We show that Yaus conjecture on the inequalities for (n-1)-th Griffiths number and (n-1)-th Hironaka number does not hold for isolated rigid Gorenstein singularities of dimension greater than 2. But his conjecture on the inequality for (n-1)-th Griffiths number is true for irregular singularities.
In this paper, we classified the surfaces whose canonical maps are abelian covers over $mathbb{P}^2$. Moveover, we construct a new Campedelli surface with fundamental group $mathbb{Z}_2^{oplus 3}$ and give defining equations for Persssons surface and
Tans surfaces with odd canonical degrees explicitly.
