Quantization of Donaldson's heat flow over projective manifolds

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Consider E a holomorphic vector bundle over a projective manifold X polarized by an ample line bundle L. Fix k large enough, the holomorphic sections H⁰(E⊗ Lᵏ) provide embeddings of X in a Grassmanian space. We define the balancing flow for bundles as a flow on the space of projectively equivalent embeddings of X. This flow can be seen as a flow of algebraic type hermitian metrics on E. At the quantum limit k→ ∞, we prove the convergence of the balancing flow towards the Donaldson heat flow, up to a conformal change. As a by-product, we obtain a numerical scheme to approximate the Yang-Mills flow in that context.