Chern degree functions

We introduce Chern degree functions for complexes of coherent sheaves on a polarized surface, which encode information given by stability conditions the boundary $(\alpha, \beta)$-plane. prove that these extend to continuous real valued and we study their differentiability in terms stability. For abelian surfaces, coincide with cohomological rank defined Jiang-Pareschi. illustrate some geometrical situations general strategy compute functions.