  In this note we establish a new Fujita-type effective bound for the base
point freeness of adjoint line bundles on a compact complex projective manifold
of complex dimension $n$. The bound we obtain (approximately) differs from the
linear bound conjectured by Fujita only by a factor of the cube root of $n$. As
an application, a new effective statement for pluricanonical embeddings is
derived.
