Every twisted Higgs bundle built by pushing forward along a finite cover factors through the normalization of the spectral cover, and Gieseker stability transfers to the sheaf on that normalization.
Algebraic surfaces and holomorphic vector bundles
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Spectral coverings without embeddings
Every twisted Higgs bundle built by pushing forward along a finite cover factors through the normalization of the spectral cover, and Gieseker stability transfers to the sheaf on that normalization.