Lectures on Geodesics in the Space of Kaehler Metrics, Lecture 4: Quantization

Another facet of geodesics is GAT or “geometric approximation theory”. It is very difficult to work directly with infinite dimensional geometry and GAT is a special method in projective K’́ahler geometry to make fiinite dimensional approximations of the infinite dimensional locally symmetric space Hω by genuine finite dimensional symmetric spaces of type GC/G with G = SU(N). The compact group SU(N) is analogous to SDiff(M,ω0) and GC = SL(N,C) is analogous to Υ = {(f, ωφ) : f ωφ = ω0}. The approximating spaces are known as Bergman metric spaces of degree k and denoted by Bk. They are submanifolds Bk ⊂ Hω but not totally geodesic ones. There is also a canonical map Hilbk : Hω → Bk. The composite map Bk ⊂ Ho → Bk is a complicated map denoted by Tk : Bk → Bk. Geodesics of Bk are induced by geodesics of SL(N,C)/SU(N), which are given by the action of one parameter subgroups e. Hence Monge-Amp‘ere geodesics of Hω are limits, in some sense, of curves of Bergman Kaehler metrics induced by one parameter subgroups. This was first explored in [PS2], in which a kind of almost everywhere convergence was proved, then uniform convergence was proved in [B, B2]. In the special case of toric Kaehler manifolds, C convergence was proved in [SZ]. We assume throughout that (M,ω0) is a projective Kaehler manifold. Thus there exists a quantizing line bundle L→M and a Hermitian metric h on L with curvature form ω0. The ideas in these notes are due to Yau, Tian, Donaldson and many others, but the source of the results is not usually indicated. We refer to [PS3] for the historical and further mathematical background. Much of the notes is copied-pasted from prior articles of the author, again without attribution.