If $R:=\mathbb{R}[x_1,..,x_{n+1}]/(x_1^2+\cdots+x_{n+1}^2-1)$ and $S^n:=Spec(R)$ 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 $J^2=-Id$. 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 $k:=\mathbb{R}$ and $K:=\mathbb{C}$, it follows there is an isomorphism
$$Spec(K\otimes R)\cong Spec(B):=S^2_K$$
with
$$B:=K[u,v,w]/(uv-(w^2+1)).$$
If $J_K$ is the pull-back of $J$ to $S^2_K$ it follows
$$\phi:=\frac{1}{2}(I+iJ) \in End(T_{S^2_K})$$
is an idempotent: $\phi^2=\phi$ and you get a direct sum
$$T_{S^2_K} \cong L_1\oplus L_2$$
with $L_i \in Pic(S^2_K)$. If $J$ is algebraic it follows $L_i$ are algebraic, and Im interested in this decomposition.
In the case of the real 6-sphere $S^6$ it follows the endomorphism bundle $End(T_{S^6})$ is a real algebraic vector bundle of rank $36$. We may consider the subvariety
$$I(S^6):=\{J \in End(T_{S^6}): J^2=-Id\}$$
and the group scheme $G:=GL(T_{S^6})$. 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 $R:=C^{\infty}(S^6)$ and the $R$-module $T^{\infty}_{S^6}$ of smooth sections of $T_{S^6}$, and the projective $R$-module $End_R(T^{\infty}_{S^6})$. We may consider a smooth almost complex structure $J\in End_R(T^{\infty}_{S^6})$ - an endomorphism with $J^2=-Id$.