Calabi flow and projective embeddings

Abstract

Let X ⊂ CP N 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 0 (L k ) 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 [13]. In our proof we combine Donaldson's techniques with an asymptotic result of Liu-Ma [17] which, as we explain, describes the asymptotic behaviour of the derivative of the map FS • Hilb whose fixed points are balanced metrics. * Supported by an FNRS postdoctoral fellowship.

The Second Coefficient of the Asymptotic Expansion of the Weighted Bergman Kernel for (0,q) Forms on $\mathbb{C^{n}}$

“…More generally, if M is any complex manifold and (q) k has O(k −n 0 ) small spectral gap on an open set D ⋐ M (see Definition 1.5 in [9], for the precise meaning of O(k −n 0 ) small spectral gap), then it is known by a recent result (see Theorem 1.6 in [9]) that the Bergman kernel admits a full asymptotic expansion in k on D. The coefficients of these expansions turned out to be deeply related to various problem in complex geometry (see e.g. [3], [4], [5]). …”

“…Now the formal adjoint of ∂ s for the inner product ( | ) given by (1.4) is 4) where in view of the unitarity of the relation (3.1),…”

Quantization of Donaldson’s heat flow over projective manifolds

“…Proof The proof is already contained in [10,Appendix] and [14] where the case of E trivial is treated in details. The important point is that the full Bergman kernel expansion holds in our setting, see [4,Theorem 4.18'].…”

“…Donaldson in the study of the constant scalar curvature problem for Kähler metrics, in relation with the socalled Yau-Tian-Donaldson conjecture, see [7]. In [10], Fine introduced another balancing flow that approximates the Calabi flow, a 4th order parabolic PDE that is expected to deforms a given Kähler metric towards a constant scalar curvature one. In [3] it was introduced the Ω-balancing flow and shown that it converges towards a flow of Kähler metrics that enjoys similar properties to the Kähler-Ricci flow, providing a new approach to the classical Calabi conjecture.…”

“…A technical part of these proofs relies on the asymptotics for Bergman kernels and Bergman functions, the so-called Catlin-Tian-Yau-Zelditch expansion (in short CTYZ) that has been extended for bundles (see [17,Chapter IV,Theorem 4.1.2] for an extensive reference on the topic, and also [2,15,23]). Another technical ingredient is an asymptotics of the Q k operator introduced in [8] adapted to the bundle case (Theorem 1) from the earlier work of Liu and Ma in [10]. To obtain the main result (Theorem 4), we use a deformation argument (Sects.…”