Quantization of Donaldson's heat flow over projective manifolds

TL;DR

This paper introduces a flow on vector bundle embeddings over projective manifolds that converges to Donaldson's heat flow as the embedding parameter grows, providing a numerical approach to approximate the Yang-Mills flow.

Contribution

It establishes the convergence of the balancing flow to Donaldson's heat flow in the large embedding limit, linking algebraic and differential geometric flows.

Findings

Proves convergence of the balancing flow to Donaldson's heat flow as k approaches infinity.

Provides a numerical scheme to approximate the Yang-Mills flow.

Connects algebraic embeddings with differential geometric flows on vector bundles.

Abstract

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 $H^0(E\otimes L^k)$ provide embeddings of $X$ in a Grassmanian space. We define the \textit{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 $k\to \infty$, 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.