On the direct image of the adjoint big and nef line bundles

We investigate the positivity properties of the direct image $f_{\ast }(K_{X\mathrm{/}Y}\otimes L)$ of the adjoint line bundle associated with a big and nef line bundle $L$, under a smooth fibration $f:X\to Y$ between projective varieties. We show that the vector bundle $f_{\ast }(K_{X\mathrm{/}Y}\otimes L)$ is big.