Two homeomorphic non-diffeomorphic complex manifolds Does there exist a closed topological manifold supporting two non-diffeomorphic smooth structures both of which admit a compatible complex structure? Also the same question, but for symplectic structure.
 A: Yes there are.   There exists infinitely many  proper elliptic surfaces  that are homeomorphic and no two are diffeomorphic.  Each of them carries a Kähler structure. For details see Corollary 3.3.23 of this book.
A: Google tells me that there are these Horikawa surfaces (I don't know what they are) which posses infinitely many differentiable structures. The fact that the author says the Horikawa surface makes me think any such complex surfaces $X,Y$ have the same underlying $X_{top}=Y_{top}$, but the underlying $X_{diff}$, $Y_{diff}$ are possibly non-diffeomorphic.
It also happens for K3 surfaces by classical work by Donaldson: on the (topological) K3 surface there are non-diffeomorphic differentiable structures. I guess they each underlie (a fortiori non biholomorphic) complex structures... [Edit: no, they don't: see comment by misha below. So this paragraph is not an answer to the question]
I haven't read any of the articles I've cited though. Also I don't know the overlaps with the answer by Liviu Nicolaescu above.
