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A moduli curve for compact conformally-Einstein Kähler manifolds | | A. Derdzinski
; G. Maschler
; | | Date: |
10 Sep 2003 | | Subject: | Differential Geometry MSC-class: 53C55, 53C21 (Primary) 53C25 (Secondary) | math.DG | | Affiliation: | The Ohio State University), G. Maschler (University of Toronto | | Abstract: | We classify quadruples $(M,g,m, au)$ in which $(M,g)$ is a compact Kähler manifold of complex dimension $m>2$ with a nonconstant function $ au$ on $M$ such that the conformally related metric $g/ au^2$, defined wherever $ au
e 0$, is Einstein. It turns out that $M$ then is the total space of a holomorphic $CP^1$ bundle over a compact Kähler-Einstein manifold $(N,h)$. The quadruples in question constitute four disjoint families: one, well-known, with Kähler metrics $g$ that are locally reducible; a second, discovered by Bérard Bergery (1982), and having $ au
e 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kähler surface metrics; and a fourth family, present only in odd complex dimensions $mge 9$. Our classification uses a {it moduli curve}, which is a subset $mathcal{C}$, depending on $m$, of an algebraic curve in $R^2$. A point $(u,v)$ in $mathcal{C}$ is naturally associated with any $(M,g,m, au)$ having all of the above properties except for compactness of $M$, replaced by a weaker requirement of ``vertical’’ compactness. One may in turn reconstruct $M,g$ and $ au$ from this $(u,v)$ coupled with some other data, among them a Kähler-Einstein base $(N,h)$ for the $CP^1$ bundle $M$. The points $(u,v)$ arising in this way from $(M,g,m, au)$ with compact $M$ form a countably infinite subset of $mathcal{C}$. | | Source: | arXiv, math.DG/0309172 | | Services: | Forum | Review | PDF | Favorites | |
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