LOCALLY CONFORMAL KÄHLER MANIFOLDS AND CONFORMAL SCALAR CURVATURE

Abstract

We show that on a compact locally conformal K<TEX>$\ddot{a}$</TEX>hler manifold <TEX>$M^{2n}$</TEX> (dim <TEX>$M^{2n}\;=\;2n\;{\geq}\;4$</TEX>), <TEX>$M^{2n}$</TEX> is K<TEX>$\ddot{a}$</TEX>hler if and only if its conformal scalar curvature k is not smaller than the scalar curvature s of <TEX>$M^{2n}$</TEX> everywhere. As a consequence, if a compact locally conformal K<TEX>$\ddot{a}$</TEX>hler manifold <TEX>$M^{2n}$</TEX> is both conformally flat and scalar flat, then <TEX>$M^{2n}$</TEX> is K<TEX>$\ddot{a}$</TEX>hler. In contrast with the compact case, we show that there exists a locally conformal K<TEX>$\ddot{a}$</TEX>hler manifold with k equal to s, which is not K<TEX>$\ddot{a}$</TEX>hler.