Abstract

Let $L$ be an ample line bundle over a compact Kähler manifold $X$. For each $k$ sufficiently large one gets an embedding of $X$ into complex projective space by choosing a basis of $H^0(X,Lk)$. Pulling back the Fubini-Study metric by these embeddings defines a sequence of “projective” Kähler metrics $\omega_k& on $X$. A fundamental theorem by Tian says that for convenient choices of basis, this sequence converges back to the original Kähler metric. Now it is natural to ask if any objects studied in Kähler geometry can be approximated by objects from projective geometry. In the talk I will mainly focus on how cscK-metrics are approximated by so-called balanced metrics and how Calabi flow, whose aim is to find cscK-metrics, is approximated by balancing flow. This is work by Donaldson for the metrics and Fine for the flows.