Kirill Zainoulline
In the present notes we introduce and study the twisted gamma-filtration on K_0(G), where G is a split simple linear algebraic group over a field of characteristic prime to the order of the center of G. We apply this filtration to construct torsion elements in the gamma-ring of an inner form of the variety of Borel subgroups of G.
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We give a definition and study the basic properties of the irregular Hodge filtration on the exponentially twisted de Rham cohomology of a smooth quasi-projective complex variety.
For the Grothendieck group of a split simple linear algebraic group, the twisted gamma-filtration provides a useful tool for constructing torsion elements in gamma-rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the gamma-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline. We use this torsion element to study algebras with orthogonal involutions.
Hodge filtrations for twisted parametrically prime divisors
该研究将霍奇滤过相关工作推广到$\mathbb{Q}$-除子情形，针对特定参数素全纯函数$f$和$α\geq 0$，给出了$\mathscr{M}(f^{-α})$上霍奇滤过的显式表达式。
Filtrations on the derived category of twisted K3 surfaces
本文引入扭曲K3曲面导出范畴的Shen-Yin-Zhao滤过，构造扭曲Beauville-Voisin类，证明其与Voisin滤过一致，并应用于Bloch猜想。
Given a mixed Hodge module and a meromorphic function f on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module, which extends the construction of arXiv:1302.4537. We show the strictness of the push-forward filtered D-module through any projective morphism, by using the theory of mixed twistor D-modules of T. Mochizuki. We consider the example of the rescaling of a regular function f, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss-Manin systems of f in ter…
Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the explicit formula for the curvature tensor of these direct images, we prove that the determinant of the twisted Hodge filtration $F^p_L=\oplus_{i\geq p}R^{n-i}Ω^i_{\mathcal{X}/S}(L)$ is (semi-) positive on the base $S$ if $L$ itself is (semi-) positive on $\mathcal{X}$.
Bloch's conjecture for equivalences between twisted abelian surfaces and applications
本文通过扭曲阿贝尔曲面构造自等价在Albanese核上的自然作用，证明所有（反）辛自等价的Bloch猜想，并应用于扭曲模Kum_n型簇的辛双有理自同构，同时引入Shen-Yin-Zhao型滤子并与Voisin滤子比较。
Filtrations on the globalization of twisted D-modules over Dedekind schemes
本文在 Dedekind 概形上证明，由 K-等变闭浸入正像得到的扭曲 D-模整体化具有 K-不变正穷举滤过，且各相伴分次为有限秩局部自由模。
We prove the Künneth formula for the irregular Hodge filtrations on the exponentially twisted de Rham and the Higgs cohomologies of smooth quasi-projective complex varieties. The method involves a careful comparison of the underlying chain complexes under a certain elimination of indeterminacy.
Let X be the variety of Borel subgroups of a simple and strongly inner linear algebraic group G over a field k. We prove that the torsion part of the second quotient of Grothendieck's gamma-filtration on X is a cyclic group of order the Dynkin index of G. As a byproduct of the proof we obtain an explicit cycle that generates this cyclic group; we provide an upper bound for the torsion of the Chow group of codimension-3 cycles on X; we relate the generating cycle with the Rost invariant and the torsion of the respective generalized Rost motives; we use this cycle to obtain a uniform lower bound…
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.
For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs bundles on a Riemann surface C agrees with the weight filtration on the cohomology of the twisted G character variety of C, when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.
In this paper, we study twisted algebraic $K$-theory from a motivic viewpoint. For a smooth variety $X$ over a field of characteristic zero and an Azumaya algebra $\mathcal{A}$ over $X$, we construct the $\mathcal{A}$-twisted motivic spectral sequence, by computing the slices of the motivic twisted algebraic $K$-theory spectrum as a twisted form of motivic cohomology. This generalizes previous results due to Kahn-Levine where $\mathcal{A}$ is assumed to be pulled back from a base field. Our methods use interaction between the slice filtration and birational geometry. Along the way, we prove a …
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。
A note on weight filtrations at the characteristic
本文研究了在非可逆特征下，有理系数上射影Dedekind方案上的重量滤泡性质，并结合后续工作，证明了已知的对数上同调理论在项目ive sncd对上具有重量滤泡，同时展示了重量滤泡上同调是开部分的不变量。
Derived $V$-filtrations and the Kontsevich-Sabbah-Saito theorem
本文在导出 $\infty$-范畴中证明了代数形式扭 de Rham 复形与消失圈等价，推广了 Kontsevich 猜想，并通过引入典范 $V$-滤过及高阶范畴技术给出新证明。
In this paper, we generalize a result of Karpenko on the torsion in the second quotient of the gamma filtration for Severi-Brauer varieties to higher degrees. As an application, we provide a nontrivial torsion in higher Chow groups and the topological filtration of the associated generic variety and obtain new upper bounds for the annihilators of the torsion subgroups in the Chow groups of a large class of Severi-Brauer varieties. In particular, using the torsion in higher degrees, we show indecomposability of certain algebras.
Graded quotients of ramification groups of local fields with imperfect residue fields
该研究针对剩余特征为$p>0$的亨泽尔离散赋值域，证明其分歧群滤过的分次商为$F_p$-向量空间，并构造了分次商特征群到基于余切复形的扭曲余切空间的单射。
Unitary representations of real groups and localization theory for Hodge modules
本文证明实约化 Lie 群表示的酉性可由 Hodge 滤过判定，方法结合混合 Hodge 模穿墙、Hodge 理论化 Beilinson–Bernstein 局部化及显式滤过计算。
An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "$F$-zips with a $G$-structure", for an arbitrary reductive linear algebraic group $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F…
Chern classes from Morava K-theories to $p^n$-typical oriented theories
针对代数几何中从代数Morava K-理论到具有pⁿ-典型形式群律的定向上同调理论，构造p-局部系数下的陈类，证明这类陈类自由生成相关运算，还引入K(n)的gamma滤子，其分次因子到Chow群的陈类满射可估计簇的低余维Chow群p-挠元。
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