On the number of irreducible components of commuting varieties
It is shown that the variety of all the pairs \((A,B)\) of \(n\times n\) matrices over an algebraically closed field \(K\) such that \([A,B]=0\), rank \(A\leq k\), rank \(B\leq h\) has \(\min\{h,n-k\}+1\) irreducible components for any \(n\) and \(h,k\in\{0,...,n\}\), \(h\leq k\). It is also proved that the corresponding variety of symmetric matrices is reducible if \(h,k\in\{1,\dots,n-1\}\) and is irreducible if \(h\) is \(0\) and if char \(K\neq 2\) and \(k\) is \(n\). Similarly it is proved that the corresponding variety of antisymmetric matrices is reducible for all \(h,k\in\{2,\dots,n-1\}\) if \(n\) is even, for all \(h,k\in\{2,\dots,n-2\}\) if \(n\) is odd and is irreducible if \(h=0\), char \(K=0\) and \(k\) is \(n\) or \(n-1\).
scientific article; zbMATH DE number 1474720