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Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\dots:z_n]\in\mathbb{C}P^n$, where $\overline{z}$ denotes the complex conjugation of $z$, then we obtain a quotient space $\overline{\mathbb{C}P^n}:=\mathbb{C}P^n/[z_0:\dots:z_n]\sim[\overline{z_0}:\dots:\overline{z_n}]$.

Now I am interested in these $\overline{\mathbb{C}P^n}$'s, but I know neither if these $\overline{\mathbb{C}P^n}$'s have a formal name nor where I can find their properties.

I'd appretiate if you can provide some relavant references.

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3 Answers 3

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This answer gives information about the cohomology of $\overline{\mathbb CP^n}$. Perhaps someone will recognize this as the cohomology of a familiar space.

The conjugation is an action of $\Sigma_2$ on $\mathbb CP^n$. We are interested in the orbit space of this action. Recall that the fixed point of this action is homeomorphic to $\mathbb RP^\infty$.

The quotient space $\mathbb CP^n/\mathbb RP^n$ has a free action of $\Sigma_2$ (away from the basepoint). The fact that for free action strict quotient is equivalent to homotopy quotient implies that there is a homotopy pushout square $$ \begin{array}{ccc} \mathbb RP^n \times \mathbb RP^\infty & \to & \mathbb RP^n \\ \downarrow & & \downarrow \\ \mathbb CP^n \times_{\Sigma_2} E\Sigma_2 & \to & \overline{\mathbb CP^n} \end{array} $$

It is perhaps instructive to consider what happens when $n=\infty$. One can realize the limit diagram as a diagram of classifying spaces:

$$ \begin{array}{ccc} B\Sigma_2\times B\Sigma_2 & \to & B\Sigma_2 \\ \downarrow & & \downarrow \\ BO(2) & \to & \overline{\mathbb CP^\infty} \end{array} $$

From this square one can, in principle, calculate the cohomology of $\overline{\mathbb CP^n}$ using the Meier-Vietoris exact sequence. If one localizes away from the prime $2$, the answer is pretty simple. In this case $\mathbb RP^\infty\simeq *$, and we obtain an equivalence away from the prime $2$: $$ \mathbb CP^n \times_{\Sigma_2} E\Sigma_2 \xrightarrow{\simeq} \overline{\mathbb CP^n}. $$ If $\Lambda$ is a ring where $2$ is invertible, then $H^*(BO(2);\Lambda)\cong \Lambda[p_1]$, where $p_1$ is a class in dimension $4$ (the first Pontryagin class). The cohomology of $\mathbb CP^n \times_{\Sigma_2} E\Sigma_2$ is isomorphic to the truncation $\Lambda[p_1]/_{(p_1^{\left\lfloor \frac{n}{2}\right\rfloor+1})}$. Notice that for $n=1$ you get the cohomology of a point, and for $n=2$ you get the cohomology of a sphere, as expected.

Cohomology with mod 2 coefficients is more complicated/interesting. In the limit when $n=\infty$ you get the following diagram in cohomology

$$ \begin{array}{ccc} H^*(\overline{\mathbb CP^\infty};\mathbb Z/2) & \to & \mathbb Z/2[w_1, w_2] \\ \downarrow & & \downarrow \\ \mathbb Z/2[x] & \to &\mathbb Z/2[x, y] \end{array} $$ Where the homomorphism on the right side sends $w_1$ to $x+y$ and $w_2$ to $xy$. The Poincare series of $\tilde H^*(\overline{\mathbb CP^\infty};\mathbb Z/2)$ comes out to be $\frac{t^4}{(1-t)(1-t^2)}$.

For finite $n$, we have to take a truncation of this diagram. If I am not mistaken, the diagram in cohomology comes out to be the following

$$ \begin{array}{ccc} H^*(\overline{\mathbb CP^n};\mathbb Z/2) & \to & \mathbb Z/2[w_1, w_2]/_{(w_2^{n+1})} \\ \downarrow & & \downarrow \\ \mathbb Z/2[x]/_{(x^{n+1})} & \to &\mathbb Z/2[x, y]/_{(x^{n+1})} \end{array} $$

According to my calculations, the Poincare series of $\tilde H^*(\overline{\mathbb CP^n};\mathbb Z/2)$ turns out to be $$t^4\frac{(1-t^{n-1})(1-t^n)}{(1-t)(1-t^2)}.$$

More explicitly, this equals to $$t^4(1+t+t^2+ \cdots +t^{n-2})(1+t^2+t^4+\cdots +t^{n-2})$$ if $n$ is even, and $$t^4(1+t^2+t^4+\cdots + t^{n-3})(1+t+t^2+\cdots +t^{n-1})$$ if $n$ is odd. Once again, this seems to give the right answer for $n=1, 2$, so I hope this is a good sign.

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For $n=1$ the quotient space is $D^2$.

For $n=2$ the quotient spaces is $S^4$ , that was proved by Arnold, Kuiper and Massey (searching "Arnold-Kuiper-Massey Theorem" should yield the relevant references).

For higher dimensions note that the quotient space has an open set homeomorphic to $\mathbb{C}^{n}/\tau$ where $\tau$ is conjugation. Note that there will be an $\mathbb{RP}^{n-1}$ cone singularity along the fixed point set. I am not sure if there is somewhere where these orbifolds are studied in detail, maybe try searching papers which cite the ones mentioned above.

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    $\begingroup$ You may probably mean $\mathbb{CP}^1/conj\approx D^2$ the 2-dim disk? Nevertheless, thank you for your help! $\endgroup$
    – GiS
    Oct 16, 2021 at 3:59
  • $\begingroup$ In fact, I noticed that the natural inclusion $\mathbb{R}[X_0,\dots,X_n]\rightarrowtail\mathbb{C}[X_0,\dots,X_n]$ exactly induces the quotient map $mProj\mathbb{C}[X_0,\dots,X_n]\twoheadrightarrow mProj\mathbb{R}[X_0,\dots,X_n]$, which is $\mathbb{CP}^n\twoheadrightarrow \overline{\mathbb{CP}^n}$. Hence I am interested in the shape and properties of $\overline{\mathbb{CP}^n}$. $\endgroup$
    – GiS
    Oct 16, 2021 at 4:11
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    $\begingroup$ Thanks for the correction!, you are right its a disk. What do you mean by shape and properties? it seems like you are interested in these spaces as real algebraic varieties? (in which case I can't help much because I know nothing about real algebraic geometry) $\endgroup$
    – Nick L
    Oct 16, 2021 at 8:01
  • $\begingroup$ Recently, I am trying computing the cellular $\mathbb{Z}$-homology of these $\overline{\mathbb{CP^n}}$ to imagine the shape of them. During the computation, I rediscover $\overline{\mathbb{CP^2}}\approx S^4$. However, it is hard for me to compute some $H_q(\overline{\mathbb{CP^n}}; \mathbb{Z})$ as $n\geq 4$ and $5\leq q \leq 2n-2$ ($n$ even) or $2n-3$ ($n$ odd). By the way, the top 2 or 3-dim of homologies are periodic. $\endgroup$
    – GiS
    Oct 16, 2021 at 9:37
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This doesn't help with $\mathbb{CP}^n$, but see Atiyah and Berndt https://arxiv.org/abs/math/0206135 for a discussion of the quotient space for the other projective planes.

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