  Let $Y$ be a submanifold of dimension $y$ of a polarized complex manifold
$(X,A)$ of dimension $k\geq 3$, with $1\leq y\leq k-1$. We define and study two
positivity conditions on $Y$ in $(X,A)$, called Seshadri $A$-bigness and (a
stronger one) Seshadri $A$-ampleness. In this way we get the natural
generalization of the theory initiated by Paoletti in \cite{Pao} (which
corresponds to the case $(k,y)=(3,1)$) and subsequently generalized and
completed in
  \cite{BBF} (regarding curves in a polarized manifold of arbitrary dimension).
The theory presented here, which is new even if $y=k-1$, is motivated by a
reasonably large area of examples.
