Equivariant Chern numbers and the number of fixed points for unitary torus manifolds

Let $M^{2n}$ be a unitary torus $(2n)$-manifold, i.e., a $(2n)$-dimensional oriented stable complex connected closed $T^n$-manifold having a nonempty fixed set. In this paper we show that $M$ bounds equivariantly if and only if the equivariant Chern numbers $<(c_1^{T^n}{)}^i(c_2^{T^n}{)}^j,[M]>=0$ for all $i, j\in {\Bbb N}$,where$c_l^{T^n}$denotesthe$l$thequivariantChernclassof$M$\mathrm{.}Asaconsequence,wealsoshowthatif$M$ does not bound equivariantly then the number of fixed points is at least $\lceil \frac{n}{2}\rceil +1$.