1. Singular Kahler-Einstein Metrics
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Kahler Currents
Problem 1.1.
[Gabor Szekelyhidi] Show that singular Kahler-Einstein metrics (or more generally singular cscK metrics) are in fact Kahler currents. This was shown in [GGZ] (2022) under suitable assumptions. This is known in dimension $\leq 3$, but it would be nice to understand it more generally.
One could also ask this for singular Kahler almost-Einstein metrics, or given integral Ricci lower bounds. -
Regularity of Kahler-Einstein Metrics
Problem 1.2.
[Jian Song] Are the local potentials of a singular Kahler-Einstein metric Holder continuous (with respect to the Fubini-Study metric), given that the singularity is smoothable?
This is known in certain cases near isolated singularities by Hein-Sun and Chiu-Szekelyhidi. -
Convexity of the Regular Set
Problem 1.3.
Is the regular set of a singular Kahler-Einstein manifold $(X,\omega_{KE})$ convex? That is, is it true that for any two points $x,y\in X_{\text{reg}}$, there is a minimal geodesic in $X$ from $x$ to $y$ which lies entirely in $X_{\text{reg}}$? This is known to be true if $X$ is a Gromov-Hausdorff limit of a sequence of noncollapsed Kahler-Einstein manifolds by Colding-Naber, and more generally if $X$ is an RCD space. -
Uniform Bounds in Families
Problem 1.4.
[Jacob Sturm] Problem 5. Let $X$ be a connected component of the moduli space of Kahler-Einstein metrics on a compact complex manifold Y of general type, and let $\pi:\mathcal{X}\to X$ be the corresponding universal family. Then there exists $N\in\mathbb{N}$ and a closed embedding $\iota:\mathcal{X}\hookrightarrow X\times\mathbb{P}^{N}$ over $X$ using powers of the relative canonical bundle $K_{\mathcal{X}/X}^{\ell}$ for sufficiently large $\ell\in\mathbb{N}$. Let $\omega_{FS}$ be the Fubini-Study metric on $\mathbb{P}^{N}$, so that we can write any Kahler-Einstein metric $\omega_{KE}\in X$ (corresponding to some $z\in X$) as $\omega_{KE}=\frac{1}{\ell}\iota_{z}^{\ast}\omega_{FS}+\sqrt{-1}\partial\overline{\partial}u$,where $\iota_{z}:\pi^{-1}(z)\hookrightarrow\{z\}\times\mathbb{P}^{N}$ is the restriction of $\iota$, and $u\in PSH(Y,\frac{1}{\ell}\iota_{z}^{\ast}\omega_{FS})\cap C^{\infty}(Y)$.
Do there exist constants $C=C(X)$ and $\alpha=\alpha(X)>0$ such that $\text{Vol}_{\omega_{FS}}(\{u<-s\})\leq Ce^{-\alpha s}$ for all $\omega_{KE}\in X$?
Cite this as: AimPL: PDE methods in complex geometry, available at http://aimpl.org/pdecomplexgeom.