Let $X$ be a q-dimensional linear space through $0$ in $E^{q+N}$ where $q+N$ is the dimension of ambient vector space. Let $E^0\subset E^1\subset \dots\subset E^{q+N}$ be the filtration of ambient vector space. (All vector spaces are real vector spaces.)
Consider the set $C_i=\{X\subset E^{q+N}\mid\dim(X\cap E^{i+N-1}\geq i\}\subset Gr(q,N+q)$ for each $1\leq i\leq q$.
Notation: Let $0\leq a_0\leq a_1\leq\dots\leq a_{q-1}\leq q$ be a sequence of integers. A Schubert variety is $(a_0,\dots, a_{q-1})$ s.t. $\dim(P(X)\cap P(E)^{a_j+j})\geq j$ where $P(-)$ means projectivized vector spaces.
"Consider all $q-$dimensional linear space $X$ through $0$ in $E^{q+N}$ satisfying the Schubert condition $\dim(X\cap E^{i+N-1})\geq i,1\leq i\leq q$, where $E^{i+N-1}$ is a fixed space of dimension $i+N-1$ through $0$. They form a cycle mod 2 of dimension $qN-i$ in $\frac{O(q+N)}{O(q)\times O(N)}$"
$\textbf{Q:}$ Why $C_i$ are cycles of dimension $qN-i$? Fix $i$. $C_i$ is imposing condition that $a_{i-1}=N-1$ but other $a_l$ is not enforces for $l\neq i-1$ with condition that $0\leq a_0\leq a_1\leq \dots\leq a_{q-1}\leq q$. If the book means all condition enforced as $\{X\mid\dim(X\cap E^{i+N-1})\geq i,1\leq i\leq q\}$ which would correspond to $a_0=\dots=a_{q-1}=N-1$ which would give $q(N-1)$. However, the book seems to indicate that it gives homologies of dimension $qN-1,\dots, q(N-1)$.
Ref. Chern, Complex Manifolds without Potential Theory, Appendix, pg 99.