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The Hilbert series of Hodge ideals of hyperplane arrangements. (English) Zbl 1445.14032

In the paper under review the authors study the Hilbert series of Hodge ideals of hyperplane arrangements. Let \(X\) be a smooth complex algebraic variety and let \(D\) be a reduced effective divisor on \(X\). One can attach to \(D\) a sequence of ideals called the Hodge ideals \(I_{p}(D)\) for \(p\geq 0\). The first ideal for \(p=0\) is the multiplier ideal \(\mathcal{I}((1-\varepsilon)D)\) for \(0<\varepsilon <<1\) and the higher ideals can be viewed as similar but more refined measures of the singularities of \(D\). In the present paper the authors focus on the classes of the Hodge ideals in the Grothendieck group \(K_{0}(X)\) of coherent sheaves on \(X\) as encoded in the generating function \[\sum_{p\geq 0}[I_{p}(D)]y^{p} \in K_{0}(X)[y].\] The main observation is that this generating function can be described in terms of the motivic Chern class of the inclusion \(j : U \rightarrow X\), where \(U\) is the complement of the support of \(D\). This motivic Chern class is a group homomorphism \[mC_{y} : K_{0}(\mathrm{Var}/X) \rightarrow K_{0}(X)[y],\] where \(K_{0}(\mathrm{Var}/X)\) is the Grothendieck group of varieties over \(X\). Let \(X = V\) be a complex vector space of dimension \(n\) and \(D = D_{\mathcal{A}}\) is the divisor corresponding to an arrangement \(\mathcal{A}\) of linear hyperplanes in \(X\). We consider the standard action of \(T = \mathbb{C}^{*}\), then describing \(I_{p}(D_{\mathcal{A}})\) in the Grothendieck group \(K_{0}^{T}(X)\) of \(T\)-equivariant coherent sheaves on \(X\) is equivalent to describing the Hilbert series \(H_{I_{p}(D_{\mathcal{A}})}(t)\) of \(I_{p}(D_{\mathcal{A}})\). Due to the fact that in the setting as above the equivariant motivic Chern class is easy to compute, the first main result describes the generating function of the Hilbert series \(H_{I_{p}(D_{\mathcal{A}})}(t)\) in terms of the Poincaré polynomial \(\pi(\mathcal{A},t)\) of the arrangement.
Theorem A. If \(\mathcal{A}\) is a central hyperplane arrangement of \(d\) in an \(n\)-dimensional complex vector space \(V\) and if \(D_{\mathcal{A}} = \sum_{H \in \mathcal{A}}H\), then \[\sum_{p\geq 0}H_{I_{p}(D_{\mathcal{A}})}(t) y^{p} = \frac{t^{d}}{(1-t)^{n}(1-t^{d}y)}\cdot \pi(\mathcal{A},(1-t)/t(1-t^{d-1}y)).\] By letting \(y=0\) in the above theorem and recalling the identification of \(I_{0}(D)\) with a multiplier ideal, one obtains the following result.
Corollary. If \(\mathcal{A}\) is a central hyperplane arrangement of \(d\) hyperplanes in an \(n\)-dimensional complex vector space \(V\) and if \(D_{\mathcal{A}} = \sum_{H \in \mathcal{A}}H\), then the Hilbert series of the multiplier ideal \(I = \mathcal{I}((1-\varepsilon)D_{\mathcal{A}})\) with \(0 < \varepsilon \ll1\) is given by \[H_{I}(t) = \frac{t^{d}}{(1-t)^{n}}\cdot \pi(\mathcal{A},t^{-1}-1).\]

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14B05 Singularities in algebraic geometry
32S22 Relations with arrangements of hyperplanes
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