Sylvester–Gallai for Arrangements of Subspaces

In this work we study arrangements of k-dimensional subspaces V₁, … , V_n⊂ C^ℓ. Our main result shows that, if every pair V_a, V_b of subspaces is contained in a dependent triple (a triple V_a, V_b, V_c contained in a 2k-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on k (and not on n). The theorem holds under the assumption that V_a∩ V_b= { 0 } for every pair (otherwise it is false). This generalizes the Sylvester–Gallai theorem (or Kelly’s theorem for complex numbers), which proves the k= 1 case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. (Proc Natl Acad Sci USA 110(48):19213–19219, 2013). One of the main ingredients in the proof is a strengthening of a theorem of Barthe (Invent Math 134(2):335–361, 1998) (from the k= 1 to k> 1 case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem).