Search Results

I don't think so: A surface of constant positive curvature is diffeomorphic to a sphere, so it would be compact. If the identity map were continuous, then by composition the image of the surface woul …

Thanks Claudio, Francesco and José for your interesting answers. This is an "answer in absence" of Demailly; he doesn't use this site, but I thought his remark was nice enough to share. What follows …

Let $X$ be a compact Kahler manifold and let $\mathcal M$ denote the space of hermitian metrics on $X$. We'll identify a hermitian metric with a smooth, real and positive $(1,1)$-form $\omega$. Let $\ …

Let $X$ be a Kähler manifold. Associated to any hermitian metric $h$ on $X$ is a smooth real $(1,1)$-form $\omega = -\text{Im } h$, called the Kähler form of $h$. One of several equivalent conditions …

If you want equivalent conditions to the Nijenhuis tensor vanishing then one is that the induced $\bar \partial$ operator defines a complex, i.e. that $\bar \partial^2 = 0$. Another one is that the ex …

I really should think more about these things before asking. The answer is "yes". K. Yang considers the flag manifold $F := F_{1,2,3} := SU(3)/S(U(1)^3)$ in Invariant Kahler metrics and projective em …

The Kahler cone of any compact manifold is described by a theorem of Demailly and Paun. If $X$ is a compact Kahler manifold, then its Kahler cone is one of the connected components of the set $$ \math …

If your surface is fairly explicit you can obtain an explicit hermitian metric on it as well. For example, if we take Francesco's Hopf surface $X$, then a hermitian metric $\omega$ on $X$ can be ident …

Let $X$ be a compact Kahler manifold with $c_1(X) = 0$. Any Kahler metric $\omega$ on $X$ gives a Laplacian $\Delta_\omega$ and the $(1,1)$-form $\omega$ is harmonic with respect to this Laplacian. …

Let $(X,\omega)$ be a compact Kahler manifold, and let $\alpha$ and $\beta$ be smooth $(1,1)$-forms on $X$ that are harmonic (with respect to $\omega$). I can consider each of my $(1,1)$-forms as an a …

I am looking at a Kahler metric $g$ on a certain manifold $M$, which has the good taste to be invariant under a transitive group of isometries, and I want to say something about its holomorphic sectio …

Let $X$ be a compact Kahler manifold of complex dimension $n$. The Aubin--Calabi--Yau theorem says that if we fix a smooth form $\rho$ in the Chern class $c_1(X)$, then every Kahler class on $X$ conta …

Has anyone translated Erich Kähler's "Über eine bemerkenswerte Hermitesche Metrik" into English or French? (Preferably, but I'll take anything.)

I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something …

I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau …