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| Format: | Preprint |
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2012
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| Online Access: | https://arxiv.org/abs/1212.2859 |
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| _version_ | 1866909480893022208 |
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| author | Efimov, Alexander I. |
| author_facet | Efimov, Alexander I. |
| contents | In this paper, we will show that for a smooth quasi-projective variety over $\C,$ and a regular function $W:X\to \C,$ the periodic cyclic homology of the DG category of matrix factorizations $MF(X,W)$ is identified (unde Riemann-Hilbert correspondence) with vanishing cohomology $H^{\bullet}(X^{an},ϕ_W\C_X),$ with monodromy twisted by sign. Also, Hochschild homology is identified respectively with the hypercohomology of $(Ω_X^{\bullet},dW\wedge).$
One can show that the image of the Chern character is contained in the subspace of Hodge classes. One can formulate the Hodge conjecture stating that it is surjective ($\otimes\Q$) onto Hodge classes. For W=0 and $X$ smooth projective this is precisely the classical Hodge conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1212_2859 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Cyclic homology of categories of matrix factorizations Efimov, Alexander I. Algebraic Geometry Complex Variables 14F05, 32S30 In this paper, we will show that for a smooth quasi-projective variety over $\C,$ and a regular function $W:X\to \C,$ the periodic cyclic homology of the DG category of matrix factorizations $MF(X,W)$ is identified (unde Riemann-Hilbert correspondence) with vanishing cohomology $H^{\bullet}(X^{an},ϕ_W\C_X),$ with monodromy twisted by sign. Also, Hochschild homology is identified respectively with the hypercohomology of $(Ω_X^{\bullet},dW\wedge).$ One can show that the image of the Chern character is contained in the subspace of Hodge classes. One can formulate the Hodge conjecture stating that it is surjective ($\otimes\Q$) onto Hodge classes. For W=0 and $X$ smooth projective this is precisely the classical Hodge conjecture. |
| title | Cyclic homology of categories of matrix factorizations |
| topic | Algebraic Geometry Complex Variables 14F05, 32S30 |
| url | https://arxiv.org/abs/1212.2859 |