On The Ellipticity of Generalised Monge-Amp\`ere Equations on Vector Bundles

In this paper, we study the ellipticity of the vector bundle versions of the Monge-Amp\`ere, $J$, dHYM and $\sigma_{k}$-equations at a point. These are nonlinear geometric partial differential equations defined on a holomorphic vector bundle over a compact K\"ahler manifold. We show that when both the dimension of the manifold and the rank of the bundle are greater than or equal to three, these equations do not preserve ellipticity along continuity paths in the connected component of the trivial solution. However, the $\sigma_{2}$-equation does preserve ellipticity along continuity paths.