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.
Convexity and Zariski decomposition structure
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abstract
This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre-Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.
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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.