We generalize the Künneth formula for Chow groups to an arbitrary OBM-homology theory satisfying descent (e.g. algebraic cobordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a description for the homology groups of smooth toric varieties, that allows the calculation of algebraic cobordism groups of singular toric varieties. Some computations are carried out.
We show that, for a simplicial complex, the supported cap product operation on Borel-Moore homology coincides with the supported cap product on simplicial homology. For this purpose we introduce the supported cap product for locally finite singular homology, and compare the cap product on the three homology theories.
We study the Borel-Moore homology of stacks of representations of preprojective algebras $Π_Q$, via the study of the DT theory of the undeformed 3-Calabi-Yau completion $Π_Q[x]$. Via a result on the supports of the BPS sheaves for $Π_Q[x]$-mod, we prove purity of the BPS cohomology for the stack of $Π_Q[x]$-modules, and define BPS sheaves for stacks of $Π_Q$-modules. These are mixed Hodge modules on the coarse moduli space of $Π_Q$-modules that control the Borel-Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is p…
A W-algebra action is constructed on the equivariant Borel-Moore homology of the Hilbert scheme of points on a nonreduced plane in three dimensional affine space, identifying it to the vacuum W-module. This is based on a generalization of the ADHM construction as well as the W-action on the equivariant Borel-Moore homology of the moduli space of instantons constructed by Schiffmann and Vasserot.
Bivariant theory is a unified framework for cohomology and Borel-Moore homology theories. In this paper, we extract an $\infty$-enhanced bivariant homology theory from Gaitsgory-Rozenblyum's six functor formalism.
We introduce and study a fermionization procedure for the cohomological Hall algebra $\mathcal{H}_{Π_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson--Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel-Moore homology of the stack of representations of the $μ$-deformed preprojective algebra introduced by Crawley-…
On Some Stratifications of Affine Deligne-Lusztig Varieties for SL_3
设L是k（（ε）），其中k是q元有限域的代数闭包，ε是不定式，σ是弗罗贝尼乌斯自同构.设G是L中σ的固定域上的可裂连通约化群，I是G（L）的Iwahori子群，它与G的一个给定的博雷尔子群相联系.设W是G的扩张仿射外尔群.给定W中的x和G（L）中的b，我们有G（L）的某个子群作用在仿射Deligne-Lusztig簇X_x（b）= {gI in G（L）/I：g^{-1}bσ（g）is in IxI}上，从而得到这个子群在簇的Borel-Moore同调上的表示.我们在G是SL_2和G是SL_3的情况下研究了b的这种表示。
Homology of Local Systems on Real Line Arrangement Complements
Qiuzhen College, Tsinghua University（清华大学丘成桐大学）；Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉科学研究院）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉科学研究院）
Weight filtration on the cohomology of algebraic varieties
我们证明了Etale上同调证明了代数闭域上的代数簇$X$（具有紧支撑）的正则权滤子$W$，并证明了$X$的紧支撑上同调的中权部分是X的紧化$X'$的交上同调的子空间，或等价地，$X$的（所谓的）Borel-Moore同调的中权部分是$X'$的交上同调的商。我们被告知，这已经被A。Weber在$X$是真的（和$k=\bC$）的情况下利用G. Barthel，J.- P. Chelet，K.- H.菲泽勒岛Gabber和L.交复形之间的态射。我们表明，断言立即遵循Gabber的纯度定理的交叉复合体。
