Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then 2r ≤ N. Here we state a stronger conjecture about varieties of square-zero upper triangular N × N matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when N < 8 or r < 3 without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson’s conjecture and give new results when N > 4 and r = 2.