On the number of irreducible components of commuting varieties

We will show that for any n and h,k is an element of {0,..., n}, h less tha n or equal to k the variety of all the pairs (A,B) of n x n matrices over a n algebraically closed field K such that [A,B] = 0, rankd A less than or eq ual to k, rank B less than or equal to h has min {h, n - k} + 1 irreducible components. Similarly, the corresponding variety of symmetric matrices is reducible if h, k is an element of {1,..., n - 1} (while it is irreducible if h is 0 and if char K not equal 2 and k is n); if char K not equal 2 and k,li are even the corresponding variety of antisymmetric matrices is reduci ble if h,k is an element of {2,..., n - 1} (while it is irreducible if h is 0 and if char K = 0 and k is n or n - 1). (C) 2000 Elsevier Science B.V. A ll rights reserved. MSC. 15A30; 14L30.