arXiv subjects

G. Maschler

Publications and source records attributed to G. Maschler.

3 recordsLinked to original sources

arXiv2003

A moduli curve for compact conformally-Einstein Kähler manifolds

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 $\mathcal{C}$.

math.DG↗

arXiv2002

Special Kähler-Ricci potentials on compact Kähler manifolds

A special Kähler-Ricci potential on a Kähler manifold is any nonconstant $C^\infty$ function $τ$ such that $J(\nablaτ)$ is a Killing vector field and, at every point with $dτ\ne 0$, all nonzero tangent vectors orthogonal to $\nablaτ$ and $J(\nablaτ)$ are eigenvectors of both $\nabla dτ$ and the Ricci tensor. For instance, this is always the case if $τ$ is a nonconstant $C^\infty$ function on a Kähler manifold $(M,g)$ of complex dimension $m>2$ and the metric $\tilde g=g/τ^2$, defined wherever $τ\ne 0$, is Einstein. (When such $τ$ exists, $(M,g)$ may be called {\it almost-everywhere conformally Einstein}.) We provide a complete classification of compact Kähler manifolds with special Kähler-Ricci potentials and use it to prove a structure theorem for compact Kähler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein.

math.DG↗

arXiv2002

Local classification of conformally-Einstein Kähler metrics in higher dimensions

The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m>2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m>2$, on three real parameters along with an arbitrary Kähler-Einstein metric $h$ in complex dimension $m-1$. We provide an explicit description of all these local-isometry types, for any given $h$. That result is derived from a more general local classification theorem for metrics admitting functions we call {\it special Kähler-Ricci potentials}.

math.DG↗