AI 中文总结
本文针对复流形上的奇异埃尔米特向量丛，引入有限Gram标量化条件将高秩可积性问题归约为标量乘子理想，得到凝聚性、强开性等性质，还建立了模跳跃数理论并研究了Skoda滤过的相关性质。
AI 中文摘要
设$(E,h)$为复流形$X$上的奇异埃尔米特向量丛，令$\boldsymbol{\nabla}(h)_x=\boldsymbol{\nabla}(E)_x$中满足$|F|_h^2 \boldsymbol{\nabla} L^1_{\text{loc},x}$的元素构成其乘子模层。我们引入有限多重次调和Gram标量化条件，在此条件下，高秩可积性问题可归约为有限个标量乘子理想；该归约无需施加一般正性假设即可得到凝聚性与强开性。当标量权重及变权重具有解析奇异性时，该条件还给出了模跳跃数理论，包括实际跳跃的同时残差判据。最后，我们研究诱导的Skoda滤过：Artin-Rees引理给出最终周期性，Tor控制周期性是否始于标量阈值，而直像商则衡量下降的障碍。
英文摘要
Let $(E,h)$ be a singular Hermitian vector bundle on a complex manifold $X$, and let \[ \mathcal E(h)_x=\{F\in\mathcal O(E)_x:|F|_h^2\in L^1_{\mathrm{loc},x}\} \] be its multiplier submodule sheaf. We introduce a finite plurisubharmonic Gram scalarization condition under which the higher-rank integrability problem reduces to finitely many scalar multiplier ideals. This reduction yields coherence and strong openness without imposing a general positivity hypothesis. When the scalar weights and the varying weight have analytic singularities, it also gives a theory of module jumping numbers, including a simultaneous-residue criterion for actual jumps. Finally, we study the induced Skoda filtration: Artin--Rees yields eventual periodicity, Tor controls whether periodicity starts at the scalar threshold, and a direct-image quotient measures the obstruction to descent.
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