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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 codim>2 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";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...)

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.

  1. 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.

  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"; I guess part of the question is what would be a good notion of singular symplectic variety so that such a thing would hold...)

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...)

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Nati
Nati
  • 2k
  • 1
  • 10
  • 23

H-principle for smoothing

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.

  1. 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.

  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"; I guess part of the question is what would be a good notion of singular symplectic variety so that such a thing would hold...)