the irreducible components of the varieties $Rep(A,\bold d)$ which parametrize the representations with dimension vector $\bold d$ of a finite dimensional algebra $A$. To do so, we move back and forth between $Rep(A,\bold d)$ and a projective variety, $GRASS(A,\bold d)$, parametrizing the same set of isomorphism classes of modules. In particular, we show the irreducible components to be accessible in a highly compressed format within the projective setting. Our results include necessary and sufficient conditions for unirationality, smooth…
Given a complete nonsingular algebraic variety $X$ and a divisor $D$ with normal crossings, we say that $X$ is log homogeneous with boundary $D$ if the logarithmic tangent bundle $T_X(- \log D)$ is generated by its global sections. We then show that the Albanese morphism $α$ is a fibration with fibers being spherical (in particular, rational) varieties. It follows that all irreducible components of $D$ are nonsingular, and any partial intersection of them is irreducible. Also, the image of $X$ under the morphism $σ$ associated with $- K_X - D$ is a spherical variety, and the irreducible compon…
We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the $σ$-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.
On Subvarieties of Degenerations of Fano Varieties
本文研究Fano簇及可分有理连通簇退化上的几何不可约子概型，证明其维数存在由一般纤维Fano指数给出的下界，拓展了相关子概型存在性结论的几何性质研究。
We consider the Kisin variety associated to a $n$-dimensional absolutely irreducible mod $p$ Galois representation $\barρ$ of a $p$-adic field $K$ and a cocharacter $μ$. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if $K$ is totally ramfied with $n=3$ or $μ$ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to $\barρ$ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.
We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variet…
In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible symplectic varieties.
We show that any normal irreducible affine $n$-dimensional $SL_n$-variety $X$ is determined by its automorphism group in the category of normal irreducible affine varieties: if $Y$ is an irreducible affine normal algebraic variety such that $Aut(X) \cong Aut(Y)$ as ind-groups, then $Y \cong X$ as varieties. If we drop the condition of normality on $Y$ , then $X$ is not uniquely determined and we classify all such varieties. In case $n \ge 3$, all the above results hold true if we replace $Aut(X)$ by $U(X)$, where $U(X)$ is the subgroup of $Aut(X)$ generated by all one-dimensional unipotent sub…
We give a dimension bound on the irreducible components of the characteristic variety of a system of linear partial differential equations defined from a suitable filtration of the Weyl algebra $A_{n}(k)$. This generalizes an important consequence of the fact that a characteristic variety defined from the order filtration is involutive. More explicitly, we consider a filtration of $A_{n}(k)$ induced by any vector $(u,v) \in {\mathbb Z}^{n}\times {\mathbb Z}^{n}$ such that the associated graded algebra is the commutative polynomial ring in $2n$ indeterminates. The order filtration is the specia…
We prove that Schubert varieties are globally F-regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic D-modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple D-modules coincide with local cohomology sheaves with support in Schubert varieties. Using a local Grothendieck-Cousin complex we prove that the decomposition of local cohomology sheaves with support in Schubert cells is multiplicity free.
Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0
本文通过阿贝尔锥的不可约分支研究，给出了亏格0到光滑射影环簇的映射模空间不可约分支的组合描述，并首次描述了非射影空间情形的可光滑轨迹。
We study the behavior of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the "relative setting" (splitting one node at a time), we demonstrate that there are many non-hereditary algebras whose irreducible components…
We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …
Shimura proved that each principally polarized abelian variety over $\mathbf{C}$ admits a unique factorization into irreducible principally polarized abelian varieties. We give an exposition of his result, and generalize to an arbitrary ground field $k$. If $k$ is separably closed, the irreducible factors are in bijection with the irreducible components of a theta divisor over $k$ giving rise to the polarization.
Monodromy and Irreducibility of Igusa Varieties
该研究结合单值群相关工作与Hida方法的推广，确定了Hodge型Shimura簇的Igusa簇、中心叶及Newton层的不可约分支，并证明离散Hecke轨道猜想的强版本一般不成立。
On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity
本文研究了不可约辛流形上Boucksom-Zariski分解的性质，并证明了在光滑情况下负数有界猜想的有效版本，从而得出关于辛流形上大线包的有效应性结论。
We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set…
Matroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models
本文研究了超图品种的matroid分层及其实现空间，揭示了超图品种的不可约分解结构，并展示了其在条件独立模型中的应用。
We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane $E_6/P_1$ and the $E_7$-adjoint variety $E_7/P_1$. From this result, we see that a general hyperplane section $F_4/P_4$ of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.
Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces
本文证明了基变换型Enriques曲面上任意亏格曲线的Severi簇可数族非空，并构造了有理椭圆曲面上特殊超丰对数Severi簇的例子。
Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties
本文提出高效算法用于分解 matroid 的电路品种，通过识别极大衰减来实现显式且可计算的不可约分解。
Maximal commutative unipotent subgroups and a characterization of affine spherical varieties
本文描述了不可约仿射簇自同构群的极大交换幂幺子群，证明群同构保持幂幺元素，并利用此结果证明自同构群能检测球面性与权幺半群，进而刻画了仿射环面簇与光滑仿射球面簇由其自同构群唯一确定的性质。
We prove that for every reductive group G with a maximal torus T and the Weyl group W there is a natural normalization map chi from T^N/W to an irreducible component of the G-character variety of Z^N. We prove that chi is an isomorphism for all classical groups. Additionally, we prove that even though there are no irreducible representations in the above mentioned irreducible component of the character variety for non-abelian G, the tangent spaces to it coincide with H^1(Z^N, Ad rho). Consequently, this irreducible component has the "Goldman" symplectic form for N=2, for which the combinatoria…
This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type $A$ in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-kn…
We introduce vector bundle techniques for finding equations of secant varieties. A test is established that determines when a secant variety is an irreducible component of the zero set of the equations found. We also prove an induction theorem for varieties that are not weakly defective, that allows one to conclude that the zero set of the equations found for s_{r-1}(X) have s_{r-1}(X) as an irreducible component when s_r(X) is an irreducible component of the equations found for it. The techniques are illustrated with examples of homogeneous varieties. We give an algorithm to decompose a gener…
Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to $X(G)$ in the coordinates $X=t_x$, $Y=t_y$ and $Z=t_{xy}$. As an easy consequence, a formula for computing the number of irreducible components of $X(G)$ as a function of $m$ and $n$ is given. We provide a combinatorial description of $X(G)$ and we prove that in most cases it is possible t…
The goals of this article are as follows: (1) To determine the irreducible components of the affine varieties parametrizing the representations of $ Λ$ with dimension vector d, where $ Λ$ traces a major class of finite dimensional algebras; (2) To generically describe the representations encoded by the components. The target class consists of those truncated path algebras $ Λ$ over an algebraically closed field K which are based on a quiver Q without oriented cycles. The main result characterizes the irreducible components of the representation variety in representation-theoretic terms and pro…
Moduli spaces on Kuznetsov components are Irreducible Symplectic Varieties
本文研究立方四胞体Kuznetsov分支中具非本原非OG10型Mukai向量的Bridgeland半稳定对象模空间，证明其为具因子终端奇点的射影不可约辛簇，形变类由m和v₀²唯一确定，推广并拓展了此前相关模空间的研究结果。
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