I'm trying to find algebraic embedding of singular curves into projective varieties that can be smoothed symplectically but not algebraically.
It's not hard (e.g. using the methods in Hartshorne-Hirschowitz "Smoothing algebraic space curve" or Hartshorne's "Families of Curves in $\mathbb P^{3}$ and Zeuthen’s Problem") to construct such examples for $\mathbb{P}^3$ with locally smoothable singularities (even ADE) which cannot be the central fiber of a flat family of embedded curves with smooth generic fiber and smooth total space (i.e. space curves which are not "strongly smoothable").
1) I was wondering if there is some known h-principle which controls whether or not a symplectic embedding of of a singular algebraic curve (smooth except a finite number of ADE-singularities) into a 3-fold can become the central fiber of a symplectic fibration with smooth total space, with generically smooth fiber.
2) More vaguely , what would be the answer for the general case (smoothing a symplectic embedding $V \hookrightarrow W$ with $V$ being an appropriate notion of ''singular symplectic variety" of codimension $>2$; I guess part of the question is what would be a good notion of singular symplectic variety so that such a thing would hold...)
