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?
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$\begingroup$ Does the MO Question mathoverflow.net/questions/98366/… 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. $\endgroup$– Robert BryantCommented Mar 12, 2015 at 20:03

$\begingroup$ 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. $\endgroup$– Philip EngelCommented Mar 12, 2015 at 20:15
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1 Answer
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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), 553603,
in particular Part IV.

1$\begingroup$ Maybe Elkik, rather than Elkies? $\endgroup$ Commented Mar 13, 2015 at 16:56

$\begingroup$ Right, thanks. Corrected (and reference added). $\endgroup$ Commented Mar 13, 2015 at 17:59