Notes on Perverse Sheaves and Vanishing Cycles
David B. Massey
本文系统总结了导出范畴、反常层与消失循环的相关结果，并在2025年8月版本中对消失循环部分进行了全面修订。
本文是一位在职数学家对导出范畴、反常层和消失循环诸多结果的总结。这是2025年8月版本，其中消失循环一节已完全修订。
A working mathematician's summary of many results on the derived category, perverse sheaves, and vanishing cycles. This is the August 2025 version, with a completely revised section on vanishing cycles.
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Kapranov and schechtman gave quiver description of perverse sheaves on real hyperplane arrangements. We used this description to relate the perverse sheaves on Coxeter hyperplane arrangements of type $\mathcal A_n$ for different values of $n$. As a consequence we prove that the simple perverse sheaves whose stalk on open cells are zero are induced from the perverse sheaves on lower dimension arrangements.
Perverse sheaves on smooth toric varieties and stacks
本文将光滑环面簇及其轨道分层上的反常层范畴显式刻画为某代数上的有限维模范畴，并推广到等变情形以描述环面轨形上的反常层，同时比较了相关导出范畴。
A Note on Numerical Perverse Sheaves
该研究定义了一类反常层并阐述其性质，为基环非域场景下的相关数学研究提供了有用的工具。
Perverse sheaves on Artin stacks
本文是在“Artin栈上层的六种运算Ⅰ：有限系数”和“Artin栈上层的六种运算Ⅱ：Adic系数”（math.AG/0512097和math.AG/0603680）的基础上发展了Artin栈上的反常层理论
We determine versal non-commutative deformations of some simple collections in the categories of perverse coherent sheaves arising from tilting generators for projective morphisms.
There is a connection between the category of perverse sheaves on a disc and different notions related to spherical functors. We introduce a category whose objects are analogous to 4-periodic semiorthogonal decompositions and prove that it is equivalent to the category of perverse sheaves on a disc stratified by the origin and its complement.
We study perverse coherent sheaves on the resolution of rational double points. As examples, we consider rational double points on 2-dimensional moduli spaces of stable sheaves on K3 and elliptic surfaces. Then we show that perverse coherent sheaves appears in the theory of Fourier-Mukai transforms. As an application, we generalize the Fourier-Mukai duality for K3 surfaces to our situation.
Perverse sheaves on the stack of $G$-zips
该研究提出了$G$-zips栈上简单反常层的计算方法，并通过多个示例完成了相关计算，为代数几何领域相关研究提供方法参考。
Another introduction to perverse sheaves with some exercises. Expanded version of five lectures at the 2015 PCMI.
Inspired by symplectic geometry and a microlocal characterizations of perverse (constructible) sheaves we consider an alternative definition of perverse coherent sheaves. We show that a coherent sheaf is perverse if and only if $RΓ_Z(\mathcal{F})$ is concentrated in degree 0 for special subvarieties Z of X. These subvarieties Z are analogs of Lagrangians in the symplectic case.
Microlocal Perverse Sheaves
微局部反常层在一个复流形的余切丛上形成一个叠加，它与流形本身上的反常层叠加类似。我们给一个嵌入的堆栈的微局部反常层到一个简单的堆栈上的余切丛，我们称之为融合莫尔斯系统。融合莫尔斯系统范畴是初等的，在这个意义上，对象是向量空间的集合和它们之间满足一定合成关系的线性映射的集合。
We study perverse sheaves of categories their connections to classical algebraic geometry. We show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on $\mathbb{P}^1$ bundles, semiorthogonal decompositions, and relate them to a recent proof of Segal that all autoequivalences of triangulated categories are spherical twists. Furthermore, we show that perverse sheaves of categories can be used to represent certain degenerate Calabi--Yau varieties.
We give a remark on the wall crossing behavior of perverse coherent sheaves on a blow-up and stability condition constructed by Toda. We also explain the wall crossing of twisted stability in terms of stability condition.
We analyze irreducible perverse sheaves on abelian varieties, defined over the complex numbers or the algebraic closure of a finite field, whose Euler characteristic is zero. We give a description of such perverse sheaves under assumptions for the case of simple abelian varieties.
We give a complete (global) characterization of complex perverse sheaves on semi-abelian varieties in terms of their cohomology jump loci. Our results generalize Schnell's work on perverse sheaves on complex abelian varieties, as well as Gabber-Loeser's results on perverse sheaves on complex affine tori. We apply our results to the study of cohomology jump loci of smooth quasi-projective varieties, to the topology of the Albanese map, and in the context of homological duality properties of complex algebraic varieties.
We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new proofs of some of the basic results of this theory. Our approach is topological/arithmetic, and avoids Hodge theory.
Perverse sheaves on Grassmannians
我们给出了A型和D型埃尔米特对称空间上的反常层范畴的完整描述，这些反常层范畴是关于舒伯特分层可构造的。计算是微局部的，并使用Borel群的作用来研究余正规簇的几何。
We show that there is a perverse sheaf on a fine moduli space of stable sheaves on a smooth projective Calabi-Yau 3-fold, which is locally the perverse sheaf of vanishing cycles for a local Chern-Simons functional, possibly after taking an etale Galois cover. This perverse sheaf lifts to a mixed Hodge module and gives us a cohomology theory which enables us to define the Gopakumar-Vafa invariants mathematically.
We survey nearby and vanishing cycles for both perverse sheaves and D-modules under analytic setting. Following ideas of A. Beilinson, M. Kashiwara and M. Saito, we explain in detail the proof of the comparison theorem between them in the sense of Riemann-Hilbert correspondence.
We characterize bifurcation values of polynomial functions by using the theory of perverse sheaves and their vanishing cycles. In particular, by introducing a method to compute the jumps of the Euler characteristics with compact support of their fibers, we confirm the conjecture of Némethi-Zaharia in many cases.
Tate conjecture and mixed perverse sheaves
利用混合反常层理论，将Lefschetz和Griffiths关于Hodge猜想的讨论推广到Tate猜想的情形，证明了因子的Tate猜想与非真簇的de Rham猜想、Tate-Shafarevich群的有限性以及解析数论中的一些定理密切相关.
It is shown that for a perverse sheaf $K$ on an abelian variety $X$ the integers $i$ for which the cohomology $H^i(X,K)$ does not vanish define an interval in the number line (under certain conditions on the field of definition of $K$)
We give a Tannakian description for categories of l-adic perverse sheaves on semiabelian varieties which combines a construction of Gabber and Loeser for algebraic tori with a generic vanishing theorem for the cohomology of constructible sheaves on abelian varieties. As an application we explain how the arising Tannaka groups on abelian varieties can be studied in terms of semiabelian degenerations via the functor of nearby cycles.
We show that there is a natural perverse sheaf on the moduli space of semistable sheaves on a smooth projective Calabi-Yau 3-fold which is locally the perverse sheaf of vanishing cycles for a local Chern-Simons functional. This gives us a cohomology theory whose Euler numbers are Donaldson-Thomas invariants.
On the action of the dual group on the cohomology of perverse sheaves on the affine grassmannian
Ginzburg和Mirkovic-Vilonen证明了一个连通约化群$G$的仿射格拉斯曼群上的$G(O)$ -等变逆层构成一个等价于对偶群$G^\vee$的有限维表示的张量范畴的张量范畴。证明采用了Tannakian形式。本文的目的是明确地构造$G^\vee$在反常层的整体上同调上的作用。
The notions of consistent pairs and consistent chains of t-structures are introduced. A theorem that two consistent chains of t-structures generate a distributive lattice is proven. The technique developed is then applied to the pairs of chains obtained from the standard t-structure on the derived category of coherent sheaves and the dual t-structure by means of the shift functor. This yields a family of t-structures whose hearts are known as perverse coherent sheaves.
In their article "Elementary construction of perverse sheaves", R.MacPherson and K. Vilonen show that on a Thom-Mather space X the category PervX of perverse sheaves is equivalent to the category C(F, G, T) whose objects are data of perverse sheaves on the complementary of the closed strata S, a local system on S and some gluing data. To show this equivalence of categories, they define a functor C going from the category PervX to the category C(F, G, T). This definition is based on the notion of perverse link. They do not define a quasi-inverse of this functor. moreover they have to consider f…
We consider non-commutative deformations of sheaves on algebraic varieties. We develop some tools to determine parameter algebras of versal non-commutative deformations for partial simple collections and the structure sheaves of smooth rational curves. We apply them to universal flopping contractions of length 2 and higher. We confirm Donovan-Wemyss conjecture in the case of deformations of Laufer's flops.
Duality between $W_n$-Cartier crystals and $\mathbb{Z}/p^n\mathbb{Z}$-perverse sheaves
本文推广 [Bau23] 的结果，建立了 $W_n$-Cartier 晶体与 $\mathbb{Z}/p^n\mathbb{Z}$-反常层之间的对偶关系。
We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalizes the one on projective spaces. This is done by generalizing the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterization of these sheaves in the particular case of projective spaces.
We study the full set of solutions to the ADHM equation as an affine algebraic set, the ADHM variety. We gather the points of the ADHM variety into subvarieties according to the dimension of the stabilizing subspace. We compute dimension, and analyze singularity and reducibility of all of these varieties. We also connect representations of the ADHM quiver to coherent perverse sheaves on $\mathbb{P}^2$ in the sense of Kashiwara.
Perverse sheaves with nilpotent singular support on the stack of coherent sheaves on an elliptic curve
本文通过定义椭圆曲线上凝聚层模栈的分层，显式描述了全局幂零锥的不可约分支，建立了球面 Eisenstein 层与这些分支之间的双射，并参数化了具有幂零奇异支集的反常层。
For a reductive group over an algebraically closed field of characteristic $p > 0$ we construct the abelian category of perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod $p$ Satake isomorphism. Along the way we prove that affine Schubert varie…
The derived moduli of perverse sheaves
本文构造了参数化复代数簇和紧实解析簇上可构造层的导出Artin叠，并证明每个反常函数给出反常层的开子叠，推广了通常的特征叠，作为应用构造了与穿孔Riemann曲面相关的上同调Hall代数的新例子。
In this note, we provide a quick introduction to the study of the Milnor fibration via the derived category and perverse sheaves. This is primarily a dictionary for translating from the standard topological setting to the derived category and/or the Abelian category of perverse sheaves.
In order to study wall crossing formula of Donaldson type invariants on the blown-up plane, Nakajima-Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima-Yoshioka's diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.
The convolution powers of a perverse sheaf on an abelian variety define an interesting family of branched local systems whose geometry is still poorly understood. We show that the generating series for their generic rank is a rational function of a very simple shape and that a similar result holds for the symmetric convolution powers. We also give formulae for other Schur functors in terms of characteristic classes on the dual abelian variety, and as an example we discuss the case of Prym-Tjurin varieties.
Gluing of perverse sheaves on the basic affine space
本文研究了由Kazhdan和Laumon引入的$G/U$（$G$是一个半单群p，$U$是$G$中博雷尔子群的幂幺根）上反常层的胶合构造. Kazhdan和Laumon证明了所有在胶合范畴中的Ext-群是有限维的，并且整体上同调维数是有限的。我们证明了这个猜想的第一部分。在附录中，我们证明了对应于常数层的胶合范畴中的简单对象具有无穷的上同调维数，从而反驳了上述猜想的第二部分。
Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.
Informal lecture notes with examples on sheaf theory and the derived category of sheaves; sheaves and Morse theory; perverse sheaves, and some applications to representation theory. Added Oct 2021: cellular perverse sheaves. Proofs are outlined with references for further details.
Computations with moduli of perverse point sheaves
给定纤维维数为1且有理奇点的复射影簇的双有理修正$X \to Y$，考虑$X/Y$上的反常点层的Bridgeland模空间$W \to Y$的主分支.我们给出了标准的规范化$W$符合转换（触发器）$X^+ \to Y$的$X \to Y$。首先，如果$X \to Y$和$W \to Y$的例外基因座具有余维$>1$，则这成立。其次，如果$X \times_Y X^+$的理想层在$X^+$上是平坦的，则至少存在一个态射$X^+ \to W$。我们使用一些例子说明结果，包括表面修改和标准的弗朗西亚翻转。
Duality between Cartier crystals and perverse $\mathbb{F}_p$-sheaves, and application to generic vanishing
该研究建立了诺特$F$-有限$\u200b\mathbb{F}_p$-概形上Cartier晶体与平展反常$\u200b\mathbb{F}_p$-层的范畴反等价，构造了Cartier晶体的上尖叫函子，给出相关经典结论的新证明，并推导出正特征阿贝尔簇上反常层的一般消亡结果。
Microlocal perverse sheaves
给出了复流形的射影余切丛上的微局部反常层叠加的一个显式构造。微局部反常层将被表示为解析ind-sheaves的复形，最近由Kashiwara-Schapira进行了研究。这种描述使我们能够制定的microlocal黎曼-希尔伯特对应，以建立一个等价的堆栈与堆栈的定期完整微微分模块。
We give examples of non-perverse parity sheaves on Schubert varieties for all primes.
External tensor product of categories of perverse sheaves
在一定的假设条件下，证明了伪流形X和Y上的可构造反常层范畴的Deligne张量积是X \times Y上的可构造反常层范畴. Deligne外张量积函子等同于几何外张量积。
Spreading out perverse sheaves
本文研究几何不可约反常层在算术基上的延拓，证明算术柏原猜想并推广分解定理与硬Lefschetz定理。
We generalize the decomposition theorem for perverse sheaves to Artin stacks with affine stabilizers over finite fields.
Addendum to the paper "Characteristic Cycles of Perverse Sheaves and Milnor Fibers"
我们注意到，定理5.4“反常层和Milnor纤维的特征圈”（P.Nang和K. Takeuchi，在线发表于Math.Zeitschrift，2004年10月15日）适用于“任何”非零复数$a$。
The titular, foundational work of Beilinson not only gives a technique for gluing perverse sheaves but also implicitly contains constructions of the nearby and vanishing cycles functors of perverse sheaves. These constructions are completely elementary and show that these functors preserve perversity and respect Verdier duality on perverse sheaves. The work also defines a new, "maximal extension" functor, which is left mysterious aside from its role in the gluing theorem. In these notes, we present the complete details of all of these constructions and theorems.
Explicit models for perverse sheaves
我们考虑广义反常层范畴，放松构造性条件，通过粘合$t$ -结构的过程，我们表现出明确的阿贝尔范畴定义的标准层范畴是等价的。特别地，我们能够实现反常层范畴作为层范畴的通常有界复形的非满交换子范畴。我们的方法使用归纳法。在本文中，我们限制自己的两层的情况下，但我们的结果推广到一般情况。
In characteristic zero, Bezrukavnikov has shown that the category of perverse coherent sheaves on the nilpotent cone of a simply connected semisimple algebraic group is quasi-hereditary, and that it is derived-equivalent to the category of (ordinary) coherent sheaves. We prove that graded versions of these results also hold in good positive characteristic.
Flops, flips and perverse point sheaves on threefold stacks
我们考虑两种情况下的双有理变换的Q-Gorenstein三倍：终端触发器的情况下，一个弗朗西亚翻转。在第一种情况下，我们表明，如果一个取代的三重由他们的规范覆盖堆栈，触发器符合模堆栈的反常点层。这增加了Kawamata的一个结果，显示了导出的等价性。我们的构造有一个缺点，即模空间是使用堆栈的表示来定义的。在第二种情况下，我们表明，如果采取一个Francia翻转$X \to Y$，并取代$X$由其规范覆盖堆栈，那么翻转符合一个版本的Bridgelan的模空间的pwrverse点层涉及一个新的反常。这是Kawamata的另一个结果。同样有一个缺点--结果的证明依赖于翻转的存在，这与布里奇兰和陈的结果形成鲜明对比，其中相关的翻转出现为构造的结果。
Perverse Euler Characteristics of Hermitian Locally Symmetric Spaces
本文证明非紧型有限体积局部Hermitian对称空间的反常欧拉示性数非负，通过建立光滑环面紧化上对数余切丛的nef性并结合正性判据，并推广到模空间。
We discuss some results on parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves
Perverse coherent sheaves on symplectic singularities
本文提出辛奇点上反常凝聚层的概念，证明其给出泊松层Grothendieck群的简单对象基，并涵盖幂零锥与仿射格拉斯曼流形的特例。
We briefly introduce the theory of perverse sheaves with special attention to the topological situation where strata can have odd dimension. This is part of a project to use perverse sheaves on the topological reductive Borel-Serre compactification of a Hermitian locally symmetric space as a tool to study perverse sheaves on the Baily-Borel compactification, a projective algebraic variety. We sketch why the decomposition theorem holds for the natural map between the reductive Borel-Serre and the Baily-Borel compactifications. We demonstrate how to calculate extensions of simple perverse sheave…
We discuss the relationship between kernels, images and cokernels of morphisms between perverse sheaves and induced maps on stalk cohomology.
Artin perverse sheaves
本文研究 Artin ℓ-adic 复形上反常 t-结构的存在性，证明在维数小于2的优概型上成立并给出三维反例，进而在有限域有限型概型上构造反常同伦 t-结构，刻画其单对象并证明无权截断函子的 t-正合性。
We study intersections of complex Lagrangian in complex symplectic manifolds, proving two main results. First, we construct global canonical perverse sheaves on complex Lagrangian intersections in complex symplectic manifolds for any pair of {\it oriented} Lagrangian submanifolds in the complex analytic topology. Our method uses classical results in complex symplectic geometry and some results from arXiv:1304.4508. The algebraic version of our result has already been obtained by the author et al. in arXiv:1211.3259 using different methods, where we used, in particular, the recent new theory of…
Period sheaves via perverse pullbacks
本文利用反常拉回函子为 Hamiltonian 空间构造周期层，通过维数约化同构揭示其对普通拉回的细化及与微茎函子的联系，并恢复余切与 Whittaker 情形的已知构造。
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