AI 中文总结
研究提出高阶向量丛的$m$-正性概念，通过关联光滑函数推广芒福德 - 竹本稳定性理论，强调全纯结构等，建立与埃尔米特 - 爱因斯坦几何联系，证明相关丛的性质并研究模空间，比较新旧稳定性概念。
AI 中文摘要
我们首先提出了高阶向量丛的$m$-正性概念，当$m = 1$时，其变体可简化为经典的格里菲斯正性。在此基础上，我们通过与给定全纯向量丛$E$的每个恰当凝聚子层${\cal F}$相关联的光滑函数，对经典的芒福德 - 竹本稳定性理论进行了推广。这强调了$E$的全纯结构和埃尔米特纤维度量，而非$E$光滑结构的数值不变量，使我们的稳定性条件成为$E$相对于其恰当凝聚子层${\cal F}$的相对逐点$m$-正性性质。我们建立了与埃尔米特 - 爱因斯坦几何的联系，证明了埃尔米特 - 爱因斯坦丛是一致半稳定的，研究了所得的模空间，并将新的概念与经典的芒福德 - 竹本（半）稳定性概念进行了比较。
英文摘要
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.
Comments51 pages
