Rationality of inner twisted flags of type $A_n$

Morava K-theory of twisted flag varieties

In the present article we prove some results about the Morava K-theory. In particular, we construct an operation from the Morava K-theory to the Chow theory analogous to the second Chern class for Grothendieck's K0-theory. Furthermore, we investigate ordinary and equivariant oriented cohomology theories in the sense of Levine-Morel of projective quadrics, and discuss the Rost motives.

A local-global principle for twisted flag varieties

We prove a local-global principle for twisted flag varieties over a semiglobal field.

Non-existence of exceptional collections on twisted flags and categorical representability via noncommutative motives

In this paper we prove that a finite product of Brauer--Severi varieties is categorical representable in dimension zero if and only if it admits a $k$-rational point if and only if it is rational over $k$. The same is true for certain isotropic involution varieties over a field $k$ of characteristic different from two. For finite products of generalized Brauer--Severi varieties, categorical representability in dimension zero is equivalent to the existence of a full exceptional collection. In this case however categorical representability in dimension zero is not equivalent to the existence of …

Chow motives of twisted flag varieties

Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer variety SB_2(A), where A is a division algebra of degree 5, into a direct sum of two indecomposable motives. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral co…

Twisted gamma filtration and algebras with orthogonal involution

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.

Twisted loop groups and their affine flag varieties

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a "twisted case"; a consequence of our results is that our construction also includes the flag varieties for Kac-Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t…

Twisted $\mathcal{D}$-module extensions of local systems on a certain subvariety isomorphic to ${\mathbb{G}_{\text{m}}}^2$ of the affine flag variety of $\text{SL}_2$

We introduce a family of rank-one local systems in the category of twisted $\mathcal{D}$-modules on a certain subvariety isomorphic to ${\mathbb{G}_{\text{m}}}^2$ of the affine flag variety of $\text{SL}_2$. We then give a criterion for these local systems, in terms of their parameters, to extend cleanly in the sense of $\mathcal{D}$-modules.

扭曲辛旗簇的格罗滕迪克类的独立性

Independence of the Grothendieck classes of twisted symplectic flag varieties

本文利用非交换动机理论，证明扭曲辛旗簇的格罗滕迪克类在线性及齐次二次意义下，在代数簇的格罗滕迪克环中具有独立性。

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Let G be a split semisimple linear algebraic group over a field k0. Let E be a G-torsor over a field extension k of k0. Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel. Consider a twisted form E/B of the variety of Borel subgroups G/B over k. Following the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an isomorphism between the Grothendieck groups of the h-motivic subcategory generated by E/B and the category of finitely generated projective modules of certain Hecke-type algebra H which depends on the root datum of G, on the torsor…

仿射旗上的扭曲惠特克范畴与混合量子群的表示范畴

Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

该研究证明仿射旗簇上的扭曲惠特克范畴与混合量子群表示范畴等价，且一般情形下量子范畴O和仿射旗上的扭曲惠特克D-模范畴等价，丰富了量子群与几何表示论的联系。

扭曲旗簇族的半正交分解

Semiorthogonal decompositions for families of twisted flag varieties

该研究将Samokhin-van der Kallen的半正交分解推广至拟分裂半单群，证明光滑扭曲广义旗簇族的导出范畴可半正交分解，实现了Panin关于扭曲旗簇Quillen K-理论计算的范畴化。

Erratum: On the torsion of Chow groups of twisted Spin-flags

In the erratum we correct a mistake (due to a wrong choice of basic polynomial invariants over Z[1/2]) in the original paper (v1). Using the correct basic polynomial invariants we improve our results and bounds on the annihilator. We also simplify some of the proofs.

On the F-rationality and cohomological properties of matrix Schubert varieties

We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the F-rationality of matrix Schubert varieties. Although it is known that such varieties are F-regular (hence F-rational) by the global F-regularity of Schubert varieties, our proof is of independent interest since it does not require the Bott-Samelson resolution. As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities.

Residues for flags of holomorphic foliations

In this work we prove a Baum-Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum-Bott type rationality conjecture in this context .

The rationality problem for forms of $\overline{M_{0, n}}$

Let $X$ be a del Pezzo surface of degree $5$ defined over a field $F$. A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree $5$ is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree $5$ over a field $F$ are precisely the twisted $F$-forms of the moduli space $\overline{M_{0, 5}}$ of stable curves of genus $0$ with $5$ marked points. Suppose $n \geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq 2$. It is easy to see that every twisted $F$-form of $\overline{M_{0, n}}$ is unira…

Amitsur subgroup and noncommutative motives

This paper addresses the problem of calculating the Amitsur subgroup of a proper $k$-scheme. Under mild hypothesis, we calculate this subgroup for proper $k$-varieties $X$ with $\mathrm{Pic}(X)\simeq \mathbb{Z}^{\oplus m}$, using a classification of so called absolutely split vector bundles ($AS$-bundles for short). We also show that the Brauer group of $X$ is isomorphic to $\mathrm{Br}(k)$ modulo the Amitsur subgroup, provided $X$ is geometrically rational. Our results also enable us to classify $AS$-bundles on twisted flags. Moreover, we find an alternative proof for a result due to Merkurje…

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。

旗簇扭曲积的幺半范畴化

Towards Monoidal Categorifications of Twisted Products of Flag Varieties

通过构造量子仿射代数$U_q(\widehat{\mathfrak{g}})$的表示幺半范畴，其Grothendieck环包含以旗簇扭曲积坐标环初始种子为基的簇代数，统一了辫簇和约化双Bruhat细胞等变体。

扭曲旗流形中的完全正性

Total positivity in twisted flag varieties

本文研究了扭曲旗流形中完全非负部分的结构，证明其分解为细胞并具有正则CW复形性质，扩展了原有研究并解决了开放问题。

Kac-Moody 群旗簇扭积的全正性

Total positivity in twisted product of flag varieties

本文证明 Kac-Moody 群旗簇扭积的全非负部分具有胞腔分解和显式参数化，并证明双旗簇、辫子簇及双 Bruhat 胞腔链环均为同胚于闭球的正则 CW 复形，解决了 Fomin-Zelevinsky 公开问题。

A Riemann-Roch type theorem for twisted fibrations of moment graphs

In the present paper we extend the Riemann-Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and pushforwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann-Roch theorem for moment graphs. As an application, we obtain the Riemann-Roch type theorem for equivariant $K$-theory of some Kac-Moody flag varieties.

Chow-Witt 环与旗簇的拓扑

Chow-Witt rings and topology of flag varieties

本文利用子商丛的 Pontryagin 类计算 A 型部分旗簇的 Witt 层上同调环，证明了实旗流形整上同调挠均为 2-挠，并描述了旗簇的 Chow-Witt 环及其枚举应用。

On the rationality problem for forms of moduli spaces of stable marked curves of positive genus

Let $M_{g, n}$ (respectively, $\overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $\overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n \geqslant 0$ if $g \geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ wit…

A chiral Borel-Weil-Bott theorem

We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of $G$-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld-Sokolov reduction as a derived functor.