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Tom
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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&Kodaria'sMorrow&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?

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&Kodaria'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?

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?

Source Link
Tom
Tom
  • 471
  • 4
  • 19

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&Kodaria'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?