On the topological structure of complex projective complete intersections with quadratic singularities

$n$-Dimensional complete intersections of “sufficiently high multidegree” in $\mathbb C P^{n+k}$, $n\ge3$, with fixed number and, possibly, position of singular points are studied. In the case where all singularities are quadratic, we give a topological description of such a variety in terms of a connected sum decomposition of special type. In this case, the diffeomorphism type of the variety is determined by the dimension, multidegree, and the number of singular points.