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Are there examples of algebraic singularities which may be smoothed analytically but not algebraically? It certainly seems possible, but if not, why? Are there conditions under which this becomes true, e.g. what if the singularity is isolated?

Philip Engel
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  • Does the MO Question http://mathoverflow.net/questions/98366/when-is-a-singular-point-of-a-variety-smooth have any bearing on your question? This appears to be an algebraic singularity that is already smooth analytically, but maybe a careful definition will rule this example out. – Robert Bryant Mar 12 '15 at 20:03
  • Interesting example, but as Francesco's answer shows, this cannot occur over $\mathbb{C}$, which I should have mentioned was the setting of this question. – Philip Engel Mar 12 '15 at 20:15

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For isolated singularities, the answer is no.

In fact, it follows by a result of Elkik that any deformation of an isolated singularity is algebraizable. See

R. Elkik, Solutions d'équations à coefficients dans un anneau hensélien, Ann. Sci. Ecole Norm. Sup. 6 (1973), 553-603,

in particular Part IV.

Francesco Polizzi
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