On the number of irreducible components of commuting varieties

We will show that for any n and h,k {0,…,n}, h≤k the variety of all the pairs (A,B) of n×n matrices over an algebraically closed field K such that [A,B]=0, rank A≤k, rank B≤h has min{h,n−k}+1 irreducible components. Similarly, the corresponding variety of symmetric matrices is reducible if h,k {1,…,n−1} (while it is irreducible if h is 0 and if char K≠2 and k is n); if char K≠2 and h,k are even the corresponding variety of antisymmetric matrices is reducible if h,k {2,…,n−1} (while it is irreducible if h is 0 and if char K=0 and k is n or n−1).