The cohomology of line bundles of splice-quotient singularities

We provide several results on splice-quotient singularities: (1) a combinatorial expression of the dimension of the first cohomology of all ‘natural’ line bundles (involving the Seiberg–Witten invariants of the singularity link); (2) an equivariant Campillo–Delgado–Gusein-Zade type formula about the dimension of relative sections of line bundles, extending former results about rational and minimally elliptic singularities; (3) in particular, we prove that the equivariant, divisorial multi-variable Hilbert–Poincaré series is topological; (4) a combinatorial description of divisors of analytic function-germs; (5) and an expression for the multiplicity of the singularity from its resolution graph (in particular solving Zariskiʼs Multiplicity Conjecture for splice-quotient hypersurfaces).From topological point of view, we get a new combinatorial expression for the Seiberg–Witten invariants of links of splice-quotient singularities.