Finite Gram Scalarization and Further Properties of Multiplier Submodule Sheaves

Jingcao Wu

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arXiv · 2608.08620

Finite Gram Scalarization and Further Properties of Multiplier Submodule Sheaves

Jingcao Wu

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Abstract

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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BibTeX RIS

Jingcao Wu. 2026-08-09. Finite Gram Scalarization and Further Properties of Multiplier Submodule Sheaves. https://arxiv.org/abs/2608.08620

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