15 problems

Let $E$ be an ALE degeneration with end modeled on $S^3/{\mathbb Z}_k$, where $k$ is an integer, and suppose the signature and curvature estimates yield…

Let $k$ be a positive integer, let $a_1,\ldots,a_k$ be parameters, and let $G(\eta,\zeta)$ be the function … Here $\eta$ and $\zeta$ are the variables in the generalized Legendre t…

ALE gluing conjecture. The resulting closed $4$-manifold should have intersection form equivalent to that of

Half-conformal-flatness conjecture. Every Ricci-flat ALE $4$-manifold is half-conformally flat; with the appropriate orientation, this means

Let $\kappa:\mathbb{M}\to\Theta_{\operatorname{Im}(\mathbb{C})}$ be the universal Kähler ALE bundle, with vertical forms $\omega_{\mathbb{M};RF}$ and…

Let $M_\zeta$ be the Calabi–Yau ALE resolution in complex dimension $3$, let $g_\zeta$ be its quotient Kähler metric, and let $(\mathcal{H}^{0,1}_\alpha)^\Gamma$ denote the invaria…

Let $\mathfrak{p}_\zeta:\mathbb{P}\to\mathbb{M}$ be the universal bundle with connection $\mathbb{A}$, and let $p_\zeta:P_\zeta\to M_\zeta$ denote its restriction to a fibre. The f…

Let $M_\zeta$ be the ALE resolution constructed from the quotient $V/\Gamma$, and let $\omega_\zeta$ denote its Kähler form. The Calabi–Yau ALE metric conjecture. The ALE space…

A Ricci-flat ALE metric is a Ricci-flat metric on an asymptotically locally Euclidean space. Kählerness conjecture. The only Ricci-flat ALE metrics are Kähler, and hence integrable…

A Ricci-flat ALE space is a complete Ricci-flat manifold that is asymptotically locally Euclidean; the paper refers to gravitational instantons and their Kähler quotients as the tw…

A Ricci-flat ALE space is a complete Ricci-flat Riemannian manifold that is asymptotically locally Euclidean. Kählerness conjecture. All Ricci-flat ALE spaces are Kähler. Ricci-fla…

The crepant resolutions of $\mathbb{C}^2/\mathbb{Z}_n$ are toric Calabi–Yau $2$-folds, also called Gibbons–Hawking spaces or ALE spaces of type A. The paper studies phase-change ph…

Let $X_k$ be an A-type ALE space, and let … , … . Let … , and let … . Higher-rank moduli–quiver variety conjecture. The moduli spaces … for a suitable choice of dimension vector…

The AGT correspondence for gauge theories on ALE spaces involves the moduli spaces of framed torsion-free sheaves and a torus action whose equivariant parameters determine a parame…

Let $G$ be an ADE group, let $m$ be a positive integer, and consider the AGT correspondence on $\mathbb R^4/\mathbb Z_m$. On the two-dimensional side, consider the $m$-th para-…