Rotationally symmetric pseudo-Kähler metrics of constant scalar curvatures

Abstract

We study the ordinary differential equations related to rotationally symmetric pseudo-Kähler metrics of constant scalar curvatures. We present various solutions on various holomorphic line bundles over projective spaces and their disc bundles, and discuss the phase change phenomenon when one suitably changes initial values.

On Geometry and Symmetry of Kepler Systems. I

“…This will leads to a discussion of the Kähler metrics on the two-dimensional resolved conifold. We will take a more general point of view and follow the approach of construction of Kähler metrics with U (n)-symmetry for n ≥ 2 developed in Duan-Zhou [13,14], generalizing a construction by LeBrun [28]. This will lead us to the application of symplectic coordinates and Hessian geometry in the noncompact case.…”

“…This was introduced by LeBrun [28]. See also [13,14]. Integrating the above differential equation, we get the Kähler potential φ as a function of u:…”

Hessian Geometry and Phase Change of Gibbons-Hawking Metrics

“…These techniques were generalized and applied to the Kepler problem in [20]. First, in §6 of that work, we use the explicit construction of Kähler metrics with U (n)symmetry [12,5,6] to obtain applications of symplectic coordinates and Hessian geometry. Next, these are applied to the A 1 case of Gibbons-Hawking metrics [8] (i.e., the Eguchi-Hanson metrics [7]), as the Kepler metric on the Kepler manifold K 2 , in [20, §7.2].…”

“…The notion of a phase change for Kähler metrics were first introduced in [5,6]. In [20, §10.6], the author presented a new method to describe the flop of Kähler Ricciflat metrics on the resolved conifold [3] by embedding them in a two-parameter family of Kähler Ricci-flat metrics on the canonical line bundle of P 1 × P 1 .…”

“…In [20, §10.6], the author presented a new method to describe the flop of Kähler Ricciflat metrics on the resolved conifold [3] by embedding them in a two-parameter family of Kähler Ricci-flat metrics on the canonical line bundle of P 1 × P 1 . It then becomes natural to reexamine the phase change phenomena introduced in [5,6] from the point of view of [20]. This has been carried out recently for local P 1 , local P 2 and local P 1 × P 1 cases in [18].…”