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Hodge--Tjurina Numbers, Tjurina Spectra, and Thom--Sebastiani Formulas
Abstract
We study Hodge--Tjurina numbers and the Tjurina subspectrum of isolated hypersurface singularities. We relate these invariants to Hodge ideals and microlocal V-filtrations and investigate their behavior under Thom--Sebastiani sums. For separated sums, we obtain a filtered defect formula and, in particular, give an affirmative answer to the question raised by Almirón concerning the Thom--Sebastiani property of the Tjurina subspectrum: it holds if and only if the Tjurina number is multiplicative. We also discuss rational sampling, spectral reconstruction, symmetry bounds, and examples illustrating the information captured and lost by integer Hodge--Tjurina data.
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BibTeX
@misc{zheng2026hodgetjurina,
title = {{Hodge--Tjurina Numbers, Tjurina Spectra, and Thom--Sebastiani Formulas}},
author = {Shijie Zheng and Huaiqing Zuo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08290},
eprint = {2610.08290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08290},
}
RIS
TY - UNPB
TI - Hodge--Tjurina Numbers, Tjurina Spectra, and Thom--Sebastiani Formulas
AU - Shijie Zheng
AU - Huaiqing Zuo
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08290
UR - https://arxiv.org/abs/2610.08290
AB - We study Hodge--Tjurina numbers and the Tjurina subspectrum of isolated hypersurface singularities. We relate these invariants to Hodge ideals and microlocal V-filtrations and investigate their behavior under Thom--Sebastiani sums. For separated sums, we obtain a filtered defect formula and, in particular, give an affirmative answer to the question raised by Almirón concerning the Thom--Sebastiani property of the Tjurina subspectrum: it holds if and only if the Tjurina number is multiplicative. We also discuss rational sampling, spectral reconstruction, symmetry bounds, and examples illustrating the information captured and lost by integer Hodge--Tjurina data.
ER -