On smooth subcanonical varieties of codimension 2 in Pn, n >= 4

We study subcanonical codimension 2 subvarieties ofP n, n ⩾ 4, using as our main tool the rank 2 vector bundle canonically associated to them. With this method we prove first that every smooth canonical surface inP 4 is a complete intersection. Next we study smooth varieties of codimension 2in P n,n⩾6; it is well known that all of them are subcanonical and R. Hartshorne conjectured that they are always complete intersections, if n⩾7. We prove this conjecture in the particular case of a variety X for which the integer e such that Ωx=θx(e) is 0 or negative. This result, togheter with a strong result by Z. Ran, provides a quadratic bound for the degree of a non-complete intersection variety of codimension 2in P n, n⩾6.