On the Griffiths numbers for higher dimensional singularities (Q748363)

Let \(V\) be a Stein analytic space of dimension \(n \geq 2\) having a single normal singularity at a point \(o \in V\). There are several interesting numbers that may be attached to the pair \((V,o)\). For instance, for \(p=1, \ldots, n\), the numbers \(g^{(p)}\) (Griffiths), \(h^{(p)}\) (Hironaka), \(\delta ^{(p)}\), \(s^{(p)}\). Their definition involves various coherent sheaves on \(V\) that extend the locally free sheaf of holomorphic differentials \({\Omega _U}^{(p)}\) to \(V\), where \(U=V \setminus \{o\}\). Concretely, \(\delta ^{(p)}\) measures how many linearly independent holomorphic \(p\)-differential forms on \(U\) extend to \(V\), \(g^{(p)}\) has a similar interpretation but using \(L^2\)-integrable holomorphic \(p\)-forms instead. In the early 1980's, S. S.-T. Yau (who investigated these invariants) conjectured that: (i) \(g^{(n-1)} \geq n-1\) and (ii) \(\delta ^{(n-1)} \geq h^{(n-1)} +n -1\). Yau proved the conjecture for certain singularities, in particular when \(n=2\) (surfaces). The authors show that for \(n \geq 3\) neither (i) nor (ii) is always valid. Indeed, if \((V,o)\) is a rigid Gorenstein singularity and \(n \geq 3\), then both (i) and (ii) fail to hold. It is known that such singularities exist. On the positive side, they prove that if \((V,o)\) (of dimension \(n \geq 2\)) satisfies \(q=s^{(n-1)} >0\), then \(g^{(n-1)} \geq n-1\) (the number \(q\) is called the \textit{irregularity} of \((V,o)\)). The proof is rather involved, it improves methods developed by Yau.