The $\log$ symplectic geometry of Poisson slices

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with adjoint group $G$. An $\mathfrak{sl}_2$-triple $\tau=(\xi,h,\eta)\in\mathfrak{g}^{\oplus 3}$ and a Poisson Hamiltonian $G$-variety $X$ together yield a distinguished Poisson transversal $X_{\tau}:=\nu^{-1}(\mathcal{S}_{\tau})$, where $\nu:X\longrightarrow\mathfrak{g}$ is the moment map and $\mathcal{S}_{\tau}:= \xi+\mathfrak{g}_{\eta}$ is the Slodowy slice associated to $\tau$. We refer to $X_{\tau}$ as the Poisson slice determined by $X$ and $\tau$. Prominent examples include the universal centralizer $\mathcal{Z}_{\mathfrak{g}}^{\tau}$ and hyperkähler slice $G\times\mathcal{S}_{\tau}$. These have natural log symplectic completions $\overline{\mathcal{Z}_{\mathfrak{g}}^{\tau}}$ and $\overline{G\times\mathcal{S}_{\tau}}$ arising from the wonderful compactification $\overline{G}$. The variety $\overline{\mathcal{Z}_{\mathfrak{g}}^{\tau}}$ partially compactifies $\mathcal{Z}_{\mathfrak{g}}^{\tau}$, while $\overline{G\times\mathcal{S}_{\tau}}$ partially compactifies $G\times\mathcal{S}_{\tau}$ if $\tau$ is a principal $\mathfrak{sl}_2$-triple.

Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry. To address the partial compactification aspect, we associate to each Hamiltonian $G$-variety $X$ and $\mathfrak{sl}_2$-triple $\tau$ the Hamiltonian reduction $\overline{X}_{\tau}:=(X\times (\overline{G\times\mathcal{S}_{\tau}}))/ \negmedspace / G$. Assuming that $\overline{X}_{\tau}$ exists as a geometric quotient, we establish its Poisson-geometric features. We also show $\overline{X}_{\tau}$ to have an open log symplectic stratum if $X$ is symplectic and $X_{\tau}$ is irreducible. If $\tau$ is a principal $\mathfrak{sl}_2$-triple and the geometric quotient $X/G$ exists, we realize $\overline{X}_{\tau}$ as a partial compactification of $X_{\tau}$ over $X/G$. Our constructions specialize to yield $\overline{\mathcal{Z}_{\mathfrak{g}}^{\tau}}$ and $\overline{G\times\mathcal{S}_{\tau}}$ as partial compactifications of $\mathcal{Z}_{\mathfrak{g}}^{\tau}$ and $G\times\mathcal{S}_{\tau}$, respectively.

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Received 13 August 2020

Accepted 4 January 2021

Published 21 October 2022