# Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces

Let $G_n(V)$ the Grassmannian of $n$-dimensional subspaces of a finite-dimensional $V$, and $l<n$. I've noticed that it is easy to associate a (possibly singular) submanifold $\tilde{M}\subseteq G_n(V)$ to a manifold $M\subseteq G_l(V)$ by $$\tilde{M}:=\{W\in G_n(V)\mid W \textrm{ contains some L}\in M\}.$$ Preliminary question 1: does the procedure $M\mapsto \tilde{M}$ (or something similar to it) have a name?

Let now $D\leq V$ be a subspace, and call $M_D$ the Grassmannian $G_l(D)$, understood as a submanifold of $G_l(V)$: then, if I'm not mistaken, $\widetilde{M_D}=\{W\in G_n(V)\mid \textrm{dim} W\cap D\geq l\}$. Hence, any Schubert cell can be written as the intersection of certain $\widetilde{M_D}$'s: this observation convinced me that the submanifolds of the form $\widetilde{M}$ must be known to the experts in Algrebraic Geometry, whose counsel I'm seeking.

Preliminary question 2: is the submanifold $\widetilde{M_D}$ (and/or the equation defining it) a well-known one?

As my usage of the term "manifold" clearly shows, I'm no expert in Algebraic Geometry: I came across these "objects" while studying Monge-Ampére equations from a geometric perspective.

Actual Question: Is there any criterion to establish whether a submanifold $E\subseteq G_n(V)$, given either implicitly or parametrically, is of the form $\widetilde{M}$?

An ideal answer would read like this: "compute a prescribed set of invariants of $E$: if they take the right values, then $E=\widetilde{M}$, for some $M$, whose ideal (or parametrization) can be obtained by that of $E$ by algebraic manipulations" - but I really do not expect this much, though I'm sure that some obvious necessary conditions for $E$ to be of the form $\widetilde{M}$ can be easily found, possibly under (even severe) restrictions (i.e. $M$ to be 0-dimensional). If this problem has been considered before, please point a reference for me! Thanks a lot in advance!

Reformulating the problem in view of the feedbacks received so far: it seems that there is a way to check whether $E$ is the "double fibration transform" of some $M$, namely

1. Lift $E$ to $F_{l,n}(V)$;
2. Construct the largest full sub-bundle $\overline{E}$ of $\lambda$ contained into $\nu^{-1}(E)$; [by full I mean that the fibers are the same, but the base may be smaller]
3. Project $\overline{E}$ back to $G_n(V)$: if the result coincides with $E$, then the answer is YES, and $M$ is the base of $\overline{E}$.

Even if what I wrote is correct, I'm not yet satisfied, since I'd like to see it algebraically. I know how to formulate algebraically steps 1. and 3., but, concerning 2., I can only see an "infinitesimal" way to carry it out. More precisely, I can enlarge the ideal of $\nu^{-1}(E)$ by adding the derivatives of its elements w.r.t. the $\lambda$-vertical differential operators: with this larger ideal, I get the submanifold of $\nu^{-1}(E)$ made of the points of tangency to the $\lambda$-fibers (which correspond to the $\alpha$-distribution mentioned by @alvarezpaiva), which is the "infinitesimal counterpart" of the $\overline{E}$ I needed in step 2..

Perhaps, as @Francois Ziegler pointed out, it is more interesting to work with "infinitesimal objects" rather then mere submanifolds... Yet I'm sure that a simple example of characterization of the double fibration transform of a submanifold can be found, so I keep waiting...

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Thanks to @Francois Ziegler's partial answer now I have the keywords I was looking for, plus references! I'm going to go through the (huge) literature on this topic... but I'd be grateful if someone showed me even a simple example of characterization of the image of a double fibration transform (thus answering the Actual Question). – Giovanni Moreno Aug 5 '13 at 6:49
If you need this stuff to deal with Monge-Ampere equations I suggest you take a look at the book Penrose transform by Baston and Eastwood. (Also, some experts on this topic should be at the DGA conference in Brno in few weeks.) – Vít Tuček Aug 6 '13 at 19:33
@robot: I did not know about "Penrose transform" but, after a quick look at the book you've suggested, I realized that it deals with the kind of problems mentioned by Francois Ziegler at the end of his comment; I'll certainly deepen the topic, though I'd still like to see an example concerning submanifolds! (now it is too late to apply for DGA: if by "expert" you mean Lychagin, then I've just spoken to him about my problem; if you mean "Landsberg", I'm going to email him right now... Did you have someboby elese in your mind?) – Giovanni Moreno Aug 7 '13 at 11:18
Mike Eastwood and Colleen Robles are going to be there as well. – Vít Tuček Aug 7 '13 at 16:36

(Too long for a comment:) Re: Preliminary Question 1, it may (or may not...) pay to look up double fibration transform — your double fibration being $$\begin{matrix} & & F_{l,n}(V)\\ &\lambda\swarrow & & \searrow\nu\\ G_l(V) &&&& G_n(V) \end{matrix}$$ where $F_{l,n}(V)$ is the partial flag manifold or incidence relation $\{(L,W)\in G_l(V)\times G_n(V):L\subset W\}$ and $\lambda$, $\nu$ are the obvious projections.

In this setting your $\widetilde M$ is just $\nu(\lambda^{-1}(M))$ and is indeed known as the double fibration transform of $M$ (see e.g. page 4 of this preprint) or also the Tits transform of $M$ (see page 4 of this one, where they compute the transforms of Schubert varieties).

Re: Actual Question, I have no idea... (Rather than just subsets, it is more common to apply this pullback-pushdown process to differential forms, cohomology classes, sheaves, $\mathcal D$-modules, etc. In those settings you will find literature characterizing the range of the operation.)

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You provided me with valuable information indeed! I was aware of the "double fibration" you've depicted above from a question answered by Robert Bryant: mathoverflow.net/questions/96055/…. He also mentioned a canonical distribution on $F_{l,n}(V)$ obtained as the direct sum of the $\lambda$- and $\nu$-vertical ones, which I found very useful for my work, but he would not tell me who discovered it first! Now I'm in trouble because I need to use it and I don't even know how to call it! Do you have any clue? – Giovanni Moreno Aug 5 '13 at 6:59
Sorry, I wouldn't know how to call that distribution... – Francois Ziegler Aug 5 '13 at 12:26

I think that the geometric structure you need are the "almost grassmannian structures" of Akivis or the somewhat more geometric notion of the incidence cone (due, I think, to Plücker and, in greater generality, to Gelfand):

On each tangent space of a Grassmannian sits an algebraic cone defined as follows: take a $k$-plane $\zeta \in G_k(\mathbb{R}^n)$ and consider the canonical identification between the tanget space at $\zeta$ and the space of linear maps between $\zeta$ and the quotient space $\mathbb{R}^n / \zeta$. The incidence cone is the cone formed by those linear transformations whose rank is one. Note that any linear transformation of $\mathbb{R}^n$ induces a diffeomorphism of the Grassmannian that sends this field of cones to itself. On this cone there are two families of isotropic subspaces (tangent subspaces completely contained in the cone), which are usually called $\alpha$-planes and $\beta$-planes, a terminology coming, I think, from twistor theory. The integral manifolds of this field of planes (or generalized conformal structure in the terminology of A. Goncharov) are the manifolds of all $k$-planes containing a common $(k-1)$-plane and the manifolds of all $k$-planes contained in a $(k+1)$-plane.

I haven't studied closely the differential invariants of these structures, but seeing that $\alpha$-surfaces are pretty much what you are asking about, I would guess that this is the way to go. This should be done somewhere under the heading of "parabolic geometries" or something of the sort.

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Is this "incidence cone" precisely the Segre variety $\mathbb{P}(\zeta^*)\times\mathbb{P}(\mathbb{R}^n/\zeta)$ in $\mathbb{P}(T_\zeta G_k(\mathbb{R}^n))$? If yes, then these $\alpha$-planes and $\beta$-planes should be the fibers of the cartesian product $\mathbb{P}(\zeta^*)\times\mathbb{P}(\mathbb{R}^n/\zeta)$. I have thought about this approach, but on the flag variety $F_{k-1,k}(\mathbb{R}^n)$, where the $\alpha$-planes correspond to an actual distribution. Are you sure that the $\alpha$-surfaces are made of $k$-planes containing a common line? I would say a common $k-1$-plane... – Giovanni Moreno Aug 5 '13 at 19:39
... in which case these $\alpha$-surfaces are precisely the "double fibration transforms" (see Francois' answer) $\nu(\lambda^{-1}(\{H\}))$ of a singleton in $G_{k-1}(\mathbb{R}^n)$. I agree: tangency to the $\alpha$-distribution can be used to detect a double filtration transfom of a 0-dimensional manifold (I had figured out this by myself), but what about the next cases? – Giovanni Moreno Aug 5 '13 at 19:59
My mistake: it is indeed a common $k-1$-plane. – alvarezpaiva Aug 6 '13 at 6:18

An early appearence of double fibrations is:

• MR1026984: Goncharov, A. B.: Integral geometry on families of k-dimensional submanifolds. (Russian) Funktsional. Anal. i Prilozhen. 23 (1989), no. 3, 11--23, 96; translation in Funct. Anal. Appl. 23 (1989), no. 3, 178–189 (1990)

Review: In order to define a double fibration ...

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I assume you are in the case where all the submanifolds involved are algebraic (zeros of polynomials in each affine chart). Then it's algorithmically decidable if $E\subset G_n(V)$ is of the form $\tilde M$.

Your steps 1. and 3. are standard procedures you can do for example with Groebner bases.

Step 2. can be done with what you called the "infinitesimal" version. In fact, the construction of adding all possible derivatives wrt to the $\lambda$-vertical derivations constructs the ideal of what you called $\overline{E}$. The reason is that if $p\in \textrm{F}_{l,n}$ is a zero of the lager ideal and $f$ is any polynomial function in that ideal, when you restrict $f$ to the $\lambda$-fiber containing $p$, then point $p$ will be a zero of infinite order of $f\mid_{\textrm{fiber}}$, hence $f$ vanishes on the whole fiber. (This argument is also valid for analytic varieties).

To turn this into an algorithm start with generators of the ideal of $\nu^{-1}(E)$ and add the $\lambda$-vertical derivatives, and iterate. Because of Noetherianity the process stabilises after a finite number of steps, you could in principle check when it stabilizes using again Groebner bases.

So that answers your actual question if one interprets "criterion" as algorithm.

Note also that the problem is not purely local, so it cannot be solved by computing differential invariants alone. (I'm not sure if that is what you meant by "computing invariants").

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