Hodge--Tjurina Numbers, Tjurina Spectra, and Thom--Sebastiani Formulas

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.