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Let $M$ be a complex projective manifold with an antiholomorphic involution. Can $M$ be defined by equations with real coefficients then?

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Corrected. As Robert Bryant points out, it is not enough to show that the manifold can be realised as a submanifold of $\mathbb CP^n$ invariant under some anti-holomorphic involution of $\mathbb CP^n$. For this reason the answer is extended.


Let $\sigma: M\to M$ be the anti-holomorphic involution. Take a very ample divisor $D$ on $M$, and consider the linear system $|D+\sigma(D)|=\mathbb P_{\mathbb C}^n$. It is invariant under $\sigma$ and $\sigma$ induces on it an anti-holomorphic involution. Now, we have the embedding $M\to \mathbb P_{\mathbb C}^{n *}$ that is invariant under the induced action of $\sigma$.


However, being invariant under an anti-holomorphic involution of $\mathbb CP^n$ doesn't mean being defined by equations with real coefficients (as Robert points out).

So to answer the question, we need to solve additionally the following exercise:

Exercise. Suppose we have $\mathbb CP^n$ with an anti-holomorphic involution $\sigma$. Find an embedding $\mathbb CP^n\to \mathbb CP^N$, such that the image is invariant under the real involution of $\mathbb CP^N$ and the real involution induces $\sigma$ on $\mathbb CP^n$.

Solution. Let us show first how to do this for $\mathbb CP^1$ with the antipodal involution (the example proposed by Robert). Here we can embed it in $\mathbb CP^2$ with equation $x^2+y^2+z^2=0$.

As for general case we note the following. On an even-dimensional space each anti-holomoprhic involution is conjugate to the usual complex conjugation $(z_1:\ldots :z_{2n+1})\to (\bar z_1:\ldots :\bar z_{2n+1})$. So we don't need to do anything if $n=2m$. On the other hand, on odd dimensional projective spaces there is one more type of anti-holomorphic involutions, namely $$(z_1:\ldots: z_m:z_{m+1}:\ldots: z_{2m}\to (-\bar z_{m+1}:\ldots: -\bar z_{2m}: -\bar z_1:\ldots: -\bar z_m ).$$

So we just need to take some Segre embedding of $\mathbb CP^{2m-1}$ into an even dimensional projective space. For this we need to find $k$ such that ${2m+k-1}\choose{k}$ $-1$ is even. This can always be done.

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    $\begingroup$ Actually, it's not true that a subvariety of $\mathbb{CP}^n$ that is invariant under an anti-holomorphic involution $\sigma:\mathbb{CP}^n\to\mathbb{CP}^n$ can necessarily be defined by real equations. For example, when $n=1$, and $\sigma([x_0,x_1]) = [-\overline{x_1},\overline{x_0}]$, one can find a set $R$ of $6$ distinct points in $\mathbb{CP}^1$ invariant under $\sigma$ for which there is no meromorphic $z$ with a single pole on $\mathbb{CP}^1$ so that the 6 points are the roots of a polynomial in $z$ with real coefficients. For higher $n$ (odd), there are irreducible examples. $\endgroup$ Oct 2, 2020 at 15:20
  • $\begingroup$ Thanks Robert, when I was originally thinking about exactly the same case (of 6 points on $\mathbb CP^1$) I imagined the vertices of an Octahedron for which there is a real involution... I'll correct the answer $\endgroup$ Oct 2, 2020 at 16:34
  • $\begingroup$ A similar result holds for a connected reductive $G$ group over $\Bbb C$ with an anti-holomorphic involution $\sigma$ preserving the group structure. Any such group $G$ con be embedded into $GL(n,\Bbb C)$ such that $\sigma$ is induced by the complex conjugation in $GL(n,\Bbb C)$. See this question. $\endgroup$ Oct 3, 2020 at 14:27
  • $\begingroup$ Moreover, the connectedness hypothesis is not necessary here. See Adams, Taïbi, Galois and Cartan cohomology of real groups, Duke Math. J. 167 (2018), no. 6, 1057–1097, Lemma 3.1. $\endgroup$ Oct 3, 2020 at 14:35

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