  We study two kinds of push-forwards of $\delta$-forms and define the
pull-backs of $\delta$-forms. As a generalization of Gubler-K\"unnemann, we
prove the projection formula and the tropical Poincar\'e-Lelong formula. As an
application, we follow the idea of Gubler-K\"unnemann and generalize the notion
of $\delta$-forms on algebraic varieties, this allows us to define the first
Chern forms for any piecewise smooth metrics.
