Let $X = X_{\mathbb C}$ be a complex analytical variety. There is a natural functor from complex analytical varities to real analytical varieties $$ (X, \mathcal{O}_X) \mapsto (X_{\mathbb{R}}, \mathcal{C}^{an}_{X_{\mathbb R}}). $$ Aware that $X_{\mathbb C}$ and $X_{\mathbb R}$ are in general different topological spaces, though with the same set of points.
For both $X_{\mathbb C}$ and $X_{\mathbb R}$ one can define their Zariski tangent sheaves $\mathcal{T}X_{\mathbb C}$ and $\mathcal{T}X_{\mathbb R}$.
So my questions are the following:
- Is there any way to reconstruct $\mathcal{T}X_{\mathbb C}$ from $\mathcal{T}X_{\mathbb R}$ and vice versa?
- More concretely, for a point $x \in X$ let $\mathcal{T}_xX_{\mathbb C}$ (resp. $\mathcal{T}_xX_{\mathbb R}$) be the fiber of the Zariski tangent sheaf in $x$. Is there a canonical isomorphism $\mathcal{T}_xX_{\mathbb C} \simeq \mathcal{T}_xX_{\mathbb R} \otimes \mathbb{C}$?
- Finally, is there any way to define the sheaf of $(p,q)$-forms $\Omega^{p,q}(X)$ on $X_{\mathbb R}$ which embedds into the sheaf $\Lambda^{p+q}\mathcal{T}^*X_{\mathbb R}$ and coinsides with the sheaf of sections of the bundle of $(p,q)$-forms when $X$ is smooth?