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.



Two homeomorphic non-diffeomorphic complex manifolds
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](https://www3.nd.edu/~lnicolae/swnotes.pdf).


