Finite Gram Scalarization and Further Properties of Multiplier Submodule Sheaves

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.

Disclosure

“1 , and Theorem 5.9 then holds on all of X for every k ≥ k1 . Acknowledgments The author would like to thank Prof.Jixiang Fu for helpful discussions and valuable suggestions. The counterexample in Section 3.3 was first carried out by the ChatGPT Pro, and the author subsequently checked, refined, and wrote out the construction. The author was supported by the NSFC, Grant No. 12271275. 20”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:02:04.550818Z

Structural counts