MathOverflow Asked by user164740 on December 3, 2020

Let $M$ be a closed non-projective Kähler manifold. There are three possibilities

- there is a proper holomorphic submersion $f:Xto Delta$ with $f^{-1}(0)cong M$ such that the projective fibers define a dense set in $Delta$
- there is no proper holomorphic submersion $f:Xto Delta$ with $f^{-1}(0)cong M$ such that the projective fibers define a dense set in $Delta$ but there is a proper holomorphic submersion $f:Xto Delta$ with $f^{-1}(0)cong M$ and at least one projective fiber
- no proper holomorphic submersion $f:Xto Delta$ with $f^{-1}(0)cong M$ has projective fibers.

K3 surfaces satisfy the possibility 1. Voisin has given examples of possibility 3. Does possibility 2 ever arise?

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