On homothetic balanced metrics

Abstract

In this article, we study the set of balanced metrics given in Donaldson's terminology (J. Diff. Geometry 59:479-522, 2001) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kähler metrics. We prove that this set is finite when M admits a non-positive Kähler-Einstein metric, in the case of non-homogenous toric Kähler-Einstein manifolds of dimension ≤ 4 and in the case of the constant scalar curvature metrics found in Arezzo and Pacard (Acta. Math. 196(2):179-228, 2006; Ann. Math. 170(2):685-738, 2009). © 2011 Springer Science+Business Media B.V

The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold

“…Regular quantizations play a prominent role in the study of Berezin quantization of Kähler manifolds (see [3] , [4], [5] and [19] and references therein). From our point of view we are interested in the following result.…”

Some remarks on the Kähler geometry of the Taub-NUT metrics

“…Hence, in this case d α < ∞ and (6) is a finite sum. The two fundamental results about existence and uniqueness of balanced metrics in the compact case are summarized in the following two theorems (we refer the reader also to the recent article [3] and references therein for a link between balanced metrics and geometric invariant theory).…”

“…Finally, by (40) we get, for j, k = 1, 2, ∂ 2 g kk ∂z k ∂z k = kkkk x 2 k + 4 kkk x k + 2 kk , ∂ 2 g kj ∂z k ∂z j = ∂ 2 g kk ∂z j ∂z j = kk j j x k x j + kk j x k + k j j x j + k j , (for k = j) ∂ 2 g jk ∂z k ∂z k = ∂ 2 g kk ∂z k ∂z j = kkk j z 2 kz kz j + 2 kk j z kz j , (for k = j), = 2222 = − 16m 3 e 8m(U −V ) 8 + 43mU + 19mV + 12m 2 4U V + 5U 2 + V 2 (1 + 2m(U + V )) 5 , ∂ 4 ∂ x 3 1 ∂ x 2 = 1112 = 16m 3 e 4m(V −U ) 2 + mU + 13mV + 24m 2 V 2 (1 + 2m(U + V )) 5 ,…”

Finite TYCZ expansions and cscK metrics

“…The reader is also referred to [18] and [19] for a recursive formula for the coefficients a j 's and an alternative computation of a j for j ≤ 3 using Calabi's diastasis function (see also [25] for the case of locally Hermitian symmetric spaces). When M is noncompact, there is not a general theorem which assures the existence of an asymptotic expansion (3). Observe that in this case we say that an asymptotic expansion (3) exists if (4) holds for any compact subset of M .…”

“…if the log-term of the disk bundle D ⊂ L * vanishes then a k = 0 for k > n, where a k are the coefficients appearing in (3). A conjecture still open, due to a private communication with Z. Lu, asks if the vanishing of the a k 's for k > n implies the vanishing of the log-term.…”