On the third conjecture of K. Ogiue (Q1824220)

Let \(P_{n+p}(C)\) be an \((n+p)\)-dimensional complex projective space with the standard Kaehler structure. The conjecture mentioned in the title of this paper is the following: An n-dimensional complete positively curved Kaehler submanifold immersed in \(P_{n+p}(C)\) is totally geodesic if \(p<n(n+1)/2.\) The author proves that if the sectional curvature K of an n-dimensional complete Bochner-flat Kaehler submanifold immersed in \(P_{n+p}(C)\) satisfies \(K<1\), then \(p\geq n(n+1)/2.\)