Hodge Filtration and Operations in Higher Hochschild (Co)homology and Applications to Higher String Topology

2017 ◽  
pp. 1-104
Author(s):  
Grégory Ginot
Keyword(s):  
String Topology ◽  
Hodge Filtration

scholarly journals Hodge decomposition of string topology

2021 ◽  
Vol 9 ◽  
Author(s):  
Yuri Berest ◽  
Ajay C. Ramadoss ◽  
Yining Zhang
Keyword(s):  
Lie Algebras ◽  
General Theorem ◽  
Hodge Decomposition ◽  
String Topology ◽  
Free Loop Space ◽  
Oriented Manifold ◽  
Universal Enveloping Algebras ◽  
Finite Dimensional ◽  
Equivariant Homology ◽  
Simply Connected

Abstract Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$ -equivariant homology $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].

scholarly journals String topology for Lie groups

2010 ◽  
Vol 3 (2) ◽  
pp. 424-442 ◽  
Author(s):  
Richard A. Hepworth
Keyword(s):  
Lie Groups ◽  
String Topology

scholarly journals Twisted Hodge filtration: Curvature of the determinant

2019 ◽  
Vol 292 (11) ◽  
pp. 2452-2455
Author(s):  
Philipp Naumann
Keyword(s):  
Hodge Filtration

scholarly journals LOCAL VANISHING AND HODGE FILTRATION FOR RATIONAL SINGULARITIES

2018 ◽  
Vol 19 (3) ◽  
pp. 801-819
Author(s):  
Mircea Mustaţă ◽  
Sebastián Olano ◽  
Mihnea Popa
Keyword(s):  
Toric Variety ◽  
Exceptional Divisor ◽  
Resolution Of Singularities ◽  
Isolated Singularities ◽  
Rational Singularities ◽  
Smooth Complex ◽  
Hodge Filtration ◽  
Generation Level ◽  
Image Position

Given an $n$-dimensional variety $Z$ with rational singularities, we conjecture that if $f:Y\rightarrow Z$ is a resolution of singularities whose reduced exceptional divisor $E$ has simple normal crossings, then $$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$ We prove this when $Z$ has isolated singularities and when it is a toric variety. We deduce that for a divisor $D$ with isolated rational singularities on a smooth complex $n$-dimensional variety $X$, the generation level of Saito’s Hodge filtration on the localization $\mathscr{O}_{X}(\ast D)$ is at most $n-3$.

scholarly journals On string topology of classifying spaces

2015 ◽  
Vol 281 ◽  
pp. 394-507 ◽  
Author(s):  
Richard Hepworth ◽  
Anssi Lahtinen
Keyword(s):  
String Topology ◽  
Classifying Spaces
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