Theta Pairing of Hypersurface Rings

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DOI:

https://doi.org/10.37256/cm.5420244895

Keywords:

Hochster Theta pairing, matrix factorization, Riemann-Hodge bilinear relations, residue pairing, chern character

Abstract

In this article, we prove a conjecture on the positive definiteness of the Hochster Theta pairing over a general isolated hypersurface singularity, namely: Let R be an admissible isolated hypersurface singularity of dimension n. If n is odd, then mceclip1-f1da6351346f6c7214d048692892a633.png is positive semi-definite on mceclip1-c9316689bcd70f3a70a507167b884e34.png. The conjecture is expected to be true for the polynomial ring over any field. We prove this conjecture over any field of arbitrary characteristic. We also provide two different proofs of the above conjecture overmceclip4-f8a26796371ead59e7b26a0d09f0fcd9.pngusing the Hodge theory of isolated hypersurface singularities and structural facts about the category of matrix factorizations. The first proof overmceclip2-96030ce12ef95e392b448de0f921a286.pngis a more complete and developed version of a former work of the author. We have extended some of the former results in this article. The second proof overmceclip3-e9a934e184341fe28a50e7a6d072c116.pngis quite direct and uses a former result of the author on Riemann-Hodge bilinear relations for Grothendieck residue pairing of isolated hypersurface singularities.

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Published

2024-10-21

How to Cite

1.
Rahmati MR. Theta Pairing of Hypersurface Rings. Contemp. Math. [Internet]. 2024 Oct. 21 [cited 2024 Nov. 13];5(4):4428-40. Available from: https://ojs.wiserpub.com/index.php/CM/article/view/4895

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Research Article

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