$m$-Positive Stability of Holomorphic Vector Bundles and Moduli Spaces

We first propose a notion of $m$-positivity for higher-rank vector bundles, a variant of which reduces to the classical Griffiths positivity when $m=1$. Based on this, we go on to propose a generalisation of the classical Mumford-Takemoto theory of stability by means of a smooth function that we associate with every proper coherent subsheaf $\mathcal{F}$ of a given holomorphic vector bundle $E$. This places the emphasis on the holomorphic structure and the Hermitian fibre metric of $E$, rather than on numerical invariants of the smooth structure of $E$, making our stability conditions into relative pointwise $m$-positivity properties of $E$ with respect to its proper coherent subsheaves $\mathcal{F}$. We establish links with Hermite-Einstein geometry, prove that Hermite-Einstein bundles are uniformly semi-stable, study the resulting moduli spaces, and compare the new notions with the classical Mumford-Takemoto (semi-)stability notions.