Slope stability of reflexive sheaves with respect to big classes is equivalent to existence of T-Hermitian-Einstein metrics when the big class has a birational Zariski decomposition with semiample positive part.
Mobile Product and Zariski decomposition
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abstract
We explain the relationship between $\alpha_{1}...\alpha_{q}$ (standard cohomology product) and $(\alpha_{1}...\alpha_{q})$ (mobile intersection product) of pseudo-effective classes $\alpha_{1},...,\alpha_{q}$ on a compact K$\ddot{a}$hler manifold. We also show how to use this relationship for proving some holomorphic Morse inequalities. Then we prove a result concerning the direct image of Lelong numbers under a modification in dimension 3, deriving a continuity property for the Lelong numbers of the wedge of $(1,1)-$currents.
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Slope Stable Sheaves and Hermitian-Einstein Metrics on Normal Varieties with Big Cohomology Classes
Slope stability of reflexive sheaves with respect to big classes is equivalent to existence of T-Hermitian-Einstein metrics when the big class has a birational Zariski decomposition with semiample positive part.