Chern-Schwartz-MacPherson classes of symmetric powers and applications

Source abstract

The Chern-Schwartz-MacPherson class $c_{SM}(X)$ is defined in the Chow group $A_*(X)$ of a scheme $X$, and generalizes $c(T_X)\cap [X]$ if $X$ is smooth. In this paper we independently compute $c_{SM}(S^nX)\in A_*(S^nX)$ modulo $n!$ torsion; we use this result to compute the pushforward to $S^nX$ of $c_{SM}(Quot^n_X(E))$ (and in particular $c_{SM}(Hilb^n X)$) when $X$ is smooth and $E$ is a vector bundle on $X$. Finally, when $X$ is smooth and complete of dimension $3$ we propose a refined version of the the MNOP conjecture that computes the image of the pushforward of $[Hilb^n X]^{vir}$ to $A_0(S^nX)$ and prove it modulo $n!$ torsion when $X$ is Calabi-Yau.