On the Yau cycle of a normal surface singularity (Q436038)

Let \(\pi :(X,E)\longrightarrow (V,o)\) be a resolution of a normal surface singularity. After M. Artin's work, \((V,o)\) has been researching through considering \((X,E)\). Ph. Wagreich considered three invariants \(p_g(V,o)\) (geometric genus), \(p_a(V,o)\) (arithmetic genus), \(p_f(V,o)\) (fundamental genus); also proved following fundamental properties.NEWLINENEWLINE(i) \(p_f(V,o)\leqq p_f(V,o)\leqq p_f(V,o)\)NEWLINENEWLINE(ii) \(p_f(V,o)=0\Longleftrightarrow p_f(V,o)=0\Longleftrightarrow p_f(V,o)=0\)NEWLINENEWLINE(iii) \(p_f(V,o)=1\Longleftrightarrow p_f(V,o)=1\).NEWLINENEWLINEIf \((V,o)\) satisfies the conditions of (ii) (resp. (iii)), then it is called a rational (elliptic) singularity.NEWLINENEWLINEFor elliptic singularities, there are many deep researches due to H. Laufer, S. S. T. Yau and many others. However, from same point of view, we have a little for the case of \(p_f(V,o)\geqq 2\). The author tries to study this case.NEWLINENEWLINEFor elliptic singularities, \textit{S. S.-T. Yau} [Trans. Am. Math. Soc. 257, 269--329 (1980; Zbl 0343.32009)] gave the definition of elliptic sequences. By using it, he proved many important results. For example, he proved that if the length of the elliptic sequence equals to \(p_g(V,o)\), then \((V,o)\) is Gorenstein. Further, he proved that if \((V,o)\) is a numerical Gorenstein singularity, then \(K_V\) (canonical cycle) is written as the sum of the cycle in the elliptic sequence. In [Pac. J. Math. 170, No. 1, 271--295 (1995; Zbl 0848.14017)], the reviewer formally generalized elliptic sequences to the case of \(p_f(V,o)\geqq 1\) and call it Yau sequences. However, through several examples, he showed that we cannot expect obtaining simple and beautiful results same as elliptic case.NEWLINENEWLINEThe author defines the \textit{Yau cycle \(Y\)} as the sum of all cycles appearing in the Yau sequence. Under some conditions, he proves a relation between the canonical cycle \(K_V\) and the Yau cycle \(Y\) (3.2). Also, under some conditions, he gives an estimate of \(p_g(V,o)\) by \(p_f(V,o)\) and the length \(m\) of the Yau sequence (Theorem 3.9: \(p_g(V,o)\leqq p_f(V,o)^2m\) ). The problem to estimate \(p_g(V,o)\) is very important and essential. Then this result seems to be very fundamental. Moreover, for Gorenstein singularities with \(p_f(V,o)=2\), he proves several meaningful results on the multiplicity and embedding dimension of \((V,o)\).