In this paper, interpretations of wedge products of the geometric Segre, respectively, geometric Chern, forms of a holomorphic vector bundle \(\,E\,\) (with hermitian metric \(\,|\;\;|\) ) over a complex space \(\,Y\,\) are given by showing that: (a) the geometric Segre forms of \(\,E\,\) coincide with the Segre forms constructed by means of the Chern–Weil theory, in Sect. 6; (b) as current the Segre wedge product \(\,\widehat{s}_1(E_{\infty };|\;\;|)^{\beta _1}\wedge \cdots \wedge \widehat{s}_{p+1}(E_{\infty };|\;\;|)^{\beta _{p+1}},\,\beta _j \in {\mathbb {Z}}[0,\infty ),\,p\in {\mathbb {N}}\,\) is extendible to a generalized Schubert cycle on \(\,Y,\,\) provided \(\,E\,\) is semi-globally spanned, in Sect. 7; (c) the cup product \(\,(-1)^{w(\beta )}\widehat{s}_1(E)^{\beta _1}\cup \cdots \cup \widehat{s}_{p+1}(E)^{\beta _{p+1}},\,\) if non-vanishing, is equal to the fundamental class of an analytic intersection cycle supported by a Schubert type analytic set, provided \(\,E\,\) is globally spanned, in Sect. 8; and (d) similar results hold for the geometric Chern forms and the (analogously defined) Chern wedge products. As prerequisites multi-symbol Schubert type analytic sets and the (corresponding) Chern–Cowen forms are first introduced for a semi-globally spanned vector bundle \(\,E\rightarrow Y.\,\)
