A moduli curve for compact conformally-Einstein Kähler manifolds
| dc.creator | Derdzinski, A. | |
| dc.creator | Maschler, G. | |
| dc.date | 2003-09-10 | |
| dc.date.accessioned | 2026-07-07T06:29:51Z | |
| dc.date.available | 2026-07-07T06:29:51Z | |
| dc.description | We classify quadruples $(M,g,m,τ)$ in which $(M,g)$ is a compact Kähler manifold of complex dimension $m>2$ with a nonconstant function $τ$ on $M$ such that the conformally related metric $g/τ^2$, defined wherever $τ\ne 0$, 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 $τ\ne 0$ 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 $m\ge 9$. 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 $(M,g,m,τ)$ 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 $τ$ 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 $(M,g,m,τ)$ with compact $M$ form a countably infinite subset of $\mathcal{C}$. | |
| dc.description | 50 pages, 1 figure (via two ps files), LateX, submitted to Compositio Mathematica | |
| dc.identifier | https://arxiv.org/abs/math/0309172 | |
| dc.identifier | http://arxiv.org/abs/math/0309172 | |
| dc.identifier | Compos. Math. 141 (2005), no. 4, 1029-1080 | |
| dc.identifier | doi:10.1112/S0010437X05001612 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/98183 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53C55, 53C21 (Primary) 53C25 (Secondary) | |
| dc.title | A moduli curve for compact conformally-Einstein Kähler manifolds | |
| dc.type | text |