Higher Du Bois and higher rational singularities (Q6610102)

The paper under review proves a generalization of Theorem of \textit{P. Du Bois} [Bull. Soc. Math. Fr. 109, 41--81 (1981; Zbl 0465.14009)].\N\NTheorem 1. Let \(f:\mathcal{Y}\to S\) be a flat proper family of complex algebraic varieties, and let \(s\in S\). Suppose that the fiber \(Y_s\) has \(k\)-Du Bois lci singularities. Then, possibly after replacing \(S\) by a neighbourhood of \(s\), the higher direct image sheaves \(R^qf_*\Omega^9_{\mathcal{Y}/S}\) of the relative Kähler differentials are locally free and compatible with base change for \(0\leq p\leq k\) and all \(q\geq 0\).\N\NA result of Du Bois corresonds to the case \(k=0\). The lci assumption is used to control the sheaves of Kähler and relative Kähler differentials, e.g.,it is shown that the sheaves \(\Omega^p_{\mathcal{Y}/S}\) are flat over \(S\) for \(p\leq k\).\N\NCorollary 2: Let \(f:\mathcal{Y}\to S\) be a flat proper family of complex algebraic varieties over an irreducible base. For \(s\in S\), suppose that the fiber \(Y_s\) has \(k\)-Du Bois lci singularities. Then, for every fiber \(t\) such that \(Y_t\) is smooth, \(\dim Gr_F^p h^{p+q}(Y_t)=\dim Gr^p_F H^{p+q}(Y_s)\) for every \(q\) abd for \(0\leq p\leq k\). Equivalently, \(h^{p,q}(Y_s)=h^{p,q}(Y_t)\) for all \(p\leq k\).\N\NCorollary 3: Let \(Y\) be a canonical Calabi-Yau variety, which is additionally a scheme with \(1\)-Du Bois lic singularities (not necessarily isolated). Then the function \({\mathbf{Def}}(Y)\) is unobstructed.\N\NIn the context of rational singularities, a definition of \(k\)-rational singularities is proposed extending the definition of rational singularities.\N\NTheorem 4: Let \(X\) have either lci singularities (not necessarily isolated), or isolated singularities (not necessarily lic). If \(X\) is \(k\)-rational, then \(X\) is \(k\)-Du Bois.\N\NConjecture 5: If \(X\) has lci singularities and \(X\) is \(k\)-Du Bois, then \(X\) is \((k-1)\)-rational.\N\NIn the case of hypersurface singularities, M. Saito proved that the proposed definition of \(k\)-rational singularities is equivalent to a numerical definition for \(k\)-rational singularities.\N\NProposition 6: Let \((X,x)\) be an isolated hypersurface singularity, and let \(\tilde{\alpha}_X\) be the minimal exponent defined by \textit{M. Saito} [Math. Ann. 295, No. 1, 51--74 (1993; Zbl 0788.32025)]. Then (i) \(X\) is \(k\)-Du Bois if and only if \(\tilde{\alpha}_X\geq k+1\); (ii) \(X\) is \(k\)-rational if and only if \(\tilde{\alpha}_X>k+1\).\N\NAs applications, the behaviour of Hodge nubmers in families and the unobstructedness of singular Calabi-yau varieties is discussed.\N\NCorollary 7: Let \(F:\mathcal{Y}\to S\) be a flat proper family of complex algebraic varieties over an irreducible base. For \(s\in S\), suppose that the fiber \(Y_s\) has \(k\)-rational lci singularities. Then, for every fiber \(t\) such that \(Y_t\) is smooth, and for all \(p\leq k\), \N\[\underline{h}{p,q}(Y_s)=\underline{h}^{q,p}(Y_s)=h^{n-p,n-q}(Y_s)\N=h^{p,q}(Y_t)=h^{q,p}(Y_t) =h^{n-p,n-q}(Y_t).\] \NMoreover, for all \(p\leq k\), \N\[\underline{h}^{p,q}(Y_s)=\text{dim Gr}^p_F \text{Im}(H^{p+q}(Y_s)\stackrel{\pi_s^*}\longrightarrow H^{p+q}(\hat{Y}_s))\] \Nwhere \(\pi_s: \hat{Y}_s\to Y_s\) is an arbitrary projective resolution.