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Let $K$ be an algebraically closed field of characteristic $0$. For $m\geqn$, we define $\tau_{m,n,k}$ to be the set of $m\times n$ matrices over $K$with kernel dimension $\geq k$. This is a projective subvariety of$\bbP^{mn-1}$, and is called the (generic) determinantal variety. In most cases$\tau_{m,n,k}$ is singular with singular locus $\tau_{m,n,k+1}$. In this paperwe give explicit formulas computing the Chern-Mather class ($c_M$) and theChern-Schwartz-MacPherson class ($c_{SM}$) of $\tau_{m,n,k}$, as classes in theprojective space. We also obtain formulas for the conormal cycles and thecharacteristic cycles of these varieties, and for their generic EuclideanDistance degree. Further, when $K=\bbC$, we prove that the characteristic cycleof the intersection cohomology sheaf of a determinantal variety agrees with itsconormal cycle (and hence is irreducible).  Our formulas are based on calculations of degrees of certain Chern classes ofthe universal bundles over the Grassmannian. For some small values of $m,n,k$,we use Macaulay2 to exhibit examples of the Chern-Mather classes, theChern-Schwartz-MacPherson classes and the classes of characteristic cycles of$\tau_{m,n,k}$.  On the basis of explicit computations in low dimensions, we formulateconjectures concerning the effectivity of the classes and the vanishing ofspecific terms in the Chern-Schwartz-MacPherson classes of the largest strata$\tau_{m,n,k}\smallsetminus \tau_{m,n,k+1}$.  The irreducibility of the characteristic cycle of the intersection cohomologysheaf follows from the Kashiwara-Dubson's microlocal index theorem, a study ofthe `Tjurina transform' of $\tau_{m,n,k}$, and the recent computation of thelocal Euler obstruction of $\tau_{m,n,k}$ .

Chern classes and Characteristic Cycles of Determinantal Varieties