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 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 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 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”.
Noncommutative Hodge theory is being developed in
Ludmil Katzarkov, Maxim Kontsevich, Tony Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/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
D. Kaledin, Tokyo lectures “Homological methods in non-commutative geometry”, pdf, TeX
Claus Hertling, Christian Sevenheck, Twistor structures, $tt^*$-geometry and singularity theory, arxiv/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 doi
C. Sabbah, Non-commutative Hodge structures, pdf
Dmytro Shklyarov, Non-commutative Hodge structures: Towards matching categorical and geometric examples, Trans. Amer. Math. Soc. 366 (2014), 2923-2974 arxiv/1107.3156 doi
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