EXTREMAL RAYS AND NULL GEODESICS ON A COMPLEX CONFORMAL MANIFOLD

<jats:p> A holomorphic conformal structure on a complex manifold X is an everywhere non-degenerate section [Formula: see text] for some line bundle N. In this paper, we show that if X is a projective complex n-dimensional manifold with non-numerically effective K<jats:sub>x</jats:sub> and admits a holomorphic conformal structure, then X ≅ ℚ<jats:sup>n</jats:sup>. This in particular answers affirmatively a question of Kobayashi and Ochiai. They asked if the same holds assuming c<jats:sub>1</jats:sub> (X) > 0. As a consequence, we also show that any projective conformal manifold with an immersed rational null geodesic is necessarily a smooth hyperquadric ℚ<jats:sup>n</jats:sup>. </jats:p>