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The Hodge structure on
the singularity category of a complex hypersurface
Michael K. Brown and Mark E. Walker |
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Vol. 10 (2025), No. 2, 237–277
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Abstract |
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Given a complex affine hypersurface with isolated singularity determined by a homogeneous polynomial, we identify the noncommutative Hodge structure on the periodic cyclic homology of its singularity category with the classical Hodge structure on the primitive cohomology of the associated projective hypersurface. As a consequence, we show that the Hodge conjecture for the projective hypersurface is equivalent to a dg-categorical analogue of the Hodge conjecture for the singularity category. |
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Keywords
Hodge conjecture, hypersurface, matrix factorization,
noncommutative Hodge theory, singularity category
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Mathematical Subject Classification
Primary: 14F08
Secondary: 13D03, 13D09, 14C30, 14J70, 19D55
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Milestones
Received: 15 July 2024
Revised: 23 May 2025
Accepted: 9 June 2025
Published: 1 July 2025
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Authors |
| Michael K. Brown | |
| Department of Mathematics and
Statistics Auburn University Auburn, AL United States |
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| Mark E. Walker | |
| Department of Mathematics University of Nebraska–Lincoln Lincoln, NE United States |
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| © 2025 MSP (Mathematical Sciences Publishers). |
