On Some Rotationally Symmetric Pseudo-K(?)hler Metrics Of Constant Scalar Curvatures

In general it is diffcult to write down explicitly K(a|¨)hler metrics with special cur-vature properties, e.g., K(a|¨)hler-Einstein metrics, K(a|¨)hler metric with constant scalar cur-vatures, extremal K(a|¨)hler metrics, K(a|¨)hler-Ricci solitons, etc. This is because the con-structions involve solving complicated partial differential equations. A useful methodto construct such metrics explicitly is to exploit the symmetries to reduce to solutionsof simpler equations. In particular, when the metric has global rotational symmetries,the problem is reduced to solving ordinary differential equations. The completenessof the constructed metric is related to the problem of natural extension over u = 0 oru =∞. Sometimes, the extension is only possible after taking quotient by some cyclicsubgroups of U(n), i.e., the metric is constructed on O_Pn-1(m), or its disc subbundle, orits projective compactification P(O_Pn-1(m)⊕O_Pn-1). E.Calabi, C.LeBrun, S.R.Simanca,H.-D.Cao, etc have done excellent work on this.In this thesis, we also use the method to consider rotationnally symmetric K(a|¨)hlermetrics with constant scalar curvatures and K(a|¨)hler-Einstein metrics as the special case.The difference with the former work is that we make the treatment of the extensionsof the metrics more explicit by using natural coordinate transformation. Furthermore,we will put the constructed metrics in families parameterized by the initial values ofthe corresponding ODEs. This will leads to some interesting"phase change"phe-nomenon. In earlier constructions, usually one choose suitable parameters to ensurethe constructed metrics are positive definite. We will consider the effect of lettingparameters vary and examine the cases when the K(a|¨)hler metric may acquire some sin-gularities or become pseudo-K(a|¨)hler at some regions. This is inspired by the discussionof the Candelas-de la Ossa metrics on the resolved conifold[19], which form a familyof K(a|¨)hler Ricci-flat metrics that depend on a parameter a. When a→0⁺, the metricreduces to the one induced from a singular metric on the conifold; when a becomesnegative, one gets a metric that should be understood as K(a|¨)hler metrics on the space where P¹ is flopped. We expect this to be a very general phenomenon, not just for theresoved conifold or just for K(a|¨)hler Ricci-flat metrics. We will provide in this workmore examples of this type by studying K(a|¨)hler-Einstein metrics and constant scalarcurvature K(a|¨)hler metrics.