# Is the complex structure of $\mathbb CP^n$ unique?

Let $$\mathbb CP^n$$ denotes the complex projective space of dimension $$n$$, we have a standard complex structure of $$\mathbb CP^n$$, and my question is: is this complex structure unique?

Or equivalently, let $$X$$ be a complex manifold diffeomorphic to $$\mathbb CP^n$$, is $$X$$ biholomorphic to $$\mathbb CP^n$$?

What I know is from p45 of Morrow&Kodaira's book 《complex manifolds》:

$$\mathbb CP^n$$ is rigid.

But this fact only ensures that small deformations don't change the complex structure of $$\mathbb CP^n$$, we did not even know whether the large deformations change the complex structure of $$\mathbb CP^n$$, or more generally, whether the same diffeomorphic type of $$\mathbb CP^n$$ admits different complex structures?

For dimension 1, I have learnt from some book that the answer is yes.
For dimension 2, cited form Yau's 1977 paper 《Calabi's conjecture and some new results in algebraic geometry》, as a corollary of Yau's solution of Calabi's conjecture, the complex structure of $$\mathbb CP^2$$ is unique.

But for higher dimensions, is this problem solved? or any progress has been made?

• This is a well-known open problem, without any real progress. At the current moment no one knows if there is an exotic complex structure on $\mathbb CP^n$ for $n>2$. The question is even more famous for $\mathbb CP^3$, because it would have an exotic complex structure, would $S^6$ have (just blow up a point). And the latter is again not known: mathoverflow.net/questions/1973/…. Jan 28, 2021 at 14:32
• Note however that a Kähler manifold homeomorphic to $\mathbb{P}^n$ is isomorphic to $\mathbb{P}^n$ — this follows from Yau's theorem, plus some preious work of Kobayashi-Ochiai.
– abx
Jan 28, 2021 at 14:37
• Precisely: if a complex manifold $M$ is diffeomorphic to $\mathbb{S}^6$, the blow up of a point in $M$ is diffeomorphic to $\mathbb{CP}^3$.
– abx
Jan 28, 2021 at 15:39
• @abx, is there a reference say something detailed about "a Kähler manifold homeomorphic to $\mathbb P^n$ is isomorphic to $\mathbb P^n$"?
– Tom
Jan 28, 2021 at 16:01
• Since you mention large deformations of $\mathbb{P}^n$, these were shown to be isomorphic to $\mathbb{P}^n$ by Siu (Crelle 89, erratum Crelle 92) Jan 28, 2021 at 18:47

Let me write this too long comment as an answer.

As abx says, what we do know is

Theorem 1. If a Kähler manifold $$X$$ is homeomorphic to $$\mathbb{CP}^n$$, then $$X$$ is biholomorphic to it.

This is due to Hirzebruch and Kodaira for $$n$$ odd (but with the strongest assumption for $$X$$ to be diffeomorphic to $$\mathbb{CP}^n$$, then relaxed to homeomorphic after work of Novikov), and to Yau for $$n$$ even.

For $$n=2$$, a stronger result holds, still proved by Yau, namely

Theorem 2. If a compact complex surface $$S$$ is homotopy equivalent to $$\mathbb{CP}^2$$, then it is biholomorphic to it.

In dimension $$n\le 6$$, we have instead a result due to Libgober-Wood which is stronger than Theorem 1, but weaker than Theorem 2, that states

Theorem 3. A compact Kähler manifold of complex dimension $$n\le 6$$ which is homotopy equivalent to $$\mathbb{CP}^n$$ must be biholomorphic to it.

You can find all this (and much more) on the very beautiful survey by V. Tosatti available here.

• Thanks, it's a perfect answer.
– Tom
Jan 28, 2021 at 16:16
• You are welcome! Glad it was useful! Jan 28, 2021 at 16:44

I would also like to mention an interesting related result of T. Fujita (which is not cited in the referenced survey article). "On topological characterizations of complex projective spaces and affine linear spaces", Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 5, 231–234.

Theorem: Let X be a smooth Fano n-fold with cohomology ring isomorphic to $$H^{*}(\mathbb{CP}^n,\mathbb{Z})$$ and $$n \leq 5$$. Then $$X \cong \mathbb{CP}^n$$.