Complex projective space has fixed-point property iff it has even complex dimension

Statement
Suppose $$n$$ is a natural number. Consider the fact about::real projective space $$\mathbb{P}^n(\mathbb{C})$$ (also denoted $$\mathbb{C}\mathbb{P}^n$$) of complex dimension $$n$$ and real dimension $$2n$$. Note that this can be interpreted as the set of nonzero vectors, up to scalar multiplication equivalence, in $$\mathbb{C}^{n+1}$$, and its elements can be written in the form $$[a_1:a_2:\dots:a_n:a_{n+1}]$$ with all $$a_j \in \mathbb{C}$$ and not all of them simultaneously zero, where:

$$[a_1:a_2:\dots:a_n:a_{n+1}] = [b_1:b_2:\dots:b_n:b_{n+1}] \iff a_jb_k = a_kb_j \ \forall \ j,k \in \{ 1,2,3,\dots,n+1 \}$$

The claim is the following:


 * If $$n$$ is odd (i.e., $$n = 1,3,5,\dots$$, so the real dimension is $$2,6,10,\dots$$), then $$\mathbb{P}^n(\mathbb{C})$$ does not have the fact about::fixed-point property, i.e., we can find a continuous map $$f: \mathbb{P}^n(\mathbb{C}) \to \mathbb{P}^n(\mathbb{C})$$ such that $$f$$ does not have any fixed point. In fact, we can choose the continuous map to be a self-homeomorphism, and to be algebraic over the reals though not over the complex numbers.
 * If $$n$$ is even (i.e., $$n = 2,4,6,\dots$$, so the real dimension is $$4,8,12,\dots$$), then $$\mathbb{P}^n(\mathbb{C})$$ has the fixed-point property, i.e., for any continuous map from $$\mathbb{P}^n(\mathbb{C})$$ to itself, there is a fixed point.

Similar facts that distinguish complex projective spaces of even and odd dimension

 * Complex projective space has orientation-reversing self-homeomorphism iff it has odd complex dimension: Note that all complex projective spaces are orientable; however, the ones with odd complex dimension additionally have the property that they have an orientation-reversing self-homeomorphism. This makes connected sums involving them well defined.

Similar facts about real projective space

 * Real projective space has fixed-point property iff it has even dimension

Facts used

 * 1) uses::Cohomology of complex projective space
 * 2) uses::Lefschetz fixed-point theorem
 * 3) Complex projective spaces are compact polyhedra

Proof for even dimensions
Given: Complex projective space $$\mathbb{P}^n(\mathbb{C})$$ with $$n$$ even. A continuous map $$f: \mathbb{P}^n(\mathbb{C}) \to \mathbb{P}^n(\mathbb{C})$$.

To prove: $$f$$ has a fixed point.

Proof:

Proof for odd dimensions
In odd dimension, we can explicitly construct a real algebraic (but not complex algebraic) self-map $$f:\mathbb{P}^n(\mathbb{C}) \to \mathbb{P}^n(\mathbb{C})$$ that works:

$$[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]$$

Note that in order to make sense of the expression, we need $$n + 1$$ to be even and hence $$n$$ to be odd.