Abstract: Singular varieties occur everywhere. Two natural ways to study them are to deform (to smoothen) or to resolve the singularities. It is natural to compare the invariants of the singular variety and of its smooth fellows. This defines various invariants of singular varieties.

The fundamental topological invariant is the Milnor number (the defect of the topological Euler characteristic). The fundamental geometric invariant is the singularity genus (the defect of the holomorphic Euler characteristic). An old conjecture, [Durfee 1978], bounds the singularity genus in terms of the Milnor number.

I will start from the general introduction to non-smooth varieties and then proceed to the recent advances in the study of these invariants. Jointly with A.Nemethi, we constructed series of counterexamples, violating the conjecture, even asymptotically. Further, we succeeded to formulate a corrected version, which is asymptotically sharp, and proved this version for a big class of singularities.

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