Chern-Schwartz-MacPherson classes and the Euler characteristic of degeneracy loci and special divisors

Abstract

This concept overlaps a large family of interesting varieties (for example, the varieties of special divisors studied in Section 3). Several authors have worked out explicit formulas for the Euler characteristic of Dr(9#) in terms of different cohomological and numerical invariants under the assumption that X is nonsingular and (v is appropriately general. For instance, if (0 is a section of a vector bundle, then the formulas for the Euler characteristic X(Do(fp)) were given by Hirzebruch [H] and Navarro-Aznar [N]. If Dr(() is a curve or a surface in a nonsingular X, some explicit formulas for X(Dr(V)) were given by Harris and Tu [H-T] in terms of the Chern classes of E, F and X, but under the extra assumption Dr,1 (v) = 0 (which implies that Dr(Q) is nonsingular). In loc.cit. the authors also posed the problem of finding a general formula for X(Dr(?O))-if such exists!-under the assumption Dr 1 (() = 0 or, even stronger, without this assumption. The first problem was solved positively by the second named author [Prl, Proposition 5.7], by the use of polynomials universally supported on degeneracy