Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?

Question

Let $$ M:=\{(P,\pi)\mid P\not\in\pi\}\subset\mathbb{P}^n\times\mathbb{P}^{n\,\ast} $$ be the open and dense (and as such $2n$-dimensional) subset of non-incident point-hyperplane pairs. If $P=\mathbb{P}(\ell)$ and $\pi=\mathbb{P}(W)$, with $\ell$ and $W$ linear subspaces of $V:=\mathbb{K}^{n+1}$ of dimension $1$ and $n$, respectively, then the condition $P\not\in\pi$ is equivalent to $\ell\cap W=0$, i.e., $M$ is the complement of the flag manifold $\mathrm{Fl(1,n;V)}$.

According, there is a chain of obvious and canonical identifications: $$ T_{(P,\pi)}M=T_P \mathbb{P}^n \oplus T_\pi \mathbb{P}^{n\,\ast}=\left(\ell^\ast\otimes\frac{V}{\ell}\right)\oplus\left(\pi^*\otimes\frac{V}{\pi} \right)=(\ell^*\otimes\pi)\oplus(\pi^*\otimes\ell)=(\ell^*\otimes\pi)\oplus(\ell^*\otimes\pi)^*\, , $$ showing that $M$ is equipped with a canonical metric of split signature (insofar as each linear space of the form $Z\oplus Z^*$ carries a scalar product of split signature).

QUESTION 1: what is the origin of such a metric? does it reflect some elementary properties of points and hyperplanes in projective spaces? what is the context where it plays the most relevant role? how it depends on the ground field $\mathbb{K}$? (e.g., when $\mathbb{K}=\mathbb{C}$ is there some relationship with the Fubini-Study metric?) does it generalizes to non-incident pairs in $\mathbb{G}(k,n)\times\mathbb{G}(n-k-1,n)$?

Motivations: I have found this metric in a beautiful preprint, where it goes under the name of "dancing metric" and it is defined in an unnecessarily (at least for me) complicated way, and only for $n=2$ (the above definition - I hope it is correct - I figured out by myself). I confessed to the author that such a structure is so elementary that it should have already appeared and studied long ago, but he said that he just know that it is "somehow related to the para-Fubini-Study metric". Also searching this forum, I could not find any clue, in spite of the many posts concerning metrics on Grassmannians.

QUESTION 2: can I extend this metric to the whole of $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$ and characterize $\mathrm{Fl(1,n;V)}$ as its degeneracy locus? if yes, would this idea work also for $k>1$?

Answer 1

Regarding Question 1: Of course, this generalizes to the space $\mathrm{S}_{pq}(V)$ of all splittings of a vector space $V$ into subspaces $V = P\oplus Q$ where $\dim P = p>0$ and $\dim Q = q>0$ and, of course, $\dim V = p+q$.

The tangent space at $(P,Q)\in \mathrm{S}_{pq}(V)$ is canonically isomorphic to $(P{\otimes}Q^*)\oplus (Q{\otimes}P^*)$, for the same reasons that you list in the case where $p$ or $q$ is $1$, so the same reasoning applies.

When the ground field is $\mathbb{R}$, this is an irreducible pseudo-Riemannian symmetric space (of split type), and, as such, appears in Berger's classification of pseudo-Riemannian symmetric spaces as $\mathrm{SL}(V)/\mathrm{S}(\mathrm{GL}(P){\times}\mathrm{GL}(Q))$.

When the ground field is $\mathbb{C}$, this space can be regarded naturally as a complexification of the space $\mathrm{Gr}_p(V)$ of $p$-planes in $V$. This canonical metric then turns out to be, in an appropriate sense, the holomorphic extension of the Fubini-Study metric on $\mathrm{Gr}_p(V)$ (defined relative to any positive definite Hermitian inner product on $V$) to this complexification.

Regarding Question 2: The metric (as a quadratic form) does not extend continuously to the product $\mathrm{Gr}_p(V)\times\mathrm{Gr}_q(V)$. The reason is that the volume form of the canonical metric on $\mathrm{S}_{pq}(V)$ gives $\mathrm{S}_{pq}(V)$ infinite volume, which it could not do if the quadratic form extended continuously across the incidence hypersurface. (Just look at the case $p=q=1$ to convince yourself of this.)