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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> noncommutative Hodge structure in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' > #Contents# * table of contents {:toc} ## Idea Noncommutative Hodge theory is an extension of the classical [[Hodge theory]]. The basic notion is a **noncommutative Hodge structure**, the generalization of _[[Hodge structure]]_, which is formulated in the language of meromorphic connections. Unlike classical Hodge theory for complex varieties, noncommutative Hodge structures can be attached to a wider class of [[noncommutative algebraic geometry|noncommutative spaces]]. In the framework of Katzarkov-Kontsevich-Pantev, noncommutative spaces are represented by [[dg-categories]], or more generally, A-infinity categories. In particular, they are interested in the dg-categories which arise in [[homological mirror symmetry]]: for example, [[Fukaya category|Fukaya categories]], (dg enhanced) [[derived categories]] of (quasi-)coherent sheaves, [[matrix factorization]] categories, [[Fukaya-Seidel categories]]. The noncommutative analogue of [[Dolbeault cohomology]] is the [[Hochschild homology]] of the category. The analogue of [[de Rham cohomology]] is the periodic [[cyclic cohomology|cyclic homology]] of the category. The analogue of the Hodge-de Rham spectral sequence is the Hochschild-cyclic spectral sequence. There is work of Weibel which makes this analogy precise. There is a conjecture of Kontsevich that the Hochschild-cyclic spectral sequence degenerates for smooth and proper noncommutative spaces. (This is the analogue of the Hodge-de Rham degeneration for smooth and proper varieties.) Kontsevich's conjecture has been proven in some cases by [[Dmitri Kaledin]], who adapts Deligne-Illusie's proof of Hodge-de Rham degeneration (using reduction mod p) to the noncommutative setting. Kontsevich's conjecture is known as the "degeneration conjecture". ## References Noncommutative Hodge theory is being developed in * [[Ludmil Katzarkov]], [[Maxim Kontsevich]], [[Tony Pantev]], _Hodge theoretic aspects of mirror symmetry_, [arxiv/0806.0107](http://arxiv.org/abs/0806.0107) * [[D. Kaledin]], _Cartier isomorphism and Hodge theory in the non-commutative case_, Arithmetic geometry, 537--562, Clay Math. Proc. __8__, Amer. Math. Soc. 2009, [arxiv/0708.1574](http://arxiv.org/abs/0708.1574) * D. Kaledin, _Tokyo lectures "Homological methods in non-commutative geometry"_, [pdf](http://imperium.lenin.ru/~kaledin/tokyo/final.pdf), [TeX](http://imperium.lenin.ru/~kaledin/tokyo/final.tex) * Claus Hertling, Christian Sevenheck, _Twistor structures, $tt^*$-geometry and singularity theory_, [arxiv/0807.2199](http://arxiv.org/abs/0807.2199) * C. Hertling, C. Sabbah, _Examples of non-commutative Hodge structure_ (v1 title: Fourier-Laplace transform of flat unitary connections and TERP structures), J. Inst. Math. Jussieu __10__, Spec. Issue 3: in honour of L. Boutet de Monvel & P. Schapira (2011) 635 - 674 [arxiv/0912.2754](http://arxiv.org/abs/0912.2754) [doi](https://doi.org/10.1017/S147474801100003X) * C. Sabbah, _Non-commutative Hodge structures_, [pdf](http://www.math.polytechnique.fr/~sabbah/sabbah_grenoble10.pdf) * [[Dmytro Shklyarov]], _Non-commutative Hodge structures: Towards matching categorical and geometric examples_, Trans. Amer. Math. Soc. 366 (2014), 2923-2974 [arxiv/1107.3156](http://arxiv.org/abs/1107.3156) [doi](https://doi.org/10.1090/S0002-9947-2014-05913-8) [[!redirects noncommutative Hodge theory]] [[!redirects Noncommutative Hodge theory]] </textarea> </div>  </body> </html>