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

Let \(k\) be an algebraically closed field of characteristic \(2\) and \(A\) the polynomial algebra in \(r\) variables with coefficients in \(k\). G. Carlsson conjectured that for any \(DG\)-\(A\)-module \(M\) of dimension \(N\) that is free as an \(A\)-module, if the homology of \(M\) is nontrivial and finite dimensional as a \(k\)-vector space, then \(2^r\leq N\).

In this talk, we discuss a stronger conjecture about varieties of square-zero upper-triangular \(N\times N\) matrices with entries in \(A\). Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when \(N < 8\) or \(r < 3\). This is joint work with Özgün Ünlü.