**Edit:** According to the comment of L. Spice I changed the inclusion sign to the subset sign.


Is there a continuous map $f:\mathbb{C}P^3 \to \textrm{Gr}\_{\mathbb{C}}(2,4)$ with $x\subset f(x)$? What about a map $g$ in the opposite direction with $g(x)\subset x$? What about a holomorphic version ($f$ or $g$ holomorphic)? What about a generalization about such maps between arbitrary grassmannian spaces?



Maps between grassmannians with inclusion property
I think there is no *holomorphic* such map. Consider the incidence variety $Z=\{(p,\ell)\in \mathbb{P}^3\times \mathbb{G}(2,4)\,|\, x\in\ell\} $. The projection $p:Z\rightarrow \mathbb{P}^{3}$ is a $\mathbb{P}^2$-bundle, in fact it is the projective tangent bundle to $\mathbb{P}^3$. You are asking for a section of this bundle; that would give a line bundle $M$ on $\mathbb{P}^3$ which is a subbundle of the tangent bundle $T\_{\mathbb{P}^3}$. Computing $c\_3$ one sees that this line bundle must be $\mathcal{O}\_{\mathbb{P}^3}(2)$; but $H^0(T\_{\mathbb{P}^3}(-2))$ is zero, so $\mathcal{O}\_{\mathbb{P}^3}(2)$ does not inject into $T\_{\mathbb{P}^3}$.


I do not know if there exists a continuous section (contrary to what I wrote before editing).


