RC-positivity and scalar-flat metrics on ruled surfaces

Let X be a ruled surface over a curve of genus g . We prove that X has a scalar-flat Hermitian metric if and only if g≥ 2 and m(X)>2-2g where m ( X ) is an intrinsic number depending on the complex structure of X . As an application, we construct explicit examples of compact Kähler manifolds which have scalar-flat Hermitian metrics, but can not support scalar-flat Kähler metrics. We also classify compact complex surfaces with scalar-flat Hermitian metrics.