  Using spin$^c$ structure we prove that K\"ahler-Einstein metrics with
nonpositive scalar curvature are stable (in the direction of changes in
conformal structures) as the critical points of the total scalar curvature
functional. Moreover if all infinitesimal complex deformation of the complex
structure are integrable, then the K\"ahler-Einstein metric is a local maximal
of the Yamabe invariant, and its volume is a local minimum among all metrics
with scalar curvature bigger or equal to the scalar curvature of the
K\"ahler-Einstein metric.
