Towards Bigness equivalence

On the flag variety $\mathcal{F}{l}_s(E)$ associated to a vector bundle $E,$ , a sequence $s$ and a partition $a,$ there is a line bundle $Q_s^a$ on $\mathcal{F}{l}_s(E)\mathrm{.}$ The aim of this paper is to prove the following conjecture: $Q_s^a$ on $\mathcal{F}{l}_s(E)$ is big if only if ${\pi }_{\ast }(Q_s^a)=S_a(E)$ on X is big. The “if” part is proven here, the “only if” part is proven under the V-bigness hypothesis.