Abstract: 
Some relations between normal complex surface singularities 
and symplectic fillings of the links of the singularities are discussed. For a certain class
of singularities of general type, which are called hypersurface $K3$ singularities in
this paper, an inequality for numerical invariants of any minimal symplectic fillings of
the links of the singularities is derived. This inequality can be regarded as a symplectic/contact
analog of the $11/8$-conjecture in $4$-dimensional topology.

1. Graduate School of Mathematics
   Nagoya University
   Nagoya, 464-8602, Japan
   
2. Department of Mathematics
   Hokkaido University
   Sapporo, 060-0810, Japan