Towards Bigness equivalence

TL;DR

This paper proves a conjecture relating the bigness of certain line bundles on flag varieties to the bigness of their pushforwards, with partial results under a V-bigness assumption.

Contribution

It establishes the 'if' direction of the conjecture and proves the 'only if' direction under the V-bigness hypothesis.

Findings

Proves that $Q^a_s$ is big if $ ext{pushforward}(Q^a_s)$ is big.

Confirms the conjecture in the 'if' case.

Partial proof of the 'only if' case under V-bigness.

Abstract

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 $\it Q^a_s$ on $ \mathcal{F}l_s(E).$ The aim of this paper is to prove the following conjecture: $Q^a_s $ on $ \mathcal{F}l_s(E)$ is big if only if $\pi_*(Q^a_s)=S_a(E)$ on X is big. The "if" part is proven here, the "only if" part is proven under the V-bigness hypothesis.