On motivic decompositions arising from the method of Bialynicki-Birula

A compactly supported formula for equivariant localization, and, simplicial complexes of Bialynicki-Birula decompositions

Let X be a projective scheme carrying a circle action S with isolated fixed points. We associate a simplicial complex Delta(X,S) of "closure chains" using a refinement of its Morse/Bialynicki-Birula decomposition. If this decomposition is a stratification (e.g. when X is a flag manifold), then Delta(X,S) is just the order complex of the poset of fixed points. For X a toric variety, Delta(X,S) is a triangulation of the moment polytope. We compute some other examples, including a Bott-Samelson manifold and the punctual Hilbert scheme of 4 points in the plane. Summing over the facets of Delta(X,S…

Estratificações no espaço móduli dos fibrados de Higgs

The work of Hausel proves that the Bialynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. These two stratifications do not coincide in general. Here, we give an approach for the rank three case of the classification of the Shatz stratification in terms of the Bialynicki-Birula stratification.

Higgs 丛模空间中的实 Bialynicki-Birula 流

Real Bialynicki-Birula flows in moduli spaces of Higgs bundles

本文利用实与四元数 Hodge 丛系统研究半稳定 Higgs 丛模空间实轨迹的拓扑，并证明秩与次数互素时其连通分支数等于实 Picard 空间的分支数。

Białynicki-Birula decomposition for reductive groups in positive characteristic

We prove the existence of Bialynicki-Birula decomposition for Kempf monoids, which is a large class that contains for example monoids with reductive unit group in all characteristics. This extends the existence statements from previous works of Alper-Hall-Rydh and Jelisiejew-Sienkiewicz.

Poincaré polynomials of moduli spaces of stable maps into flag manifolds

By using Oprea's Bialynicki-Birula decomposition for the stack of genus zero stable maps to flag manifolds. We calculate the Poincaré polynomial of the moduli space in degree one and degree two.

旗流形上 Schubert 簇与光滑 $T$-稳定子簇的交集

Intersections of Schubert varieties and smooth $T$-stable subvarieties of flag varieties

本文研究了正则半单 Hessenberg 簇中 Bialynicki-Birula 胞腔闭包（即 Hessenberg Schubert 簇）的光滑性，通过考察 Schubert 簇与 Hessenberg 簇交集的不可约性与光滑性，得出了该簇光滑的充分条件。

中岛箭图簇、箭图链和星形箭图的比亚林斯基 - 比鲁拉分解

Białynicki-Birula Decompositions of Nakajima Quiver Varieties, Quiver Chains and Star-Shaped Quivers

研究中岛箭图簇相关的比亚林斯基 - 比鲁拉分解，通过箭图链描述不动点，探讨分解几何，如吸引纤维维度等，还将框架用于星形箭图，对特定维数向量分类不动点轨迹分量并计算动机类。

Comparison of motivic Chern classes and stable envelopes for cotangent bundles

We consider a complex smooth projective variety equipped with an action of an algebraic torus with a finite number of fixed points. We compare the motivic Chern classes of Białynicki-Birula cells with the $K$-theoretic stable envelopes of cotangent bundle. We prove that under certain geometric assumptions satisfied e.g. by homogenous spaces these two notions coincide up to normalization.

少点的希尔伯特概型的非约化性

Non-reducedness of the Hilbert schemes of few points

该研究运用广义Bialynicki-Birula分解、零化对偶性与障碍理论，证明了A^6上13个点的希尔伯特概型的非约化性，其论证无需计算机计算，还给出了该概型上分形状结构的实例。

The homology groups of certain moduli spaces of plane sheaves

Using the Bialynicki-Birula method, we determine the additive structure of the integral homology groups of the moduli spaces of semi-stable sheaves on the projective plane having rank and Chern classes (5, 1, 4), (7, 2, 6), respectively, (0, 5, 19). We compute the Hodge numbers of these moduli spaces.

$\mathbf{G}_m$-作用的真良商

Proper good quotients for $\mathbf{G}_m$-actions

本文借助 Alper、Halpern-Leistner 和 Heinloth 的代数栈模空间存在性定理，以纯代数方法重新证明并刻画了带 $\mathbf{G}_m$-作用的正常真簇中存在真良商的不变开子集。

Rational smoothness, cellular decompositions and GKM theory

We introduce the notion of Q-filtrable varieties: projective varieties with a torus action and a finite number of fixed points, such that the cells of the associated Bialynicki-Birula decomposition are all rationally smooth. Our main results develop GKM theory in this setting. We also supply a method for building nice combinatorial bases on the equivariant cohomology of any Q-filtrable GKM variety. Applications to the theory of group embeddings are provided.

Permutation module decomposition of the second cohomology of a regular semisimple Hessenberg variety

Regular semisimple Hessenberg varieties admit actions of associated Weyl groups on their cohomology space of each degree. In this paper, we consider the module structure of the cohomology spaces of regular semisimple Hessenberg varieties of type $A$. We define a subset of the Bialynicki-Birula basis of the cohomology space so that they become a module generator set of the cohomology module of each degree. We then use those generators to construct permutation submodules of the degree two cohomology module and show that they form a permutation module decomposition. Our construction is consistent…

Chow动机的高阶Tate迹

Higher Tate traces of Chow motives

本文通过定义动机的Tate迹这一新不变量，并利用Chernousov-Gille-Merkurjev分解与Bialynicki-Birula-Hesselink-Iversen滤过，完成了p-内半单代数群射影齐性簇Chow动机的完整分类，同时推广了内半单群动机等价性的刻画。

Three instability stratifications of the stack of Higgs bundles on a smooth projective curve

We study three instability stratifications of the stack of twisted Higgs bundle of a fixed rank and degree on a smooth complex projective curve. The first is the Harder-Narasimhan (HN) stratification, defined by the instability type of the Higgs bundle. The second is the bundle Harder-Narasimhan (bHN) stratification, defined by the instability type of the underlying bundle. While an unstable HN stratum fibres over the stack parametrising Higgs bundles which are isomorphic to their graded, this is not true for Higgs bundles of unstable bHN type. Obtaining such a fibration requires refining the …