ON WEIGHTED COMPLEX RANDERS METRICS

In this paper we introduce the weighted complex Randers metric F = h + Sigma(m)(i=1)vertical bar B-i vertical bar(1/i) on a complex manifold M, here h is a Hermitian metric on M and B-i, i = 1 , ... ,m are holomorphic symmetric forms of weights i on M, respectively. These metrics are special case of jet metric studied in Chandler Wong [6]. Our main theorem is that the holomorphic sectional curvature hbsc(F) of F is always less or equal to hbsc(h). Using this result we obtain a rigidity result, that is, a compact complex manifold M of complex dimension n with a weighted complex Randers metric F of positive constant holomorphic sectional curvature is isomorphic to P-n.