Sylvester-Gallai for Arrangements of Subspaces

In this work we study arrangements of κ-dimensional subspaces V₁, . . . , V_n ⊂ Cl. 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 2κ-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. [1]. One of the main ingredients in the proof is a strengthening of a theorem of Barthe [3] (from the k = 1 to κ>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).