A Remark on Rigidity of BN-sheaves

关于齐性空间上凝聚层的导出范畴的几点注记

Some remarks on the derived categories of coherent sheaves on homogeneous spaces

本文首先证明了光滑基上平坦族凝聚层的导出范畴的半正交分解的一个一般定理。基于math.AG/0510670的结果，我们证明了经典型旗簇上相干层的导出范畴是由完全例外集生成的。最后，我们在小秩辛群的某些齐性空间的导出范畴中找到了完全例外集。

Remarks on 1-motivic sheaves

We generalize the construction of the category of 1-motives with torsion ${}^tM_1$ (introduced by Barbieri-Viale, Rosenschon and Saito) as well as the construction of the category of 1-motivic sheaves ${\rm Shv}_1$ (defined by Barbieri-Viale and Kahn) to perfect fields $k$ (without inverting the exponential characteristic). For $k$ transcendental over the prime field we extend a result of Barbieri-Viale and Kahn, showing that ${}^tM$ and ${\rm Shv}_1$ have equivalent bounded derived categories.

A remark on the Castelnuovo-Mumford regularity of powers of ideal sheaves

We show that a bound of the Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety $X\subseteq\mathbb{P}^r$ is sharp exactly for complete intersections, provided the variety $X$ is cut out scheme-theoretically by several hypersurfaces in $\mathbb{P}^r$. This generalizes a result of Bertram-Ein-Lazarsfeld.

A few remarks concerning the geometry of the moduli spaces of semistable sheaves supported on plane curves of multiplicity four

This paper has been withdrawn by the author due to a crucial error in the proofs. The error has been corrected and the paper has been expanded in arXiv:0910.5327

关于导出弧空间的高阶同调sheaf的注记

A Remark on Higher Homotopy Sheaves of Derived Arc Spaces

本文探讨导出弧空间在光滑方案与局部完整交方案中的同调sheaf特性，指出其与经典几何空间无异。

A remark on braid group actions on coherent sheaves

A construction of braid group actions on coherent sheaves using mixed Hodge modules and some well known constructions from geometric representation theory is given.

A remark on the tensor product of SC-reciprocity sheaves

We give a description of the tensor product of SC-reciprocity presheaves with transfers in terms of $K$-group of geometric type, and we study a structure of the tensor product of $\mathbb{G}_a$ and $\mathbb{G}_a$. We apply our description to give a description of Chow group of 0-cycles with modulus of products of curves. We also show that the tensor product of $\mathbb{G}_a$ and $\mathbb{G}_m$ is isomorphic to the sheaf $Ω^1$ of Kähler differential forms as reciprocity sheaves over characteristic zero.

概率-étale 动力学与固有刚性

Pro-étale motives and solid rigidity

本文引入了概率-étale 动力学与谱，并展示其赋予六种运算。通过凝聚范畴理论定义固有过程，证明其与Fargues-Scholze的固有sheaves接近，从而获得固有动力学实现函子，扩展了ℓ-adic实现函子。

邻近周期层的 Hodge-Grothendieck 类与单值化不变量

Hodge-Grothendieck classes and monodromy invariants of nearby cycles sheaves

研究邻近周期层的 Hodge-Grothendieck 类与单值化不变量，评述邻近周期函子并证明广义局部不变周期结果。

Wall crossing of the moduli spaces of perverse coherent sheaves on a blow-up

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.

关于对数平展层化的注记

Remarks on logarithmic étale sheafification

本文研究对数概形上预层的对数平展层化问题，给出预层成为全对数平展层的判定准则，明确结构层与对数结构为对数平展层的情形，并证明满足对数平展下降的对数概形上对数皮卡群是全对数平展拓扑中的栈。

Some remarks on the semi-positivity theorems

We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the dualizing sheaves of smooth varieties. Using the weight spectral sequence of mixed Hodge modules, we then reduce the semi-positivity theorem for the higher direct images of dualizing sheaves to the smooth case where the assertion is well known. This may be used to simplify some constructions in a recent paper of Y. Kawamata. We also give a simple proof of the semi…

Some remarks of Hochschild homology and semi-orthogonal decompositions

Given a nontrivial semi-orthogonal decomposition $\Perf(\X)=\langle \mathcal{A},\mathcal{B}\rangle$, and assume that the base locus of $ω_{\X}$ is a proper closed subset, it was proved by Kotaro Kawatani and Shinnosuke Okawa that all skyscraper sheaves $\k(\x)$ with $\x\notin \mathsf{Bs}\vert ω_{\X}\vert$ belong to exactly one and only one of the components. It is natural to ask which one it is, and whether we can determine this by certain linear invariants. In this note we use Hochschild homology of derived category of coherent sheaves with support to provide another proof that if the $-\n^þ$…

交错层的拟遗传性

On the quasi-hereditary property for staggered sheaves

设G是一个代数闭域上的代数群，作用在一个具有n个轨道的簇X上。交错层是X上的G-等变相干层的某些复合体，它们似乎具有许多显著的性质。本文在交错层范畴中构造了“标准”和“共标准”对象，并证明了交错层范畴中有足够的投射和内射。

Some remarks on phantom categories and motives

A phantom category is an admissible subcategory with vanishing Grothendieck group of the bounded derived category of coherent sheaves on a smooth projective variety. The goal of this paper is to study the abstract situation when such a category appears and establish some results which provide evidence for the idea that these categories are invisible on the level of Chow motives.

关于Ricci平坦流形导出等价的注记

Remarks on derived equivalences of Ricci-flat manifolds

我们给出的结果表明，一个Ricci平坦流形在其不可约因子中的分解反映在相干层的导出范畴中。更确切地说，我们证明了一个光滑的射影簇，是派生等价于一个阿贝尔簇，分别。不可约辛簇是同一类型的。本文还证明了平凡典范丛流形的Caldararu定理，证明了霍赫希尔德同调的修正HKR同构与模结构相容.

Remarks on the nonequivariant coherent-constructible correspondence for toric varieties

We prove the following result of Bondal's: that there is a fully faithful embedding $κ$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $κ$ is an open problem.

Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles

We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.

Remarks on the Grothendieck-Teichmüller group and A. Beilinson's gluing

In this note, we study A. Beilinson's gluing for perverse sheaves in the case of the diagonal arrangement and its relation to the Grothendieck-Teichmüller group. We also explain a relation to the Etingof-Kazhdan quantisation

关于三角范畴的生成元和维数的注记

Remarks on generators and dimensions of triangulated categories

本文证明了光滑拟射影曲线上凝聚层的有界导出范畴的维数等于1。我们还讨论了这些范畴的维数谱。

A remark on Deligne's finiteness theorem

Over a connected geometrically unibranch scheme $X$ of finite type over a finite field, we show finiteness of the number of irreducible $\bar \Q_\ell$-lisse sheaves, with bounded rank and bounded ramification in the sense of Drinfeld, up to twist by a character of the finite field. On $X$ smooth, with bounded ramification in the sense of bounding the Swan conductors on curves, this is Deligne's theorem. Version 2: We prove also that for Deligne's finiteness theorem, it is enough to assume $X$ normal. However, the proof uses the smooth case, unlike the proof in the case of bounded ramification …

不稳定 étale motives

Unstable étale motives

本文在有界 étale 上同调维数的基概形上证明 anima 的 p-完备 étale A¹-不变层刚性，并用于构造不稳定 étale 实现函子。

Hall algebras in the derived category and higher rank DT invariants

We remark that the combination of the works of Ben-Bassat-Brav-Bussi-Joyce and Alper-Hall-Rydh imply the conjectured local description of the moduli stacks of semi-Schur objects in the derived category of coherent sheaves on projective Calabi-Yau 3-folds. This result was assumed in the author's previous papers to apply wall-crossing formulas of DT type invariants in the derived category, e.g. DT/PT correspondence, rationality, etc. We also show that the above result is applied to prove the higher rank version of DT/PT correspondence and rationality.

Rational rigidity for E_8(p)

We prove the existence of certain rationally rigid triples E8 in good characteristic and thereby show that these groups over the prime field occur as Galois groups over the field of rational numbers. We show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0. As a byproduct we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a classification of possible overgroups in exceptional groups containing regular unipotent elements.

复动力系统中的同宿轨道、乘子谱与刚性定理

Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

本文利用复动力学与非阿基米德动力学工具，给出 McMullen 乘子谱刚性定理的新证明，并将其推广到长度谱情形，同时解决 Milnor 关于例外映射的两个猜想。

关于扇的组合交上同调的注记

Remarks on the combinatorial intersection cohomology of fans

我们回顾了由Barthel-Schuelet-Fieseler-Kaup，Bressler-Lunts和Karu提出的扇形的组合交上同调理论。这个理论给出了一个替代的交集上同调的环面品种，具有所有预期的形式属性，但即使是非理性球迷，这并不定义一个环面品种的意义。因此，复曲面$g$和$h$多项式的一些有趣的结果已从有理多面体推广到一般多面体。我们提出了明确的复杂计算的组合IH度1和2;度2复杂的刚性复杂的卡莱以前使用的研究$g_2$。我们提出了几个新的结果，遵循这些方法，以及以前未发表的证据卡莱$g_k(P) = 0$意味着$g_k(P^*) = 0$和$g_{k+1}(P) = 0$。