Equality cases for matrix spaces with bounded-rank commutators

Let \(0 \leq k < n\), and let \(V \subseteq M_n(\mathbb{C})\) be a complex linear subspace satisfying\(\operatorname{rank}[S,T] \leq k\) for all \(S,T \in V\).Omladič, Radjavi, and Šivic proved the sharp bound\[\dim V \leq nk + \left\lfloor \frac{(n-k)^2}{4} \right\rfloor + 1\]and conjectured a classification of the equality cases. We prove their conjecture.If equality holds, then, after a similarity and possibly transposition, \(V\) consists of all block upper-triangular matrices with arbitrary upper-left and upper-right blocks and with lower-right block in a maximum-dimensional commuting subspace of \(M_{n-k}(\mathbb{C})\).For \(n-k \geq 4\), these commuting subspaces are the classical equality cases in Schur's theorem; in dimensions \(2\) and \(3\), the additional equality cases also occur. At the equality dimension, the rank condition defines a projective algebraic subset of a Grassmannian.For \(2 \leq k \leq n-2\), we determine all of its irreducible components.If \(n-k \geq 4\), there are exactly two components when \(n-k\) is even and exactly four when \(n-k\) is odd; if \(n-k \in \{2,3\}\), there are exactly two.When \(n-k \geq 4\), the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit.When \(n-k \in \{2,3\}\), the two components are obtained by varying the invariant \(k\)-dimensional subspace and the maximum-dimensional commuting subspace on the quotient, together with their transposes.