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For an irreducible complex algebraic variety which is non-singular there is a naturally associated tangent bundle with tangent spaces as fibers. For such bundles the late Professor Chern Shingshen introduced the so-called Chern classes and Chern numbers which revolutionized the modern mathematics. Moreover, Chern pointed out that for algebraic varieties these Chern classes enjoy an algebrico-geometrical character, cf. his classical papers [CH1,2]. Now for an irreducible complex algebraic variety with singularities no such tangent bundle is defined. To meet the difficulties mathematicians had applied some blow-up process to get some non-singular varieties and Chern classes and Chern numbers are then introduced in this ingenious way, cf. notably [MacPh]. In any way it seems that there are few ways of determining such generalized Chern classes and Chern numbers in concrete cases. In the present article dedicated to the late Professor Ch. Ehresmann we shall give a simple method of introducing generalized Chern classes and Chern numbers for irreducible complex algebraic varieties with arbitray singularities which are computable in some natural sense and may be easily determined in concrete cases. To see this let us consider an irreducible complex algebraic variety Vd of dimension d in a complex projective space P C of dimension n. Let us take an arbitrary generic point G of Vd. Then G is a regular point of Vd and has the well-defined tangent space TG of dimension d through that point. The pair (G, TG) may be considered as a point in the composite Grassmannian GR(n; 0, d) of composite elements consisting of pairs of an arbitrary point as well as an arbitrary d-dimensional linear space through that point in the complex projective space P C . The pair (G, TG) will then determine a subvariety of

On generalized Chern classes and Chern numbers of irreducible complex algebraic varieties with arbitrary singularities