  Theorems of Barth-Lefschetz type describe restrictions on the topology of
varieties of small codimension. R. Schoen and J. Wolfson, using Morse theory on
a path space, have described a technique to prove theorems of this kind for
complex submanifolds of K\"ahler manifolds of non-negative holomorphic
bisectional curvature. In this paper this program is carried out for the
compact Hermitian symmetric spaces. The key technical point is to define and
compute an invariant, called the {\it complex positivity}, that measures the
``amount'' of positive curvature, in a suitable sense.
