2 problems

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Kobayashi–Hitchin conjecture for holomorphic vector bundles on compact Kähler manifolds
Kobayashi–Hitchin conjecture. The equivalence between stability and the existence of a Hermitian–Einstein structure should hold for holomorphic vector bundles over compact Kähler m…
Conjectureopenhitchin-kobayashi-correspondence holomorphic-vector-bundles kahler-geometryarXiv posted 2 days ago
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Kobayashi's conjecture on approximate Hermitian–Yang–Mills structures
Kobayashi's conjecture. These three conditions should be equivalent in general, independently of whether $X$ is algebraic:
Conjectureopenhermitian-yang-mills hitchin-kobayashi-correspondence kahler-geometryarXiv posted 2 days ago

Kobayashi–Hitchin conjecture for holomorphic vector bundles on compact Kähler manifolds