# Deformation equivalent vs diffeomorphic to projective manifold

Let $$M$$ be a closed complex manifold that is not deformation equivalent to a complex projective manifold.

Can $$M$$ be orientedly diffeomorphic to a complex projective manifold? What if $$M$$ is moreover Kähler?

I believe the answer is yes and follows from the combination of Theorem 4.6 here https://arxiv.org/pdf/math/0111245.pdf and Theorem 1.3 here https://arxiv.org/pdf/math/0111245.pdf

The first result shows that deformations of standard complex tori are complex tori (i.e. $$\mathbb C^n/\Gamma$$ where $$\Gamma\cong \mathbb Z^{2n}$$). The second result shows that on $$T^6$$ there is an infinite dimensional family of complex structures.

PS. As for the second version of the question, where $$M$$ is asked additionally to be Kahler, I would guess that it can be safely counted as an open problem. Remember in https://link.springer.com/article/10.1007/s00222-003-0352-1 Voisin solved negatively Kodaira problem by constructing the first ever example of a Kahler manifold that is not deformation equivalent to a projective one. It seems to me that since then no new examples were found of such a phenomenon. And as you pointed out in a different post, it was proven recently that in dimension $$3$$ every Kahler manifold can be deformed to a projective one, but this is hard work (strongly relying on dim $$3$$). So in order to advance in your question one has to advance in one of these two directions - trying to extend the $$3$$-dimensional result to dimension $$4$$ and trying to look for new constructions of Kahler manifolds...

Yes.

As everyone knows, flat complex structures on $$\Bbb R^4$$ are parametrised by two copies of $$\Bbb CP^1$$. Take your favourite elliptic curve $$E$$; map $$e \mapsto -e$$ gives two-sheeted ramified covering $$p: E \to \Bbb CP^1$$. Consider a complex structure on $$E \times T^4$$ where complex structure on the fiber over $$e \in E$$ corresponds to the point $$p(e)$$. This almost complex structure is integrable (somewhat easy exercise) and non-Kahler (if it was, then corresponding Kahler metric would be flat and globally pluriclosed, but computation shows that such metric could not be compatible).

There is a paper by Catanese from early 00s where he shows that any deformation of a flat complex 3-dim torus is again a flat torus (and same result for products of curves of higer genus with torus $$\Sigma \times T^4$$). (UPD: it's the first link in Dmitri's answer)

I'm not aware of any Kahler examples.