Microlocalization of Ind-sheaves
M. Kashiwara, Pierre Schapira, F. Ivorra, I. Waschkies
设X是一个C-无穷流形。我们从X上的ind层的有界复形的导出范畴到X的余切丛上的有界复形，构造了一个微局部化函子$μ_X$.这个函子推广了经典的微局域化理论。
Let X be a C-infinity manifold. We construct a microlocalization functor $μ_X$ from the derived category of bounded complexes of ind-sheaves on X to the one on the cotangent bundle of X. This functor generalizes the classical theory of microlocalization.
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Microlocalization of subanalytic sheaves
本文定义了次解析层的特化和微局部化。将这些函子应用于调和函数和Whytney全纯函数的层，我们利用次解析层理论得到了调和微局部化和形式微局部化的统一描述。
Ind-Sheaves, distributions, and microlocalization
这是一篇正在进行的长篇论文的公告。在局部紧空间上，我们引入了ind-层（具有紧支撑的层范畴的ind-对象）的栈，并构造了层上通常的六种运算的类似运算.作为应用，我们对待，在ind-sheaves的形式主义，功能的增长条件，如“回火全纯函数”的复流形。我们还构造了一个从流形上的层范畴到余切丛上的ind层范畴的微局部化函子。
Characteristic cycles of constructible sheaves and microlocalization
研究正特征域上光滑概型可构造层的奇异支撑和特征圈，探讨类似超越情形的微局部化构造在代数情形的可行性，取得部分肯定成果。
DG-methods for microlocalization
对于复流形$X$，微微分算子环$\E_X$作用在微局部化$μhom(F,Ø_X)$上，对于$F$在$X$上的导出范畴中。Kashiwara，Schapira，Ivorra，Waschkies证明了，作为他们新的ind-sheaves微局部化函子$μ_X$的副产品，$μhom(F,Ø_X)$实际上可以被定义为$\E_X$ -模的导出范畴的对象：这是从$μ_X Ø_X$集中在一个度的事实得出的。本文证明了回火微局部化也是$\E_X$ -模的导出范畴的一个对象。由于我们不知道是否回火版本的$μ_X Ø_X$是集中在一个程度，我们介绍了一种方法来建立合适的决议，其中$\E_X$的行动是实现的范畴内的配合物。我们定义了一个版本的de Rham代数上的次解析网站，这是拟内射的，我们的工作在范畴的DG-模在这个de Rham代数，而不是导出范畴的层。
Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we show how Sato's specialization and microlocalization functors have a natural enhancement, and discuss some of their properties.
Singular supports, characteristic cycles, and microlocalization of constructible sheaves
本文证明Saito微局部奇异支集包含于Beilinson奇异支集，回答Saito问题，并解决相关支集估计猜想，证明特征类猜想。
Three lectures on Algebraic Microlocal Analysis
这三个讲座介绍了微观局部分析的一些基本和经典方面。从Sato的微局部化函子和层的微支撑出发，构造了层的霍赫希尔德同调的微局部模拟，并应用它恢复了D-模和椭圆对的指标定理.在第三讲中，我们构造了温带和Whitney全纯函数的ind层，并给出了它在非正则完整D-模研究中的一些应用。
Coherent sheaves on general K3 surfaces and tori
设M是K3曲面或偶维紧环面。我们证明了M上相干层的范畴与复结构的选择无关，如果这个复结构是一般的。
Ind-coherent sheaves
我们发展了概型和栈上的独立相干层理论。ind-coherent层的范畴与拟凝聚层的范畴密切相关，但并不等价，这种差异对于范畴几何朗兰兹对应的表述至关重要。
Note on Relation between Enhanced Ind-Sheaves and Enhanced Subanalytic Sheaves
本文通过阐明增强归纳层与增强亚解析层之间的关系，解释了D'Agnolo–Kashiwara与Kashiwara关于全纯D-模及不规则全纯D-模的Riemann-Hilbert对应定理之间的联系。
Functorial properties of the microsupport and regularity for ind-sheaves
M。Kashiwara和P. Schapira在“ind-sheaves I的微局部研究：微支持和规律性”中。在本文中，我们研究了几个函子操作下的微支撑的行为和特征的“微局部”的ind-sheaves是正规沿对合流形。作为应用，我们证明了如果凝聚D-模M沿对合流形V正则（在“具有正则奇点的微分方程组及其边值问题“的意义下）则M的温带全纯解的内层沿V是正则的。另一个应用是层在次解析位置上的微支撑的概念，它可以直接给出，而不使用更复杂的IND-槽的语言。
Sheaves on subanalytic sites
在Asterisque 271中，作者引入了ind-sheaf的概念，并在这个框架中定义了六个Grothendieck运算。他们定义了次解析层，并通过将次解析层纳入ind-层的范畴，获得了Grothendieck运算的六个形式。本文的目的是在次解析格点的框架下直接构造六个Grothendieck运算，避免了ind-sheaves的重理论。作为应用，我们展示了如何恢复温带和惠特尼全纯函数的次解析层。
A. D'Agnolo and M. Kashiwara proved that their enhanced solution functor induces a fully faithful embedding of the triangulated category of holonomic D-modules into the one of R-constructible enhanced ind-sheaves. In this paper, we define C-constructible enhanced ind-sheaves and show that the triangulated category of them is equivalent to its essential image. Moreover we show that there exists a t-structure on it whose heart is equivalent to the abelian category of holonomic D-modules.
Coherent sheaves on generic compact tori
设T是紧复环面，dim T>2.我们证明了T上相干层的范畴与复结构的选择无关，如果这个复结构是一般的。该证明独立于数学AG/0205210，其中对K3曲面和偶数维环面证明了相同的结果。
Microlocal homology
本文在 l.c.i. 概形的移位余切丛上构造反常层 μ_Z 以精细化 Nadler 微局部同调，并证明其等价于 DT 层，从而为移位余切丛的 DT 层提供新构造。
Finiteness properties of Ind-Sheaves with Ring Actions
本文研究带环作用的 ind-层的有限性，证明凝聚 ind-$β\mathcal{A}$-模范畴与凝聚 $\mathcal{A}$-模范畴等价，其中 $\mathcal{A}$ 为域 $\Bbbk$ 上的 k-代数层。
Microlocal study of ind-sheaves I: micro-support and regularity
定义了ind层的微支撑和正则性概念，并利用切触变换证明了它们的不变性。我们将所得结果应用于D-模的温带全纯解的ind层。我们证明了这样的ind层的微支撑是相应的D-模的特征簇，并且如果D-模是正则完整的，则ind层是正则的。
It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particular, weak equivalences between these monoids induce Quillen equivalences between the corresponding categories of modules. On the other hand, it is also proved that the functor of pre-compact bornology applied to simplicial $C^\infty$-rings preserves and reflects weak equivalences…
Recently, the Riemann-Hilbert correspondence was generalized in the context of general holonomic D-modules by A. D'Agnolo and M. Kashiwara. Namely, they proved that their enhanced de Rham functor gives a fully faithfully embedding of the derived category of cohomologically holonomic complexes of D-modules into the derived category of complexes of real constructible enhanced ind-sheaves. In this paper, we study a condition when a complex of real constructible enhanced ind-sheaves $K$ is induced by a cohomologically holonomic complex of D-modules. We characterize such $K$ in terms of the restric…
Ind-abelian categories and quasi-coherent sheaves
研究了当一个ind-对象范畴是阿贝尔范畴时的问题。我们的答案允许进一步推广的概念弱Tannakian范畴介绍了作者。作为应用，我们证明了在适当的条件下，具有分解性质的两个格式的乘积上的相干层范畴由两个因子的相干层范畴的Deligne张量积给出.为此，我们证明了具有分解性质的拟紧半分离格式类在纤维积下是闭的。
We develop the notion of indscheme in the context of derived algebraic geometry, and study the categories of quasi-coherent sheaves and ind-coherent sheaves on indschemes. The main results concern the relation between classical and derived indschemes and the notion of formal smoothness.
Some examples of isomorphisms induced by Fourier-Mukai functors
给出了由Fourier-Mukai函子诱导的层模同构的一些例子。作为应用，我们给出了交换曲面和K3曲面上层的某些模空间的变形类型的另一个证明。
We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way we produce a derived equivalence between two compact hyperkaehler 2g-folds that are not birational, for every g >= 2. We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived-equivale…
Relative Tate Objects and Boundary Maps in the K-Theory of Coherent Sheaves
我们调查的性质，相对类似物的容许的Ind，Pro，和基本泰特对象对的确切范畴，并给出这些类别是阿贝尔的标准。引入了相对指数映射，作为应用，我们推导出了相干层K理论中边界态射在Noether格式上的描述。
We consider the category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves where $\mathbb{X}$ is a weighted noncommutative regular projective curve over a field $k$. This category is a hereditary, locally noetherian Grothendieck category. We classify all indecomposable pure-injective sheaves and all cotilting sheaves of slope $\infty$. In the cases of nonnegative orbifold Euler characteristic this leads to a classification of pure-injective indecomposable sheaves and a description of all large cotilting sheaves in $\operatorname{Qcoh}\mathbb{X}$.
Coherent sheaves, sheared D-modules, and Hochschild cochains
本文证明拟光滑概形上的归纳凝聚层范畴与其移位余切丛上的剪切D-模范畴之间存在自然的Morita等价，并推广到拟光滑Artin叠，从而为函数域上非分歧自守函数在算术Arthur参数叠上的局部化提供了精确表述。
Coherent Sheaves on Quiver Varieties and Categorification
构造了Nakajima簇上凝聚层的导出范畴上的几何范畴李代数作用。这些行动分类Nakajima的建设卡茨-穆迪代数表示的K-理论上的K-簇。我们在这些导出范畴上定义了一个诱导仿射辫群作用。
The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let $X$ be a non-singular irreducible complex surface and let $E$ be a vector bundle of rank $n$ on $X$. We use the $m$-elementary transformation of $E$ at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;c_1,c_2)$ to $Hilb_{\, \ma…
Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann-Hilbert correspondence for holonomic D-modules of A. D'Agnolo and M. Kashiwara to describe the Stokes phenomenon topologically. Using this description, we perform a topological computation of the Fourier-Laplace transform of a D-module of pure Gaussian type in this framework, recovering and generalizing a result of C. Sabbah.
On Banica sheaves and Fano manifolds
在本文中，我们讨论秩> 1的凝聚层，其投射化产生光滑簇-这种类型的簇也称为光滑涡卷。我们证明了这些品种的一些基本性质，我们给出了一些相关的例子。我们对这种类型的大指数法诺流形进行分类。
Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes
本文在带平坦仿射态射到具有对偶化复形的ind-诺特ind-概形的ind-概形上，构造了拟凝聚挠层半导出范畴的半无限张量结构，其中半张量积混合了基底余张量积与纤维导出张量积，对偶化复形的逆像为单位对象，部分实现了半无限代数几何纲领。
We consider the derived category of coherent sheaves on a complex vector space equivariant with respect to an action of a finite reflection group G. In some cases, including Weyl groups of type A, B, G_2, F_4, as well as the groups G(m,1,n), we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of G. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces V^g/C(g), where C(g) is the centralizer subgroup of g in G. In the case of the Weyl groups the construction uses some key results about the…
Temperate holomorphic solutions and regularity of holonomic D-modules on curves
设$X$是复流形.在“Ind-sheaves的微局部研究I：微支撑和规则性”中，M。Kashiwara e P. Schapira猜想一个完整D-模$\shm$是正则完整模当且仅当$R\mathcal{I}{hom}_{β_X\shd_X}(β_X\shm,\sho_X^t)$是正则的（在“Ind-sheaves I的微局部研究：微支撑和正则性”的意义上），这个猜想的“仅当”部分紧接着“Ind-sheaves I的微局部研究：微支撑和正则性”。我们的目的是证明这个猜想在一维。
The theory of integral, or Fourier-Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General "representation theorems" identify all reasonable linear functors between categories of perfect complexes (or their "large" version, quasi-coherent sheaves) on schemes and stacks over some fixed base with integral kernels in the form of sheaves (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of c…
We define functors on the derived category of the moduli space M of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the K-theory of the moduli space M, thus generalizing the action studied by Nakajima, Grojnowski and Baranovsky in cohomology.
Geometric Langlands in positive characteristic from characteristic zero
本文在正特征域上证明了几何朗兰兹猜想的部分结论，即幂零奇异支撑的自守层范畴与朗兰兹参数栈部分分量上的ind-凝聚层范畴等价。
Sheaves of Higher Categories on Generalised Spaces
本文研究同伦层沿Yoneda嵌入的右Kan扩张，证明其保持同伦层并诱导等价，应用于微分几何向量丛下降、丛格拉斯曼下降及光滑场论值提取。
Degenerations of shifted symplectic moduli stacks of sheaves
本文通过推广 Brav--Dyckerhoff 的构造，证明了 Calabi--Yau 族上凝聚层的导出模叠具有相对平移辛结构，并识别其为伪完美对象模的子叠，为枚举几何提供了潜在应用。
A note on sheaves without self-extensions on the projective $n$-space
设${\bf P}^n$是复数上的射影$n-$空间。本文证明了${\bf P}^n$上不可分解的刚性凝聚层有平凡自同态代数。这推广了Drezet关于$n=2.$的一个结果
Moduli of Flags of Sheaves and their K-theory
我们引入了P^2上层的标志模空间，并利用它们得到了P^2上层的通常模空间的导出范畴之间的函子。这些函子通过某些明确的公式诱导了shuffle代数对K-理论的作用。
We introduce a notion of compatibility for families $(\mathcal{F}_{\ell})_{\ell}$ of bounded constructible $\ell$-adic complexes of étale sheaves on schemes. For schemes of finite type over a field, this notion is preserved by the usual six functors. We prove that the compatibility of a family is preserved by the nearby cycles functor and by the linearized $\varepsilon$-factors introduced recently by the author. We establish independence of $\ell$ for the characteristic cycles and characteristic $\varepsilon$-cycles of compatible families.
Quasicoherent sheaves for dagger analytic geometry
本文基于 Ind-Banach 空间建立 dagger 解析簇上的拟凝聚层理论，给出解析拓扑下降、紧支集前推伴随 $f_! \dashv f^!$、切除序列及 Grothendieck 对偶定理。
Quasicoherent sheaves on complex noncommutative two-tori
在复非对易二环面$T$上引入拟相干层的概念，作为$T$上全纯向量丛范畴中的一个ind-object.推广了math.QA/0211262和math.QA/0308136的结果，证明了$T$上的拟相干层的导出范畴等价于相应的椭圆曲线上的通常拟相干层的导出范畴.我们定义了可以取任意非负实值的拟相干层的秩。我们研究范畴$\Qcoh(η_T)$，它是通过取拟相干层范畴与秩为零的对象的子范畴（称为挠层）的商而得到的。我们表明，有限秩的投射对象在$\Qcoh(η_T)$分类到同构的秩。我们还证明了$\Qcoh(η_T)$中有限秩对象的子范畴等价于半遗传代数上的有限表示模范畴。
The étale fundamental group and $F$-divided sheaves in characteristic $p>0$
本文证明特征 p 下光滑射影簇态射诱导的 étale 基本群满射或同构，可分别保证 F-divided 层拉回函子完全忠实或范畴等价，推广 Esnault–Mehta 定理。
Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds
这封信是为纪念克里斯特·基塞尔曼教授而写的。乘子理想层确定关键估计失败的位置和程度。在这篇笔记中，我们将讨论和解释乘数理想层概念的历史演变，特别是在终止或限制无限过程的背景下，从不稳定子层的观点来解释。我们还将讨论利用动态乘子理想层和奇异放大复Monge-Ampere方程构造Fano流形中的有理曲线的方法。这一办法仍在制定之中，细节正在拟订之中。我们将指出仍需制定的细节。
Quantum McKay Correspondence and Equivariant Sheaves on the Quantum Projective Line
本文利用量子McKay对应，构造了单位根上量子投影线上G-等变层的“导出范畴“。更准确地说，我们使用U_{q} sl（2）在单位根上的表示论来构造对称代数和结构层的类似物。结构层的类似物实际上是一个复形，而且它是一个dg-代数。我们的导出范畴通过这个dg-代数的G-等变dg-模的三角范畴而产生。然后，我们将其与图G的表示（Γ，\Om）相关联，其中Γ是通过量子麦凯对应与G相关联的A、D、E图，并且\Om是Γ的定向。作为推论，我们的范畴范畴化了相应的根格，不可分解层给出了相应的根系。
The goal of this work is to construct a perverse t-structure on the infinity-category of l-adic LG-equivariant sheaves on the loop Lie algebra Lg and to show that the affine Grothendieck-Springer sheaf S is perverse. Moreover, S is an intermediate extension of its restriction to the locus of ``compact" elements with regular semi-simple reduction. Note that classical methods do not apply in our situation because LG and Lg are infinite-dimensional ind-schemes.
Serre functor and $\mathbb{P}$-objects for perverse sheaves on $\mathbb{P}^n$
本文证明了 $\mathbb{P}^n$ 上可构造导出范畴的逆 Serre 函子可由开层简单反常层的 $\mathbb{P}$-扭转实现，并证明所有不可分解反常层均为 $\mathbb{P}$-like 对象且显式构造了其自同态空间的生成态射。
Derived categories of functors and Fourier--Mukai transform for quiver sheaves
设C是小范畴，A是任意范畴.考虑范畴C（A），它的对象是从C到A的函子，它的态射是自然变换。给定一个函子F：A --> B，得到一个导出函子F_C：C（A）--> C（B）。若A和B是阿贝尔范畴，则C（A）和C（B）也是阿贝尔范畴，且有两个函子R（F_C）：D（C（A））--> D（C（B））和（RF）_ C：C（D（A））--> C（D（B））.本文的目的是：1）找到D（C（A））与C（D（A））之间的关系; 2）把函子R（F_C）与（RF）_C联系起来。作为应用，我们证明了一个版本的Mukai定理的层。
Tensor triangular geometry of filtered objects and sheaves
本文利用∞-范畴与无点方法，通过分配格和半格等工具计算了滤过或分次对象所成张量三角范畴中紧对象的 Balmer 谱，并应用于概形滤过导出范畴与滤过谱同伦范畴。
Wall And Chamber Structure For A Special Biserial Algebra Coming From Perverse Sheaves on $\mathbb{P}^n$
研究来自\(\mathbb{P}^n\)上反常层的特殊双列代数的壁和腔结构，通过特殊双列代数不可分解模分类推断其为有限表示型并明确壁的描述，且该结构与\(\mathbb{P}^n\)上可构造层有界导出范畴稳定性条件空间开子集的腔结构一致。
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