If $R\mathrel{:=}\mathbb{R}[x_1,\dotsc,x_{n+1}]/(x_1^2+\dotsb+x_{n+1}^2-1)$ and $S^n\mathrel{:=}\operatorname{Spec}(R)$ is the real $n$-spere, a classical result of Borel and Serre says that the only real spheres with an almost complex structure is $S^2$ and $S^6$. An almost complex structure is an endomorphism of the tangent bundle $$J: T_{S^2} \rightarrow T_{S^2}$$ with $J^2=-\operatorname{Id}$. In the case of the real 2-sphere it follows the tangent bundle $T_{S^2}$ is a real algebraic vector bundle of rank 2.

**Question 1:** I'm looking for an example of a real algebraic (even dimensional) manifold $M$ with an almost complex structure
$J:T_M \rightarrow T_M$, where $J$ is not algebraic. The problem of constructing a holomorphic structure on $S^6$ — is this still an open problem?

**Note:** If you let $k\mathrel{:=}\mathbb{R}$ and $K\mathrel{:=}\mathbb{C}$, it follows there is an isomorphism
$$\operatorname{Spec}(K\otimes R)\cong \operatorname{Spec}(B)\mathrel{:=}S^2_K$$
with
$$B\mathrel{:=}K[u,v,w]/(uv-(w^2+1)).$$

If $J_K$ is the pull-back of $J$ to $S^2_K$ it follows that $$\phi\mathrel{:=}\frac{1}{2}(I+iJ) \in \operatorname{End}(T_{S^2_K})$$ is an idempotent: $\phi^2=\phi$ and you get a direct sum $$T_{S^2_K} \cong L_1\oplus L_2$$ with $L_i \in \operatorname{Pic}(S^2_K)$. If $J$ is algebraic it follows $L_i$ are algebraic, and I'm interested in this decomposition.

In the case of the real 6-sphere $S^6$ it follows the endomorphism bundle $\operatorname{End}(T_{S^6})$ is a real algebraic vector bundle of rank $36$. We may consider the subvariety $$I(S^6)\mathrel{:=}\{J \in \operatorname{End}(T_{S^6}): J^2=-\operatorname{Id}\}$$ and the group scheme $G\mathrel{:=}\operatorname{GL}(T_{S^6})$. There is a canonical action $$\sigma: G \times I(S^6) \rightarrow I(S^6)$$ and a "parameter space" $I(S^6)/G$ parametrizing algebraic almost complex structures on $S^6$. Is this construction used in the study of the Hopf problem — the problem of constructing a holomorphic structure on $S^6$? If there is a holomorphic structure on $S^6$ — is this neccessarily algebraic? I ask for references.

**Note:** We may also consider the ring $R\mathrel{:=}C^{\infty}(S^6)$ and the $R$-module $T^{\infty}_{S^6}$ of smooth sections of $T_{S^6}$, and the projective $R$-module $\operatorname{End}_R(T^{\infty}_{S^6})$. We may consider a smooth almost complex structure $J\in \operatorname{End}_R(T^{\infty}_{S^6})\cong \Omega^{1,\infty}_R\otimes_R T^{\infty}_{S^6}$. We get a topological subspace $I^{\infty}(S^6)$ of smooth almost complex structures on $S^6$ equipped with an action of the group of smooth automorphisms of $T^{\infty}_{S^6}$. This is a topological subspace of a smooth vector bundle of rank $36$, and the orbit space parametrize all smooth almost complex structures on $S^6$. Hence if there is a holomorphic structure on $S^6$ it "lives" in this orbit space.

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