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We consider two classical extensions for singular varieties of the usual Chern classes of complex manifolds, namely the total Schwartz–MacPherson and Fulton–Johnson classes,c   S  M(  X  )andc   F  J(  X  )    . Their difference (up to sign) is the total Milnor class    M  (  X  )   , a gener‐alization of the Milnor number for varieties with arbitrary singular set. We get first Verdier‐Riemann–Roch type formulae for the total classesc   S  M(  X  )andc   F  J(  X  )    , and use these to prove a surprisingly simple formula for the total Milnor class when   X   is defined by a finite number of local complete intersectionX  1   ,  .  …  .    ,   X  rin a complex manifold, satisfying certain transversality conditions. As applications, we obtain a Parusiński–Pragacz type formula and an Aluffi type formula for the Milnor class, and a description of the Milnor classes of   X   in terms of the global Lê classes of the    X  i   .

On the Chern classes of singular complete intersections