Moment Maps, Scalar Curvature and Quantization of K�hler Manifolds

Symmetries, quotients and Kähler-Einstein metrics

“…Let γ M and γ N be continuous hermitian forms on M and N respectively. Since π * γ N is continuous and γ M is positive definite, there is c 1 …”

Stable bundles and the first eigenvalue of the Laplacian

“…On the other hand a stronger notion of "balanced" metric has been deeply studied in the last few years for line bundles (see e.g. [6], [1], [14] and references therein). While in general more rare (they in fact measure a non vacuous stability property of line bundles), it would be very interesting to know whether these metrics, when they exist, can give stronger informations on the spectrum of the laplacian.…”

On homothetic balanced metrics

“…On the other hand, the flat metric on an elliptic curve and the hyperbolic metric on a Riemann surface of genus ≥ 2 cannot be projectively induced (see [27] for a proof) and hence M is forced to be biholomorphic to CP 1 (and g B isometric to an integer multiple of the Fubini-Study metric). (1) mg is not projectively induced for all m; (2) there is at least a non-constant coefficient a j 0 , with j 0 ≥ 2, of the TYZ expansion (5) of the Kempf distortion function T mg (x).…”

“…recent results of [33] and of Della Vedova and the third author [13] show that there exist a large class of polarized manifolds (M, L) such that M admits a constant scalar curvature metric in the class c 1 (L) and such that (M, L m ) is not polystable, for all m sufficiently large. Regarding the uniqueness of balanced metrics the first and the second author [2] have shown the following: Theorem 2.2 Let g andg be two balanced metrics whose associated Kähler forms are cohomologous Then g andg are isometric, i.e., there exists F ∈ Aut(M) such that F * g = g.…”