Hodge ideals and spectrum of isolated hypersurface singularities
Let \(X\) be a complex algebraic variety and let \(D=\alpha Z\) be a \(\mathbb{Q}\)-divisor on \(X\). \textit{M. Mustaţǎ} and \textit{M. Popa} defined in [J. Éc. Polytech., Math. 6, 283--328 (2019; Zbl 1427.14045); Forum Math. Sigma 8, Paper No. e19, 41 p. (2020; Zbl 1451.14055)] a series of Hodge ideals \(\mathcal{I}_p(D)\subseteq\mathcal{O}_X\), which conform an interesting, finer invariant of \(D\) than multiplier ideals. Although this notion is of algebraic nature, the authors of the paper under review show that it can be extended to the analytic case of \(X\) being a complex manifold. Their first goal is to study such ideals in the case that \(Z\) is defined locally by a holomorphic function \(f\) with an isolated singularity at the origin and which either is semi-weighted-homogeneous or has non-degenerate Newton boundary. In fact, they give an explicit formula for the Hodge ideals, improving the general ones obtained by Mustaţă and Popa and generalizing the work of \textit{M. Zhang} [Asian J. Math. 25, No. 5, 641--664 (2021; Zbl 1498.14043)] showing also that, in the non-degenerate Newton boundary case, the assumption that \(f\) is convenient is not necessary. The second goal of the authors of the paper under review is to relate the usual spectrum \(\mathrm{Sp}_f(t)\) of \(f\) with a new one, called Hodge ideal spectrum, denoted by \(\mathrm{Sp}_f^{\mathrm{HI}}(t)\) and defined by means of the Hodge ideals \(I_p(D)\) modulo the Jacobian ideal of \(f\). Both spectra coincide in the case \(f\) is weighted-homogeneous but not in general under the same assumptions on \(f\) as above. This is shown by several results that they prove about the difference between the exponents of \(f\) given by both spectra. All those last results are clearly stated in the introduction and proved later on in the paper after the discussion about Hodge ideals. After that, the authors consider some enlightening examples, carefully chosen to answer certain questions about the most important concepts throughout the text.
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