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.
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3$\begingroup$ This is a well-known problem, which has been solved by Donaldson in the early 80's. See for instance the book Compact complex surfaces by Barth, Hulek, Peters, Van de Ven, ch. XI, §5: "Infinitely many homeomorphic surfaces which are not diffeomorphic". $\endgroup$– abxCommented Dec 4, 2018 at 11:06
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1$\begingroup$ It would be interesting to have examples of complex dimension greater than two. $\endgroup$– mmeCommented Dec 5, 2018 at 2:48
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2 Answers
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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.
answered Dec 4, 2018 at 11:01
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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.
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1$\begingroup$ I do not think that K3 example is correct. By classification of complex surfaces, we know that a complex surface whose underlying topological manifold is a topological K3 has to be a K3 surface; the underlying smooth manifolds of any two K3 surfaces are diffeomorphic. Therefore, exotic smooth structures on topological K3 can not be compatible with complex structure $\endgroup$– mishaCommented Dec 4, 2018 at 20:14
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$\begingroup$ @misha: I see, thank you for the clarification! $\endgroup$– QfwfqCommented Dec 4, 2018 at 20:16
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$\begingroup$ @misha: oh well, so what about the Hirokawa surfaces? If the exotic ones are non-complex then maybe I'll delete my answer. $\endgroup$– QfwfqCommented Dec 4, 2018 at 20:18
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$\begingroup$ about Horikawa surfaces I am not so sure. $\endgroup$– mishaCommented Dec 4, 2018 at 20:25
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1$\begingroup$ Horikawa surfaces are of general type as far as I can remember. In particular they are not elliptic. $\endgroup$ Commented Dec 4, 2018 at 20:27