Calabi flow and projective embeddings

Let X ⊂ CP be a smooth subvariety. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X which attempts to deform the given embedding into a balanced one. If L→ X is an ample line bundle, considering embeddings via H(L) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldson’s theorem relating balanced embeddings to metrics with constant scalar curvature [11]. In our proof we combine Donaldson’s techniques with an asymptotic result of Liu–Ma [15] which, as we explain, describes the asymptotic behaviour of the derivative of the map FS ◦Hilb whose fixed points are balanced metrics.