Put another way, is the complex structure on $\mathbb C\mathbb P^n$ unique? I know that this is the case for $n\in\{1,2\}$, so I'm curious as to whether it's the case in general. If it's not known, is it believed to be true, and what progress has been made on this question?
Asked
Active
Viewed 98 times
2
-
1https://arxiv.org/abs/1510.02636 : This might help. – Partha Apr 05 '19 at 04:16
-
Actually I have not read the paper, both the writers are my professors, so I happen to know about their work. You should contact them for detailed answers. – Partha Apr 05 '19 at 05:15
1 Answers
2
The following theorem is due to Hirzebruch, Kodaira, and Yau:
Theorem: Let $M$ be a compact Kähler manifold homeomorphic to $\mathbb{CP}^n$. Then $M$ is biholomorphic to $\mathbb{CP}^n$.
Moreover, Libgober and Wood showed that the result remains true for $n \leq 6$ if $M$ is only assumed to be homotopy equivalent to $\mathbb{CP}^n$.
This still leaves open the possibility that there is a compact complex manifold $M$ homeomorphic to $\mathbb{CP}^n$ which is not biholomorphic to it, but such an $M$ is necessarily non-Kähler. For $n = 1$ and $2$, this is known not to occur, but is open for $n \geq 3$. In particular, if $S^6$ admits a complex structure, then there is a compact complex manifold diffeomorphic to $\mathbb{CP}^3$ which is not biholomorphic to it.
For details on all of these results, see the excellent survey Uniqueness of $\mathbb{CP}^n$ by Valentino Tosatti.
Michael Albanese
- 99,526