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Journal articles on the topic 'DG-Manifolds'

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Author: Grafiati

Published: 8 June 2024

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1

Cheng, Jiahao, Zhuo Chen, and Dadi Ni. "Hopf algebras arising from dg manifolds." Journal of Algebra 584 (October 2021): 19–68. http://dx.doi.org/10.1016/j.jalgebra.2021.05.004.

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2

Ciocan-Fontanine, Ionuţ, and Mikhail Kapranov. "Virtual fundamental classes via dg–manifolds." Geometry & Topology 13, no. 3 (March 16, 2009): 1779–804. http://dx.doi.org/10.2140/gt.2009.13.1779.

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3

Stiénon, Mathieu, and Ping Xu. "Fedosov dg manifolds associated with Lie pairs." Mathematische Annalen 378, no. 1-2 (July 26, 2020): 729–62. http://dx.doi.org/10.1007/s00208-020-02012-6.

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4

Laurent-Gengoux, Camille, Mathieu Stiénon, and Ping Xu. "Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds." Advances in Mathematics 387 (August 2021): 107792. http://dx.doi.org/10.1016/j.aim.2021.107792.

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5

Seol, Seokbong, Mathieu Stiénon, and Ping Xu. "Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras." Communications in Mathematical Physics 391, no. 1 (February 24, 2022): 33–76. http://dx.doi.org/10.1007/s00220-021-04265-x.

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6

Uribe, Bernardo. "Group Actions on DG-Manifolds and Exact Courant Algebroids." Communications in Mathematical Physics 318, no. 1 (January 23, 2013): 35–67. http://dx.doi.org/10.1007/s00220-013-1669-2.

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7

Bernardara, Marcello, Matilde Marcolli, and Gonçalo Tabuada. "Some remarks concerning Voevodsky’s nilpotence conjecture." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 738 (May 1, 2018): 299–312. http://dx.doi.org/10.1515/crelle-2015-0068.

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Abstract In this article we extend Voevodsky’s nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky’s original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds.

8

Lupercio, Ernesto, Camilo Rengifo, and Bernardo Uribe. "T-duality and exceptional generalized geometry through symmetries of dg-manifolds." Journal of Geometry and Physics 83 (September 2014): 82–98. http://dx.doi.org/10.1016/j.geomphys.2014.05.012.

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9

GRIBACHEVA, DOBRINKA. "A NATURAL CONNECTION ON A BASIC CLASS OF RIEMANNIAN PRODUCT MANIFOLDS." International Journal of Geometric Methods in Modern Physics 09, no. 07 (September 7, 2012): 1250057. http://dx.doi.org/10.1142/s0219887812500570.

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A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

10

Pridham, J. P. "Representability of derived stacks." Journal of K-Theory 10, no. 2 (January 31, 2012): 413–53. http://dx.doi.org/10.1017/is012001005jkt179.

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AbstractLurie's representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack. We establish several variants of Lurie's theorem, making the hypotheses easier to verify for many applications. Provided a derived analogue of Schlessinger's condition holds, the theorem reduces to verifying conditions on the underived part and on cohomology groups. Another simplification is that functors need only be defined on nilpotent extensions of discrete rings. Finally, there is a pre-representability theorem, which can be applied to associate explicit geometric stacks to dg-manifolds and related objects.

11

FIORENZA, DOMENICO, CHRISTOPHER L. ROGERS, and URS SCHREIBER. "A HIGHER CHERN–WEIL DERIVATION OF AKSZ σ-MODELS." International Journal of Geometric Methods in Modern Physics 10, no. 01 (November 15, 2012): 1250078. http://dx.doi.org/10.1142/s0219887812500788.

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Chern–Weil theory provides for each invariant polynomial on a Lie algebra 𝔤 a map from 𝔤-connections to differential cocycles whose volume holonomy is the corresponding Chern–Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and to dg-bundles and that the Chern–Simons action functional associated this way to an n-symplectic manifold is the action functional of the AKSZ σ-model whose target space is the given n-symplectic manifold (examples of this are the Poisson σ-model or the Courant σ-model, including ordinary Chern–Simons theory, or higher-dimensional Abelian Chern–Simons theory). Here we show how, within the framework of the higher Chern–Weil theory in smooth ∞-groupoids, this result can be naturally recovered and enhanced to a morphism of higher stacks, the same way as ordinary Chern–Simons theory is enhanced to a morphism from the stack of principal G-bundles with connections to the 3-stack of line 3-bundles with connections.

12

TURAEV, D. "ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS." International Journal of Bifurcation and Chaos 06, no. 05 (May 1996): 919–48. http://dx.doi.org/10.1142/s0218127496000515.

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An analogue of the center manifold theory is proposed for non-local bifurcations of homo- and heteroclinic contours. In contrast with the local bifurcation theory it is shown that the dimension of non-local bifurcational problems is determined by the three different integers: the geometrical dimension dg which is equal to the dimension of a non-local analogue of the center manifold, the critical dimension dc which is equal to the difference between the dimension of phase space and the sum of dimensions of leaves of associated strong-stable and strong-unstable foliations, and the Lyapunov dimension dL which is equal to the maximal possible number of zero Lyapunov exponents for the orbits arising at the bifurcation. For a wide class of bifurcational problems (the so-called semi-local bifurcations) these three values are shown to be effectively computed. For the orbits arising at the bifurcations, effective restrictions for the maximal and minimal numbers of positive and negative Lyapunov exponents (correspondingly, for the maximal and minimal possible dimensions of the stable and unstable manifolds) are obtained, involving the values dc and dL. A connection with the problem of hyperchaos is discussed.

13

Merkulov, Sergei, and Thomas Willwacher. "Classification of universal formality maps for quantizations of Lie bialgebras." Compositio Mathematica 156, no. 10 (October 2020): 2111–48. http://dx.doi.org/10.1112/s0010437x20007381.

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We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props \[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \] satisfying certain boundary conditions, where $\mathcal {A}\textit{ssb}_\infty$ is a minimal resolution of the prop of associative bialgebras. We prove that the set of such formality morphisms is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator $\mathfrak{A}$ there is an associated ${\mathcal {L}} ie_\infty$ quasi-isomorphism between the ${\mathcal {L}} ie_\infty$ algebras $\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$ and $\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$ controlling, respectively, deformations of the standard bialgebra structure in $\odot V$ and deformations of any given Lie bialgebra structure in $V$. We study the deformation complex of an arbitrary universal formality morphism $\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$ and prove that it is quasi-isomorphic to the full (i.e. not necessary connected) version of the graph complex introduced Maxim Kontsevich in the context of the theory of deformation quantizations of Poisson manifolds. This result gives a complete classification of the set $\{F_\mathfrak{A}\}$ of gauge equivalence classes of universal Lie connected formality maps: it is a torsor over the Grothendieck–Teichmüller group $GRT=GRT_1\rtimes {\mathbb {K}}^*$ and can hence can be identified with the set $\{\mathfrak{A}\}$ of Drinfeld associators.

14

Bondal, Alexey Igorevich, and Alexei Andreevich Rosly. "Coherent sheaves, Chern classes, and superconnections on compact complex-analytic manifolds." Izvestiya: Mathematics 87, no. 3 (2023): 439–68. http://dx.doi.org/10.4213/im9386e.

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A twist-closed enhancement of the bounded derived category $\mathcal{D}^b_{\mathrm{coh}} (X)$ of complexes of $\mathcal{O}_X$-modules with coherent cohomology is constructed by means of the DG-category of $\overline\partial$-superconnections. The machinery of $\overline\partial$-superconnections is applied to define Chern classes and Bott-Chern classes of objects in the category, in particular, of coherent sheaves.

15

Beckmann, Thorsten. "Atomic objects on hyper-Kähler manifolds." Journal of Algebraic Geometry, April 19, 2024. http://dx.doi.org/10.1090/jag/830.

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We introduce and study the notion of atomic sheaves and complexes on higher-dimensional hyper-Kähler manifolds and show that they share many of the intriguing properties of simple sheaves on K3 surfaces. For example, we prove formality of the dg algebra of derived endomorphisms for stable atomic bundles. We further demonstrate the characteristics of atomic objects by studying atomic Lagrangian submanifolds. In the appendix, we prove nonexistence results for spherical objects on hyper-Kähler manifolds.

16

Chen, Zhuo, Maosong Xiang, and Ping Xu. "Hochschild Cohomology of dg Manifolds Associated to Integrable Distributions." Communications in Mathematical Physics, September 12, 2022. http://dx.doi.org/10.1007/s00220-022-04473-z.

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17

Behrend, Kai, Hsuan-Yi Liao, and Ping Xu. "Differential Graded Manifolds of Finite Positive Amplitude." International Mathematics Research Notices, February 27, 2024. http://dx.doi.org/10.1093/imrn/rnae023.

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Abstract We prove that dg manifolds of finite positive amplitude, that is, bundles of positively graded curved $L_{\infty }[1]$-algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved $L_{\infty }[1]$-algebras. As an application, we study the derived intersections of manifolds.

18

Wei, Zhaoting. "Descent of dg cohesive modules for open covers on complex manifolds." European Journal of Mathematics 9, no. 3 (August 22, 2023). http://dx.doi.org/10.1007/s40879-023-00675-4.

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19

Chang, Hua-Shin, and Hsuan-Yi Liao. "Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair." Journal of Geometry and Physics, March 2024, 105169. http://dx.doi.org/10.1016/j.geomphys.2024.105169.

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