2026/07/19 by Dan Popovici
#math.DG #math.AG #math.CV
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 \cal 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 \cal 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.