Actions of diagonalizable $p$-groups and Chern numbers modulo $p$

We obtain lower bounds for the dimension of fixed loci of diagonalizable $p$-groups acting on smooth projective varieties. Those bounds depend on the modulo $p$ Chern numbers of the ambient variety, and are expressed in a natural way by introducing an appropriate filtration on the "modulo $p$ cobordism ring" (for $p=2$ this is Thom's unoriented cobordism ring $MO^*$). They are obtained using equivariant localization methods, via the concentration theorem for the Chow ring, and by a technique of "partition dividing". As applications we derive statements in the spirit of Boardman's Five-Halves Theorem for involutions on manifolds.