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 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 was wondering if there is some known h-principle which controls whether or not a symplectic embedding of codim>2 of a singular algebraic curve (smooth except a finite number of ADE-singularities) can become the central fiber of a symplectic fibration with smooth total space, with generically smooth fiber.
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"; I guess part of the question is what would be a good notion of singular symplectic variety so that such a thing would hold...)