Related papers: Localization of Atiyah classes
Review of localization in geometry: equivariant cohomology, characteristic classes, Atiyah-Bott formula, Atiyah-Singer equivariant index formula, Mathai-Quillen formalism
We develop a formalism involving Atiyah classes of sheaves on a smooth manifold, Hochschild chain and cochain complexes. As an application we prove a version of the Riemann--Roch theorem.
We generalize Illusie's definition of the Atiyah class to complexes with quasi-coherent cohomology on arbitrary algebraic stacks. We show that this gives a global obstruction theory for moduli stacks of complexes in algebraic geometry…
We derive equivariant localization formulas of Atiyah--Bott and cohomological field theory types in the Batalin-Vilkovisky formalism and discuss their applications in Poisson geometry and quantum field theory.
Equivariant cohomology, a captivating fusion of symmetry and abstract mathematics, illuminates the profound role of group actions in shaping geometric structures. At its core lies the Atiyah-Bott Localization Theorem, a mathematical jewel…
We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization…
We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah…
In analogy to the classical holomorphic setting, Lang, Jia and Liu introduced the notion of the Atiyah class for a generalized holomorphic vector bundle using three different approaches: leveraging $\rm{\check{C}}$ech cohomology, employing…
For a Lie algebroid pair $A\hookrightarrow L$ we study cocycles constructed from the extension to $L$ of the higher connection forms of a representation up to homotopy $E$ of the Lie algebroid $A$. We show that there exists a cohomology…
We prove an Atiyah-Bott-Berline-Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern-Simons type invariants, for example the volume of Sasakian…
We develop a global cohomology theory for number fields by offering topological cohomology groups, an arithmetical duality, a Riemann-Roch type theorem, and two types of vanishing theorem. As applications, we study moduli spaces of…
We study cohomology of morphisms of Lie-Yamaguti algebras. As an application, we establish that this cohomology `controls' the formal deformations. Additionally, we demonstrate its connection to the abelian extension of morphisms of…
The purpose of this paper is to present a ``Cech-De Rham'' model for the cohomology of leaf spaces. This model lends itself to the construction of characteristic classes (in the cohomology of classifying spaces) by explicit geometrical…
Cobordism categories have played an important role in classical geometry and more recently in mathematical treatments of quantum field theory. Here we will compute localisations of two-dimensional discrete cobordism categories. This allows…
The cohomology and deformation theory of 3-Lie algebras are revisited. The theory of extending structures and unified product for 3-Lie algebras are developed.It is proved that the extending structures of 3-Lie algebras can be classified by…
We use Hodge theory and a construction of Merkulov to construct $A_{\infty}$ structures on de Rham cohomology and Dolbeault cohomology.
The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to…
To address the need for a unified framework that incorporates Lie algebroid connections on both vector and principal bundles, this paper investigates a generalized Atiyah algebroid structure and its short exact sequence. Building on this…
In this paper, we establish a kind of Dolbeault type cohomology groups for the purpose of studying the varying of complex structure invariants in infinitesimal deformations of any order. We give a concrete description of the higher order…
The locality conditions for the vanishing of local anomalies in field theory are shown to admit a geometrical interpretation in terms of local equivariant cohomology, thus providing a method to deal with the problem of locality in the…