The Milnor number and the singularity genus are important invariants of
(complex, smoothable) singularities. The Milnor number measures the
defect of the topological Euler characteristic. The singularity genus
measures the defect of the holomorphic Euler characteristic. The two invariants are related in a complicated way.
In 1978 A.Durfee stated a conjectural bound on the singularity genus of
smoothable surface singularities in terms of the Milnor number.
Despite many works the conjecture has been verified in some specific cases
only. Even for Newton-non-degenerate singularities of surfaces in C^3
the conjecture is a highly non-trivial combinatorial statement on
the number of integral points inside an integral polytope. And remains open.
We prove a much stronger bound for "high enough" Newton-non-degenerate
singularities of hypersurfaces. Similarly, the stronger bound is proved
for absolutely isolated hypersurface singularities. On the other hand,
the conjecture fails totally for isolated locally complete
intersections (ICIS). For a big class of ICIS we prove a weaker bound.
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