Curvature of higher direct image sheaves and its application on   negative-curvature criterion for the Weil-Petersson metric
  We shall show that $q$-semipositivity of the vector bundle $E$ over a K\"ahler total space $\mathcal X$ implies the Griffiths-semipositivity of the $q$-th direct image of $\mathcal O(K_{\mathcal X/B}\otimes E)$. As an application, we shall give a negative-curvature criterion for the generalized Weil-Petersson metric on the base manifold. 
Xu Wang
math.CV math.AG math.DG