On smooth subcanonical varieties of codimension 2 in \(P^ n\), n\(\geq 4\) (Q800443)

Let \(X\subset {\mathbb{P}}_ N({\mathbb{C}})\) be a codimension 2 smooth submanifold with \(\omega_ X\cong {\mathcal O}_ X(e),\) \(e\leq 0\). Here we prove that X is a complete intersection, proving that it has the degree of a complete intersection and then applying a recent result of \textit{Z. Ran} [Invent. Math. 73, 333-336 (1983; Zbl 0521.14018)]. If \(e<0\) the assertion about the degree was proved independently, simultaneously and in the same way by \textit{Y. Sakane} in Saitama Math. J. 1, 9-27 (1983; Zbl 0544.14031).