The canonical filtration of higher dimensional purely elliptic singularity of a special type (Q1176642)

Over a field of characteristic 0, the author considers the following class of normal, \(d\)-dimensional singularities \((A,m)\), which he refers to as ``very good'' elliptic singularities: They are purely elliptic (i.e. have plurigenera \(\delta_ m=1\) for \(m\geq 1)\), Cohen Macaulay and of Hodge type \((0,d-1)\) [in the sense of \textit{S. Ishii}, Math. Ann. 270, 541-554 (1985; Zbl 0541.14002) and in Complex analytic singularities, Proc. Semin., Ibaraki/Jap. 1984, Adv. Stud. Pure Math. 8, 165-198 (1987; Zbl 0628.14002)]. For dimension 2, this means just ``simply elliptic''. The author shows: If further \((A,m)\) is a quasihomogeneous hypersurface singularity and \(\dim A\geq 2\), then \(A\) has a canonical model; this is a direct proof, independent of the recent ``minimal model program''. As an application of this result, the following theorem on the Newton boundary of a quasihomogeneous polynomial is shown: ``Let \(f\in\mathbb{C}[X_ 1,\ldots,X_{d+1}]\) be a quasi-homogeneous polynomial with respect to the weight \(q=(q_ 1,\ldots,q_{d+1})\), where \(0<q_ i\in\mathbb{Q}\) for \(i=1,\ldots,d+1\). Assume that (1) \(\sum^{d+1}_{i=1}q_ i=\deg_ qf=1\), and (2) \(\{f=0\}-\{0\}\) has only rational singularities. -- Then \(\dim_ \mathbb{R}\Gamma(f)=d\), and the point \((1,\ldots,1)\) is contained in the relative interior of \(\Gamma(f)\).'' Further, \(f\) is shown to be the initial form of a ``very good'' elliptic singularity with respect to the canonical filtration.