The equivariant Atiyah class

Abstract Holomorphicity of vector bundles can stabilize complex structure moduli of a Calabi-Yau threefold in N = 1 supersymmetric heterotic compactifications. In principle, the Atiyah class determines the stabilized moduli. In this paper, we study how this mechanism works in the context of elliptically fibered Calabi-Yau manifolds where the complex structure moduli space contains two kinds of moduli, those from the base and those from the fibration. Defining the bundle with spectral data, we find three types of situations when bundles’ holomorphicity depends on algebraic cycles exist only for special loci in the complex structure moduli, which allows us to stabilize both of these two moduli. We present concrete examples for each type and develop practical tools to analyze the stabilized moduli. Finally, by checking the holomorphicity of the four-flux and/or local Higgs bundle data in F-theory, we briefly study the dual complex structure moduli stabilization scenarios.

Tangent Lie algebra of derived Artin stacks

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0065 ◽

2018 ◽

Vol 2018 (741) ◽

pp. 1-45 ◽

Cited By ~ 5

Author(s):

Benjamin Hennion

Keyword(s):

Algebraic Geometry ◽

Lie Algebra ◽

Algebraic Variety ◽

Category Theory ◽

Smoothness Assumption ◽

Coherent Sheaves ◽

Derived Algebraic Geometry ◽

Atiyah Class ◽

Artin Stacks

Abstract Since the work of Mikhail Kapranov in [Compos. Math. 115 (1999), no. 1, 71–113], it is known that the shifted tangent complex \mathbb{T}_{X} [-1] of a smooth algebraic variety X is endowed with a weak Lie structure. Moreover, any complex of quasi-coherent sheaves E on X is endowed with a weak Lie action of this tangent Lie algebra. This Lie action is given by the Atiyah class of E. We will generalise this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to singular schemes. This work uses tools of both derived algebraic geometry and {\infty} -category theory.

The Atiyah class of a dg-vector bundle

Comptes Rendus Mathematique ◽

10.1016/j.crma.2015.01.019 ◽

2015 ◽

Vol 353 (4) ◽

pp. 357-362 ◽

Cited By ~ 9

Author(s):

Rajan Amit Mehta ◽

Mathieu Stiénon ◽

Ping Xu

Keyword(s):

Vector Bundle ◽

Atiyah Class

The Atiyah class and complex structure stabilization in heterotic Calabi-Yau compactifications

Journal of High Energy Physics ◽

10.1007/jhep10(2011)032 ◽

2011 ◽

Vol 2011 (10) ◽

Cited By ~ 40

Author(s):

Lara B. Anderson ◽

James Gray ◽

Andre Lukas ◽

Burt Ovrut

Keyword(s):

Complex Structure ◽

Atiyah Class ◽

Structure Stabilization

THE BIG CHERN CLASSES AND THE CHERN CHARACTER

International Journal of Mathematics ◽

10.1142/s0129167x08004856 ◽

2008 ◽

Vol 19 (06) ◽

pp. 699-746 ◽

Cited By ~ 7

Author(s):

AJAY C. RAMADOSS

Keyword(s):

Vector Bundle ◽

Lie Algebra ◽

Representation Theorem ◽

Tangent Bundle ◽

Chern Character ◽

Free Lie Algebra ◽

Chern Classes ◽

Enveloping Algebra ◽

Atiyah Class ◽

Bounded Below

Let X be a smooth scheme over a field of characteristic 0. The Atiyah class of the tangent bundle TX of X equips TX[-1] with the structure of a Lie algebra object in the derived category D +(X) of bounded below complexes of [Formula: see text] modules with coherent cohomology [6]. We lift this structure to that of a Lie algebra object [Formula: see text] in the category of bounded below complexes of [Formula: see text] modules in Theorem 2. The "almost free" Lie algebra [Formula: see text] is equipped with Hochschild coboundary. There is a symmetrization map [Formula: see text] where [Formula: see text] is the complex of polydifferential operators with Hochschild coboundary. We prove a theorem (Theorem 1) that measures how I fails to commute with multiplication. Further, we show that [Formula: see text] is the universal enveloping algebra of [Formula: see text] in D +(X). This is used to interpret the Chern character of a vector bundle E on X as the "character of a representation" (Theorem 4). Theorems 4 and 1 are then exploited to give a formula for the big Chern classes in terms of the components of the Chern character.

Symplectic structures on moduli spaces of sheaves via the Atiyah class

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2009.03.008 ◽

2009 ◽

Vol 59 (7) ◽

pp. 843-860 ◽

Cited By ~ 16

Author(s):

A. Kuznetsov ◽

D. Markushevich

Keyword(s):

Moduli Spaces ◽

Symplectic Structures ◽

Moduli Spaces Of Sheaves ◽

Atiyah Class

ATIYAH CLASS AND CHERN CHARACTER FOR GLOBAL MATRIX FACTORISATIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748020000614 ◽

2021 ◽

pp. 1-26

Author(s):

Bumsig Kim ◽

Alexander Polishchuk

Keyword(s):

Lie Algebra ◽

Chern Character ◽

Local Case ◽

Atiyah Class

Abstract We define the Atiyah class for global matrix factorisations and use it to give a formula for the categorical Chern character and the boundary-bulk map for matrix factorisations, generalising the formula in the local case obtained in [12]. Our approach is based on developing the Lie algebra analogies observed by Kapranov [7] and Markarian [9].

The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem

Journal of the London Mathematical Society ◽

10.1112/jlms/jdn064 ◽

2008 ◽

Vol 79 (1) ◽

pp. 129-143 ◽

Cited By ~ 25

Author(s):

Nikita Markarian

Keyword(s):

Hochschild Cohomology ◽

Atiyah Class ◽

Roch Theorem

Direct images of sheaves of differentials and the Atiyah class

Mathematische Zeitschrift ◽

10.1007/bf01179269 ◽

1987 ◽

Vol 196 (1) ◽

pp. 75-85 ◽

Cited By ~ 4

Author(s):

Th. Peternell ◽

J. le Potier ◽

M. Schneider

Keyword(s):

Atiyah Class

Atiyah Classes of Strongly Homotopy Lie Pairs

Algebra Colloquium ◽

10.1142/s1005386719000178 ◽

2019 ◽

Vol 26 (02) ◽

pp. 195-230

Author(s):

Zhuo Chen ◽

Honglei Lang ◽

Maosong Xiang

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Module Structure ◽

Algebra Structure ◽

Atiyah Class ◽

Graded Lie Algebra ◽

Infinitesimal Deformations ◽

The Subject ◽

Intrinsic Character ◽

Algebra Module

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is extended to L. In fact, with such an SH Lie pair (L, A) and any A-module E, there is associated a canonical cohomology class, the Atiyah class [αE], which generalizes the earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [αL/A] induces a graded Lie algebra structure on [Formula: see text], and the Atiyah class [αE] of any A-module E induces a Lie algebra module structure on [Formula: see text]. Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.