This paper is concerned with the relationship between the Du Bois complex \({\underline{\Omega}}^\bullet_Z\) of a reduced hypersurface \(Z\) inside of a smooth complex algebraic variety \(X\) and the minimal exponent \(\widetilde{\alpha}(Z)\). The proofs of the results use methods from the the theory of mixed Hodge modules.
The Du Bois complex \(\underline{\Omega}^\bullet_Z\) is an object in the derived category of filtered complexes of sheaves which, if \(Z\) is smooth, agrees with the complex of Kähler differentials (the algebraic de Rham complex) \(\Omega_Z^\bullet\) on \(Z\) equipped with the “stupid” filtration. In general, if \(\underline\Omega^p_Z:=\mathrm{Gr}_F^p \Omega_Z^\bullet[p]\), there exist canonical morphisms \[ \Omega_Z^p\to\underline\Omega^p_Z \] for all \(p\), and we say that \(Z\) has Du Bois singularities (resp. \(k\)-Du Bois) if the above morphism is a quasi-isomorphism for \(p=0\) (resp. for all \(p\) such that \(0\leq p\leq k\)).
On the other hand, the minimal exponent \(\widetilde\alpha(Z)\) is a refinement of the log-canonical threshold. More precisely \(\widetilde\alpha(Z)\) is defined as the oposite number of the greatest root of \(\frac{b_Z(s)}{(s+1)}\), where \(b_Z(s)\) is the Bernstein-Sato polynomial of \(Z\). For every \(x\in Z\) there is also a local version \(\widetilde\alpha_x (Z)\) such that \(\widetilde\alpha(Z)=\mathrm{min}_{x\in Z}\widetilde\alpha_x(Z)\). In terms of Mustață-Popa’s Hodge ideals, the support of the Hodge ideal \(I_p(Z)\) is the locus in \(Z\) where the (local version) of the minimal exponent is \(<p+1\).
In [M. Saito, Mosc. Math. J. 9, No. 1, 151–181 (2009; Zbl 1196.14015)] it is shown that \(Z\) has Du Bois singularities if and only if \(\widetilde \alpha (Z)\geq 1\). The main result of this paper provides a generalization of one of the directions of this result to \(p\)-Du Bois singularities.
Theorem 1.1. (Main Theorem) If \(p\) is an integer such that \(0\leq p \leq \widetilde\alpha(Z)- 1\), then the canonical morphism \[ \Omega_Z^p\to\underline\Omega^p_Z \] is a quasi-isomorphism.
The converse of this result was proven afterwards in [S.-J. Jung et al., Proc. London Math. Soc. 125, No. 3, 543–567 (2022; doi:10.1112/plms.12464)].
The main theorem has some very interesting consequences which do not involve the Du Bois complex in their statements. Exploiting the connection between the Du Bois complex and the sheaves of differential forms with logarithmic poles on a resolution [J. H. M. Steenbrink, Astérisque 130, 330–341 (1985; Zbl 0582.32039)], the authors obtain a vanishing result (Corollary 1.2 in the paper) for the derived pushforwards of those sheaves via the resolution morphism, extending results of J. H. M. Steenbrink [Astérisque 130, 330–341 (1985; Zbl 0582.32039), Proposition 3.3] and K. Schwede [Compos. Math. 143, No. 4, 813–828 (2007; Zbl 1125.14002), Theorem 4.3]. The main theorem also allows the authors to translate the Akizuki-Nakano vanishing theorem for the graded pieces of the Du Bois complex [F Guillén et al., Hyperrésolutions cubiques et descente cohomologique. (La plupart des exposés d’un séminaire sur la théorie de Hodge-Deligne, Barcelona (Spain), 1982). Berlin etc.: Springer-Verlag (1988; Zbl 0638.00011), Theorem V.5.1] to a version for the sheaves of Kähler differentials under the appropriate assumptions on the minimal exponent (Corollary 1.3 in the paper).
Part of the proof of the main theorem involves showing that \(\mathcal H^i(\underline\Omega^p_Z)=0\) for all \(i\neq 0\) under the given hypothesis. In fact, the authors are able to prove a couple more theorems about the vanishing of \(\mathcal H^i(\underline\Omega^p_Z)=0\) under different hypotheses. Under the weaker assumption that \(0\leq q\leq\widetilde \alpha(Z)\), the authors prove the vanishing of \(\mathcal H^{n-q-1}(\underline\Omega^q_Z)\) provided that \(q\neq n-1\) (Theorem 1.4 in the paper). Moreover, the paper contains the following more general vanishing result of the individual cohomologies of the graded pieces of the Du Bois complex, which also applies to the non-Du Bois case (that is, when there exists \(x\in Z\) such that \(\alpha_x(Z)<1\)).
Theorem 3.7. Let \(p\) be a nonnegative integer. If the locus \(Z_p\) of points \(x\in Z\) with \(\widetilde\alpha_x(Z)<p+1\) satisfies \(\mathrm{codim}_X Z_p > i + p + 2\) for some \(i \geq 1\), then \(\mathcal H^i(\underline\Omega^p_Z)=0\).
Note that, since \(Z_p\) is contained in the singular locus of \(Z\), the previous theorem implies (Corollary 3.8 in the paper) that \[ \mathcal H^i(\underline\Omega^p_Z)=0\quad\text{for all }p \text{ and for }1\leq i<n-s-p-2, \] where \(n\) is the dimension of \(X\) and \(s\) is the dimension of the singular locus of \(Z\).
On top of the aforementioned vanishing results, the paper contains an important non-vanishing result for the cohomology of certain graded pieces of the Du Bois complex when the minimal exponent is large (Theorem 1.5 in the paper), whose proof uses the \(V\)-filtration and the duality between nearby and vanishing cycles, unlike the other results in the paper. The non-vanishing result, combined with the vanishing of \(\mathcal H^i(\underline\Omega_Z^p)\) for all \(i\geq 1\) and all \(p\) if \(Z\) has toroidal or quotient singularities (see [P. Du Bois, Bull. Soc. Math. Fr. 109, 41–81 (1981; Zbl 0465.14009)], Section 5 and [F Guillén et al., Hyperrésolutions cubiques et descente cohomologique. (La plupart des exposés d’un séminaire sur la théorie de Hodge-Deligne, Barcelona (Spain), 1982). Berlin etc.: Springer-Verlag (1988; Zbl 0638.00011), Chapter V.4]) yields the following bound on the minimal exponent as an application.
Corollary 1.6. If \(Z\) is singular and has quotient of toroidal singularities, then \(1< \widetilde \alpha(Z)\leq 2\).
Another interesting application of the non-vanishing result given by the authors is an example (Example 1.7) showing that \(\mathcal H^i(\underline\Omega_Z^p)\) is not upper semicontinuous in families. They do so by constructing a pencil of hypersurfaces with singular points at \(0\) in affine space from a hypersurface with quotient singularities (in which all the higher cohomology sheaves of all the graded pieces of the Du Bois complex vanish) and a hypersurface with high minimal exponent at \(0\) (to which the non-vanishing result applies for certain \(i\) and \(p\)), and employing the fact that the minimal exponent is lower semicontinuous in families.