"**Motto**" Moduli space of Calabi-Yau varieties can be connected by using [Symplectic surgery theory][1].

***[Miles Reid’s Fantasy][2]:“There is only one Calabi-Yau space”*** 
i.e "All CY connected through conifold transitions $S^3→S^2$"

Such surgery are very important for study of Hermitian–Yang–Mills metrics under
conifold transitions

A conifold transition is a surgery on a (real) six-dimensional manifold $X$
which replaces a three-sphere with trivial normal bundle by a two-sphere with
trivial normal bundle, cutting out $S^3 × D^3$ and replacing it with $D^4 × S^2$.






 [Miles Reid][3] conjectured that the moduli
space of Calabi-Yau spaces $\mathscr M_{CY}$ is connected after allowing the conifold transitions
, i.e., both resolutions/contractions and deformations/degeneration.


> **Question** :Is there any progress about this conjecture (what about in
> higher dimension), A reference or information is welcomed . See [some
> very partial result][4]


  [1]: https://spiral.imperial.ac.uk/bitstream/10044/1/231/1/Symplectic%20conifold%20transitions.pdf
  [2]: http://link.springer.com/article/10.1007%2FBF01458074
  [3]: https://arxiv.org/pdf/0904.4487.pdf
  [4]: http://link.springer.com/article/10.1007%2FBF01218081