Morse theory and tilting sheaves

Parity sheaves and tilting modules

We show that tilting modules and parity sheaves on the affine Grassmannian are related through the geometric Satake correspondence, when the characteristic is bigger than an explicit bound.

Modular perverse sheaves on flag varieties I: tilting and parity sheaves

In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.

Representing the big tilting sheaves as holomorphic Morse branes

We introduce Morse branes in the Fukaya category of a holomorphic symplectic manifold, with the goal of constructing tilting objects in the category. We give a construction of a class of Morse branes in the cotangent bundles, and apply it to give the holomorphic branes that represent the big tilting sheaves on flag varieties.

加权射影线倾斜理论的几何-组合方法

Geometric-combinatorial approaches to tilting theory for weighted projective lines

本研究针对(2,2,n)型加权射影线的凝聚层范畴，构建带标记点柱面的几何-组合模型，建立不可分解层与斜曲线、伪三角剖分与倾斜层的双射，证明倾斜图连通性，为倾斜理论提供新组合视角。

Whittaker平均中心层的倾斜性质

The tilting property of Whittaker averaged central sheaves

通过微局部描述Iwahori-Whittaker平均的核，推广了Faergeman和Raskin关于反温和性等价于不规则奇点支撑的定理，并利用Radon变换将Whittaker平均中心层的倾斜性质推广到整数系数。

Zariski locality of quasi-coherent sheaves associated with tilting

A classic result by Raynaud and Gruson says that the notion of an (infinite dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for n = 0) of the locality of more general notions of quasi-coherent sheaves related to (infinite dimensional) n-tilting modules and classes. Here, we prove the latter locality for all n and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is noetherian).

Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$. The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even…

Tilting modules in category O and sheaves on moment graphs

We describe tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of category O. We prove that they map to Braden-MacPherson sheaves constructed along the reversed Bruhat order under Fiebig's localization functor. By this means, we get character formulas for tilting modules and explain how Soergel's result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration.

普遍单值大倾斜层

Universal monodromic tilting sheaves

本文在基础仿射空间上构造了普遍单值大倾斜层并计算其自同态，恢复了Soergel的pro-unipotent Endomorphismensatz，推导出Eberhardt关于不完备Koszul对偶的猜想。

The nonequivariant coherent-constructible correspondence and tilting

The coherent-constructible correspondence is a relationship between coherent sheaves on a toric variety X, and constructible sheaves on a real torus T. This was discovered by Bondal, and explored in the equivariant setting by Fang, Liu, Treumann and Zaslow. In this paper we collect partial results towards a proof of the non equivariant coherent-constructible correspondence. Also, we give applications to the construction of tilting complexes in the derived category of toric DM stacks.

Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture

Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good chara…

coherent sheaves on the projective line的KLR-Schur代数：反演与PBW基

KLR-Schur algebra of coherent sheaves on the projective line: Tilting and PBW bases

本文研究了$\mathbb{P}^1$上凝聚sheaves的KLR-Schur代数，通过反演等价和稳定性条件构造代数插值，并构建PBW基。

Free-monodromic mixed tilting sheaves on flag varieties

In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties, see [P. Achar, S. Makisumi, S. Riche, and …

Tilting exercises

This is an application of the theory of tilting objects to the geometric setting of perverse sheaves. We show that this theory is a natural framework for Beilinson's gluing of perverse sheaves construction. In the special case of Schubert stratification of a flag variety we get a short proof of Soergel's "Struktursatz", and describe (following a conjecture of Kapranov) Serre functor for category O. Some of our results were obtained independently by Rouquier.

Tilting on non-commutative rational projective curves

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show the dimension of the bounded d…

通过理查德森品种研究翻转组合系列的几何学

The geometry of tilting composition series via Richardson varieties

研究翻转sheaves在旗形簇上的(加权)乔丹-霍尔乘数通过理查德森品种在朗兰兹对偶旗形簇上的超上同调几何解释，并提供基于ℓ-卡兹德-卢斯滕布希多项式的显式公式。

Filtrations of tilting modules and costalks of parity sheaves

Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski constructed a filtration of weight spaces of a G module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. The goal of this article is to generalize these results to positive characteristics.

Morse groups in symmetric spaces corresponding to the symmetric group

We describe the Morse groups of the nearby cycles sheaves on the nilcones in three classical symmetric spaces.

Tilting bundles and the "missing part" on the weighted projective line of type $(2, 2, n)$

This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type $(2, 2, n)$, and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.

Lecture notes on sheaves and perverse sheaves

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.

Silting complexes of coherent sheaves and the Humphreys conjecture

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and each indecomposable tilting vector bundle $T$ on $C$ a certain complex $S(C,T)$ of $G \times \mathbb{G}_m$-equivariant coherent sheaves on $\mathcal{N}$. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-$t$-structure. We then show that if $p$ is larger than the Coxeter number, then the hypercoho…

凝聚层导出范畴上的 Co-$t$-结构与倾斜模的上同调

Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules

本文在幂零锥及抛物 Springer 解消的凝聚层导出范畴上构造 co-$t$-结构，用以刻画外尔奇偶对象推进的不可分解性，并证明 $A$ 型、$p>h$ 时相对 Humphreys 猜想的概形论形式。

扭曲仿射旗簇上的反常层与Langlands对偶

Perverse sheaves on twisted affine flag varieties and Langlands duality

本文用Langlands对偶数据描述温和分歧约化群仿射旗簇上的Iwahori-Whittaker等变反常层，推广了Arkhipov-Bezrukavnikov的工作，并通过扩展Wakimoto层理论及建立倾斜性质解决了非分裂群的中心函子缺失问题。

型为 $(2,2,2,p)$ 的 Geigle–Lenzing 射影平面上的 ACM 倾斜丛

ACM tilting bundles on a Geigle-Lenzing projective plane of type $(2,2,2,p)$

本文研究型 $(2,2,2,p)$ 的 Geigle–Lenzing 射影平面上的 ACM 倾斜丛，刻画线丛倾斜丛、给出构造程序，并获得其分类结果。

Parity Sheaves

Given a stratified variety X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of "parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain r…

Stacks similar to the stack of perverse sheaves

We introduce, on a topological space X, a class of stacks of abelian categories we call "stacks of type P." This class of stacks includes the stack of perverse sheaves (of any perversity, constructible with respect to a fixed stratification), and is singled out by fairly innocuous axioms. We show that some basic structure theory for perverse sheaves holds for a general stack of type P: such a stack is locally equivalent to a MacPherson-Vilonen construction, and under certain connectedness conditions its category of global objects is equivalent to the category of modules over a finite-dimension…

$(p,q)$型加权射影线的几何模型

Geometric model for weighted projective lines of type $(p,q)$

本研究构建$(p,q)$型加权射影线凝聚层范畴的环面几何模型，建立不可分解层与曲线同伦类的双射，给出倾斜图组合描述，证明自同构群与映射类群同构及作用相容性，并给出垂直范畴的几何描述。

Calculations with graded perverse-coherent sheaves

In this paper, we carry out several computations involving graded (or $\mathbb{G}_{\mathrm{m}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the $\mathbb{G}_{\mathrm{m}}$-action on certain normalized (or "canonical") simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the coho…

仿射旗流形上的混合模 perverse sheaves 与 Koszul 对偶

Mixed modular perverse sheaves on affine flag varieties and Koszul duality

本文基于技术假设，证明了仿射旗流形上不可分解 tilting perverse sheaves 的重数公式，并解释了连接混合模 perverse sheaves 与普通 perverse sheaves 的“degrading functor”的构造。

反常层与薄厚仿射旗 manifold 上的 t-结构

Perverse sheaves and t-structures on the thin and thick affine flag varieties

本文研究了与裂简约群 G 关联的薄厚仿射旗 manifold 上的 Iwahori-等变反常层范畴，通过扩展前期工作，给出了厚范畴的反球形商描述，并证明了图像在 Iwahori-Whittaker 分类中的卷积-精确反常层是投射的。

Frobenius 核的块的几何模型

A geometric model for blocks of Frobenius kernels

本文基于 Steinberg 张量积公式的几何对应，定义并研究了 Langlands 对偶群的仿射 Grassmannian 上的反常层范畴，为代数群 Frobenius 核的表示块提供几何模型，并证明其具有射影、内射对象且为最高权范畴。

通过一对对偶例外序列实现最高权范畴

Highest weight categories via pairs of dual exceptional sequences

本文通过双偶例外序列提出判断abelian范畴为最高权的标准，并应用于几何场景，证明了某些范畴的最高权性质及相关特殊对象的几何描述。