On isolated log canonical singularities with index one (Q2902459)

Let \(P \in X\) be an \(n\)-dimensional isolated log canonical singularity with index one which are not log terminal. Let \(f: Y \rightarrow X\) be a projective resolution such that \(f\) is an isomorphism outside \(P\) and that Supp\(f^{-1}(P)\) is a simple normal crossing divisor on \(Y\). Denote NEWLINE\[CARRIAGE_RETURNNEWLINEK_Y=f^* K_X+F-E,CARRIAGE_RETURNNEWLINE\]NEWLINE \noindent where \(E\) and \(F\) are effective Cartier divisors and have no common irreducible components. In the paper under review, the author proves that NEWLINE\[CARRIAGE_RETURNNEWLINER^i f_{*} \mathcal{O}_Y \simeq H^i (E, \mathcal{O}_E)CARRIAGE_RETURNNEWLINE\]NEWLINE \noindent for every \(i>0\) and that NEWLINE\[CARRIAGE_RETURNNEWLINER^{n-1} f_{*} \mathcal{O}_Y \simeq \mathbb{C} (P).CARRIAGE_RETURNNEWLINE\]NEWLINE Similar results have been obtained in [\textit{S. Ishii}, Math. Ann. 270, 541--554 (1985; Zbl 0541.14002)] by the theory of Du Bois singularities, more specifically, Ishii proved NEWLINE\[CARRIAGE_RETURNNEWLINER^i f_{*} \mathcal{O}_Y \simeq H^i (f^{-1} (P)_{red}, \mathcal{O}_{f^{-1} (P)_{red}})CARRIAGE_RETURNNEWLINE\]NEWLINE \noindent for every \(i>0\) and that NEWLINE\[CARRIAGE_RETURNNEWLINER^{n-1} f_{*} \mathcal{O}_Y \simeq H^{n-1} (E, \mathcal{O}_E) \simeq \mathbb{C}.CARRIAGE_RETURNNEWLINE\]NEWLINE In [Zbl 0541.14002], the singularity \(P \in X\) is said to be of type \((0, i)\) if NEWLINE\[CARRIAGE_RETURNNEWLINEGr_k^{W} H^{n-1} (E, \mathcal{O}_E)=\begin{cases} \mathbb{C} \;\text{if} \;k=i \\ 0 \text{ otherwise} \end{cases}CARRIAGE_RETURNNEWLINE\]NEWLINE \noindent where \(W\) is the weight filtration of the mixed Hodge structure on \(H^{n-1} (E, \mathbb{C})\). In this paper, the author defines \(\mu (P \in X)\) as NEWLINE\[CARRIAGE_RETURNNEWLINE\mu=\mu (P \in X)=\min \{\dim W| W \quad \text{is a stratum of} \quad E\}.CARRIAGE_RETURNNEWLINE\]NEWLINE \noindent He proves that this invariant coincides with Ishii's Hodge theoretic one, that is, for \(P \in X\) an isolated log canonical singularity with index one which are not log terminal, \(P \in X\) is of type \((0, i)\) if and only if \(\mu (P \in X)=i\).NEWLINENEWLINEIn the proof of his main results, the author uses the minimal model program with scaling, a technique originally developed in [\textit{C. Birkar} et al., J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)]. It is interesting to see that the minimal model program meshes well with Hodge theory in the investigation of isolated log canonical singularities. It seems that the minimal model program reveals more information on the divisor \(E\). We refer the interested reader to the cited paper for details.