A complete intersection is a subvariety cut out by as many equations as its codimension, with the equations forming a regular sequence in the scheme-theoretic
definition. A projective-space complete intersection of codimension 
is the common zero scheme
of 
homogeneous equations in projective space with
the expected dimension. A hypersurface is the codimension-one
example. Smooth projective-space complete intersections are targets of ordinary Gromov-Witten
theory.
The degrees of the equations form its multidegree. The full descendant Virasoro constraints concern its entire cohomology, including classes not obtained by restriction from the ambient projective space.
