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In an answer of this MO question, Frank Loray constructed an example of analytic singularity which is not algebraic. On the other hand, as I learned from one of Joël's comments in that question, Artin showed that locally any analytic isolated singularity is algebraic. This leads to the following question: Which analytic singularities are algebraic? To be more precise, let $x$ be a point in a complex space $X$, when is the germ at $x$ isomorphic to the analytic germ at some point of an algebraic variety?

Clearly finite quotient singularities are algebraic in the above sense. What about singularities that people study in the minimal model program (terminal, canonical, klt, etc.)? The singularity in Frank Loray's example is not normal. How to construct a normal singularity which is not algebraic?

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    $\begingroup$ Let $X,Y\subset \mathbb{P}^2\times \Delta$ be two smooth families of plane cubics over the disk $\Delta$ (over the complex numbers). Let $CX,CY\subset \mathbb{P}^3\times \Delta$ be the projective cones over these families. Consider the fiber product $CX\times_\Delta CY$ with its projection to $\Delta$. If this were algebraic, then the $j$-functions of $X$ and $Y$ as holomorphic functions on $\Delta$ would satisfy an algebraic equation. I believe that elliptic surface singularities are log canonical . . . $\endgroup$
    – Jason Starr
    Commented Aug 3, 2017 at 14:47

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