Local inequalities for $cD$ and $cE$ singularities

Abstract

We obtain inequalities for isolated $c{D}_n,c{E}_6,c{E}_7$ and $c{E}_8$ singularities, analogues of the $4{\mu }^2/(n+1)$-inequality for isolated $c{A}_n$ singularity in \cite{KOP}. These inequalities are sharp. This enables us to prove the birational rigidity of certain families of Fano $3$-fold weighted hypersurfaces, which contain only terminal quotient singularities and one $c{D}_n$ singularity.

22 pages, comments welcome