Characterization of isolated complete intersection singularities with \(\mathbb{C}^{\ast}\)-action of dimension \(n \geq 2\) by means of geometric genus and irregularity (Q376083)

For singularities of complex spaces, there are many kinds of invariants (topological, holomorphic etc). Given singularities, the problems to compute those invariants and to find relations between them are important. Moreover, the converse problems (namely, characterizing of singularities by invariants) are also standard and important.NEWLINENEWLINEFor a complex singularity \((V,0)\), the authors discuss two invariants: the geometric genus \(p_g(V,0)\) and the irregularity \(q(V,0)\). Let \(\pi: (\tilde V,E) \longrightarrow (V,0)\) be a resolution of a normal singularity of dimension \(n\) (\(n\geqq 2\)). Then, \(p_g(V,0)\) and \(q(V,0)\) are defined as follows: NEWLINENEWLINE\[NEWLINEp_g(V,0)=\dim_{\mathbb C}R^{n-1}\pi_*{\mathcal O}_{\tilde V},\quad q(V,0)=\dim_{\mathbb C}H^0(\Omega_{{\tilde V}-E}^{n-1})/ H^0(\Omega_{{\tilde V}}^{n-1}),NEWLINE\]NEWLINENEWLINENEWLINEwhere \(\Omega_{{\tilde V}}^{i}\) is the sheaf of germs of holomorphic \(i\)-forms on \(V\). NEWLINENEWLINENEWLINENEWLINE In 1983, the first author proved that \(q(V,0) \geqq p_g(V,0)-h^{n-1}({\mathcal O}_E)\). The first main result of this paper is the following. NEWLINENEWLINENEWLINENEWLINE \vskip 2truemm { Theorem A.} Let \((V,0)\) be a normal singularity of dimension \(n\), \(n\geqq 2\), with \({\mathbb C}^*\)-action, and \(\pi: (\tilde V,E) \longrightarrow (V,0)\) be a good resolution with \(E=f^{-1}(0)_{\mathrm{red}}\). Then \(q=p_g-h^{n-1}({\mathcal O}_E)\).NEWLINENEWLINENEWLINENEWLINE\vskip 2truemm Namely, for a singularity with \({\mathbb C}^*\)-action, the gap between \(p_g\) and \(q\) is determined only by the exceptional set \(E\). Next, the authors consider the inverse problem for the above. Here we recall the definition of Du Bois singularities.NEWLINENEWLINENEWLINENEWLINE\vskip 2truemm { Definition.} A normal isolated singularity \((V,0)\) is called a Du Bois singularity if the canonical map \((R^i\pi_*{\mathcal O}_{\tilde V})_0 \longrightarrow H^i(E,{\mathcal O}_E)\) is an isomorphism for each \(i\), where \(\pi: (\tilde V,E) \longrightarrow (V,0)\) is a good resolution with \(E=f^{-1}(0)_{\mathrm{red}}\). NEWLINENEWLINENEWLINENEWLINE \vskip 2truemm Therefore, the almost structure of a Du Bois singularity seems to be determined by the structure of \(E\). Actually, in 1985, S. Ishii proved that for a 2-dimensional Gorenstein singularity, it is Du Bois if and only if it is rational (\(p_g=0\)), simple elliptic or cusp. The second result of this paper is the following.NEWLINENEWLINE\vskip 2truemm { Theorem B.} Let \((V,0)\) be a normal isolated complete intersection singularity of dimension \(n\), \(n\geqq 2\), and \(\pi: (\tilde V,E) \longrightarrow (V,0)\) be a good resolution with \(E=f^{-1}(0)_{\mathrm{red}}\). If \(q=p_g-h^{n-1}({\mathcal O}_E)\), then either \((V,0)\) has a \({\mathbb C}^*\)-action or \((V,0)\) is a Du Bois singularity.