The main goal of this paper is to generalize a part of the relationship between mean
curvature and Harder–Narasimhan filtrations of holomorphic vector bundles to
arbitrary polarized fibrations. More precisely, for a polarized family of complex
projective manifolds, we establish lower bounds on a fibered version of Yang–Mills
functionals in terms of the Harder–Narasimhan slopes of direct image sheaves
associated with high tensor powers of the polarization. We discuss the optimality of
these lower bounds and, as an application, provide an analytic characterisation of a
fibered version of generic nefness. As another application, we refine the existent
obstructions for finding metrics with constant horizontal mean curvature. The study
of the semiclassical limit of Hermitian Yang–Mills functionals lies at the heart of our
approach.
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