The log symplectic geometry of Poisson slices

The Poisson slice construction yields a number of varieties relevant to geometric representation theory and symplectic geometry. One begins with a complex semisimple linear algebraic group with Lie algebra $\mathfrak{g}$. Let us also consider a Hamiltonian $G$-variety $X$, i.e. a smooth Poisson variety with a Hamiltonian action of $G$ and moment map $\nu:X\longrightarrow\mathfrak{g}$. Each $\mathfrak{sl}_{2}$-triple $\tau=(\xi,h,\eta)\in\mathfrak{g}^{\oplus 3}$ determines a Slodowy slice

and the preimage

is a Poisson transversal in $X$. The variety $X_{\tau}$ is thereby Poisson, and we call it the Poisson slice determined by $X$ and $\tau$. To a certain extent, Poisson slices are complex Poisson-geometric counterparts of symplectic cross-sections [31, 38, 27, 28] in real symplectic geometry.

Noteworthy examples of Poisson slices include the product $G\times\mathcal{S}_{\tau}$, a hyperkähler and Hamiltonian $G$-variety studied by Bielawski [5, 6], Moore–Tachikawa [41], and several others [11, 14, 13, 1, 15]. A second example is a Coulomb branch [7] called the universal centralizer

where $\tau$ is a fixed principal $\mathfrak{sl}_{2}$-triple in $\mathfrak{g}$. This hyperkähler variety has received considerable attention in the literature [4, 7, 12, 40, 51, 55, 56], and it features prominently in Bălibanu’s recent paper [2]. Bălibanu assumes $G$ to be of adjoint type. She harnesses the geometry of the wonderful compactification $\overline{G}$ and constructs a fibrewise compactification $\overline{\mathcal{Z}_{\mathfrak{g}}^{\tau}}\longrightarrow\mathcal{S}_{\tau}$ of $\mathcal{Z}_{\mathfrak{g}}^{\tau}\longrightarrow\mathcal{S}_{\tau}$, where the latter map is projection onto the $\mathcal{S}_{\tau}$-factor. She subsequently endows $\overline{\mathcal{Z}_{\mathfrak{g}}^{\tau}}$ with a log symplectic structure.

The preceding discussion gives rise to the following rough questions.

• •

  Is there a coherent and systematic approach to the partial compactification of Poisson slices that is related to $\overline{G}$ and specializes to yield $\overline{\mathcal{Z}_{\mathfrak{g}}^{\tau}}\longrightarrow\mathcal{S}_{\tau}$ as a fibrewise compactification of $\mathcal{Z}_{\mathfrak{g}}^{\tau}\longrightarrow\mathcal{S}_{\tau}$?

• •

  If the previous question has an affirmative answer and $X_{\tau}$ is symplectic, does the partial compactification of $X_{\tau}$ carry a log symplectic structure?

Our inquiry stands to benefit from two observations. One first notes the universal or atomic nature of $G\times\mathcal{S}_{\tau}$ as a Poisson slice, i.e. the existence of a canonical Poisson variety isomorphism

for each Hamiltonian $G$-variety $X$ and $\mathfrak{sl}_{2}$-triple $\tau$ in $\mathfrak{g}$. These atomic Poisson slices have counterparts in the theories of symplectic cross-sections [31], symplectic implosion [27], symplectic contraction [34], hyperkähler implosion [17, 16], and Kronheimer’s hyperkähler quotient with momentum [37]. A second observation is that $G\times\mathcal{S}_{\tau}$ sits inside of a larger log symplectic variety $\overline{G\times\mathcal{S}_{\tau}}$ as the unique open dense symplectic leaf; the construction of $\overline{G\times\mathcal{S}_{\tau}}$ assumes $G$ to be of adjoint type and exploits the geometry of $\overline{G}$.

The preceding considerations motivate us to define

and conjecture that $\overline{X}_{\tau}$ is the desired partial compactification of $X_{\tau}$. While this naive conjecture needs to be refined and made more precise, it inspires many of the results in our paper.

Our paper develops a detailed theory of Poisson slices and addresses the questions posed above. The following is a summary of our results. We work exclusively over $\mathbb{C}$ and take all Poisson varieties to be smooth. We use the Killing form to freely identify $\mathfrak{g}^{*}$ with $\mathfrak{g}$, as well as the left trivialization and Killing form to freely identify $T^{*}G$ with $G\times\mathfrak{g}$.

Suppose that $X$ is a Poisson Hamiltonian $G$-variety with moment map $\nu:X\longrightarrow\mathfrak{g}$. Let $\tau$ be an $\mathfrak{sl}_{2}$-triple in $\mathfrak{g}$ and consider the Poisson transversal

The following are some first properties of the Poisson slice $X_{\tau}$. Such properties are well-known in the case of a symplectic variety $X$ (see [5]).

We also consider some special cases of the Poisson slice construction, including the following well-known result in the symplectic category.

Now suppose that the above-mentioned Poisson variety $X$ is log symplectic [25, 23, 50, 47], by which the following is meant: $X$ has a unique open dense symplectic leaf, and the degeneracy locus of the Poisson bivector is a reduced normal crossing divisor. We establish the following log symplectic counterpart of Observation 1.2.

Now assume $G$ to be of adjoint type. One may consider the De Concini–Procesi wonderful compactification $\overline{G}$ of $G$ [18], along with the divisor $D:=\overline{G}\setminus G$. The data $(\overline{G},D)$ determine a log cotangent bundle $T^{*}\overline{G}(\log(D))$, which is known to have a canonical log symplectic structure. Its unique open dense symplectic leaf is $T^{*}G$, and the canonical Hamiltonian ($G\times G$)-action on $T^{*}G$ extends to such an action on $T^{*}\overline{G}(\log(D))$. The moment maps

can be written in explicit terms. This leads to the following straightforward observations, whose proofs use Observation 1.2 and Proposition 1.3. To this end, recall that a principal $\mathfrak{sl}_{2}$-triple is an $\mathfrak{sl}_{2}$-triple consisting of regular elements.

In light of these observations, it is natural to consider the Poisson slice

One has an inclusion $G\times\mathcal{S}_{\tau}\subseteq\overline{G\times\mathcal{S}_{\tau}}$, while $G\times\mathcal{S}_{\tau}$ and $\overline{G\times\mathcal{S}_{\tau}}$ carry residual Hamiltonian actions of $G=G\times\{e\}\subseteq G\times G$. The respective moment maps are

and they feature in the following result.

Our paper subsequently discusses the relation of (1.1) to Bălibanu’s fibrewise compactification

We next study Hamiltonian reductions of the form

where $X$ is a Hamiltonian $G$-variety and $\tau$ is an $\mathfrak{sl}_{2}$-triple in $\mathfrak{g}$. The special case $\tau=0$ features prominently in our analysis, and we write $\overline{X}$ for $\overline{X}_{\tau}$ if $\tau=0$. This amounts to setting

with $G$ acting as $G=G\times\{e\}\subseteq G\times G$ on $T^{*}\overline{G}(\log D)$.

One has the preliminary issue of whether the quotients $\overline{X}_{\tau}$ and $\overline{X}$ exist. The following result provides some sufficient conditions.

The variety $\overline{X}_{\tau}$ enjoys certain Poisson-geometric features. A first step in this direction is to set

where $(X\times\overline{G\times\mathcal{S}_{\tau}})^{\circ}$ is the open set of points in $(X\times(\overline{G\times\mathcal{S}_{\tau}}))^{\circ}$ with trivial $G$-stabilizers. The variety $\overline{X}_{\tau}^{\circ}$ exists as a geometric quotient if $\overline{X}$ exists as a geometric quotient, in which case one has inclusions

Our final main result addresses the extent to which $\overline{X}_{\tau}$ partially compactifies $X_{\tau}$. We begin by assuming that both $\overline{X}_{\tau}$ and $X/G$ exist as geometric quotients. This allows us to construct canonical maps

It is then straightforward to deduce that

commutes, where the horizontal arrow is inclusion. This leads to the following theorem.

In the case of a principal $\mathfrak{sl}_{2}$-triple $\tau$, we realize the fibrewise compactifications (1.1) and (1.2) as special instances of Theorem 1.9.

In Section 2, we introduce the concepts from Lie theory and Poisson geometry that form the foundation for our work. Section 3 details the theory of Poisson slices and provides complete proofs of Propositions 1.1 and 1.3. Section 4 subsequently considers the Poisson slice enlargements $\overline{X}_{\tau}$ mentioned above, and it contains the proofs of Theorems 1.6, 1.8 and 1.9. We conclude with Section 5, which is devoted to illustrative examples. A list of recurring notation appears after Section 5.

The first author gratefully acknowledges the Natural Sciences and Engineering Research Council of Canada for support [PDF–516638].

We work exclusively over $\mathbb{C}$ and understand all group actions as being left group actions. We also write $\mathcal{O}_{X}$ for the structure sheaf of an algebraic variety $X$, as well as $\mathbb{C}[X]$ for the coordinate ring $\mathcal{O}_{X}(X)$. The dimension of $X$ is understood to be the supremum of the dimensions of the irreducible components. We understand $X$ to be smooth if $\dim(T_{x}X)=\dim(X)$ for all $x\in X$. Note that this convention forces a smooth variety to be pure-dimensional.

Let $K$ be a linear algebraic group. We adopt the term $K$-variety in reference to a variety $X$ endowed with an algebraic $K$-action.

Let $X$ be a $K$-variety admitting a geometric quotient $\pi:X\longrightarrow Y$, and write $X/K$ for the set of $K$-orbits in $X$. One then has a canonical bijection $Y\cong X/K$, through which $X/K$ inherits a variety structure. Any two geometric quotients $\pi:X\longrightarrow Y$ and $\pi^{\prime}:X\longrightarrow Y^{\prime}$ induce the same variety structure on $X/K$, and this structure makes the set-theoretic quotient map $X\longrightarrow X/K$ a geometric quotient. With this in mind, we shall sometimes write “$X/K$ exists” or “the geometric quotient $X\longrightarrow X/K$ exists” to mean that $X$ admits a geometric quotient.

We will need the following algebro-geometric notion of a principal bundle appearing in [8, Definition 2.3.1].

A principal $K$-bundle is necessarily a geometric quotient (e.g. by [8, Proposition 2.3.3]). We understand “$X$ is a principal $K$-bundle” as meaning that $X$ admits a geometric quotient $\pi:X\longrightarrow X/K$, and that $\pi$ is a principal $K$-bundle.

Let $X$ be a smooth variety. Suppose that $P$ is a global section of $\Lambda^{2}(TX)$, and consider the bracket operation defined by

for all $f_{1},f_{2}\in\mathcal{O}_{X}$. One calls $P$ a Poisson bivector if this bracket renders $\mathcal{O}_{X}$ a sheaf of Poisson algebras. We use the term Poisson variety in reference to a smooth variety $X$ equipped with a Poisson bivector $P$. In this case, $\{\cdot,\cdot\}$ is called the Poisson bracket. Let us also recall that a variety morphism $\phi:X_{1}\longrightarrow X_{2}$ between Poisson varieties $(X_{1},P_{1})$ and $(X_{2},P_{2})$ is called a Poisson morphism if

for all one-forms $\alpha$ defined on any open subset of $X_{2}$. Our convention is to have $(X_{1}\times X_{2},P_{1}\oplus(-P_{2}))$ be the Poisson variety product of $(X_{1},P_{1})$ and $(X_{2},P_{2})$.

Let $(X,P)$ be a Poisson variety. Contracting the bivector with cotangent vectors allows one to view $P$ as a bundle morphism

whose image is a holomorphic distribution on $X$. One refers to the maximal integral submanifolds of this distribution as the symplectic leaves of $X$. The symplectic form $\omega_{L}$ on a symplectic leaf $L\subseteq X$ is constructed as follows. One defines the Hamiltonian vector field of a locally defined function $f\in\mathcal{O}_{X}$ by

This gives rise to the tangent space description

for all $x\in L$, and one has

for all $x\in L$ and $f_{1},f_{2}\in\mathcal{O}_{X}$ defined near $x$.

We conclude by discussing log symplectic varieties, which have received considerable attention in recent years (e.g. [2, 30, 23, 50, 29, 32, 10, 47, 25, 43, 42, 24, 26, 49]). To this end, one calls a Poisson variety $(X,P)$ log symplectic if the following conditions hold:

• (i)

  $(X,P)$ has a unique open dense symplectic leaf $X_{0}\subseteq X$;

• (ii)

  the vanishing locus of $P^{n}$ is a reduced normal crossing divisor $D\subseteq X$, where $2n=\dim(X_{0})$ and $P^{n}\in H^{0}(X,\Lambda^{2n}(TX))$ is the top exterior power of $P$.

In this case, we call $D$ the divisor of $(X,P)$. One immediate observation is that $D=X\setminus X_{0}$.

We now review the salient aspects of Hamiltonian actions in the Poisson category. To this end, let $K$ be a linear algebraic group with Lie algebra $\mathfrak{k}$. Let $(X,P)$ be a Poisson variety, and assume that $X$ is also a $K$-variety. Each $y\in\mathfrak{k}$ then determines a fundamental vector field $V_{y}$ on $X$ via

for all $x\in X$. The $K$-action on $X$ is called Hamiltonian if $P$ is $K$-invariant and there exists a $K$-equivariant morphism $\nu:X\longrightarrow\mathfrak{k}^{*}$ satisfying the following condition:

for all $y\in\mathfrak{k}$, where $\nu^{y}\in\mathbb{C}[X]$ is defined by

One then refers to $\nu$ as a moment map and calls $(X,P,\nu)$ a Hamiltonian $K$-variety. The moment map $\nu$ is known to be a Poisson morphism with respect to the Lie–Poisson structure on $\mathfrak{k}^{*}$ (e.g. [9, Proposition 7.1]).

We now briefly recall the process of Hamiltonian reduction for a Hamiltonian $K$-variety $(X,P,\nu)$. One begins by observing that $\nu^{-1}(0)$ is a $K$-invariant closed subvariety of $X$. Let us assume $X\sslash K$ exists, by which we mean that the geometric quotient

exists. Write

and note that the comorphism $\pi^{*}:\mathbb{C}[X\sslash K]\longrightarrow\mathbb{C}[\nu^{-1}(0)]$ induces an algebra isomorphism

At the same time, the canonical surjection $\mathbb{C}[X]\longrightarrow\mathbb{C}[\nu^{-1}(0)]$ restricts to a surjection

if $K$ is connected and reductive. One also knows that $\mathbb{C}[X]^{K}$ is a Poisson subalgebra of $\mathbb{C}[X]$, and that the kernel of (2.7) is a Poisson ideal $I\subseteq\mathbb{C}[X]^{K}$. It follows that $\mathbb{C}[\nu^{-1}(0)]^{K}$ inherits the structure of a Poisson algebra. One may therefore endow $\mathbb{C}[X\sslash K]$ with the unique Poisson bracket for which (2.6) is an isomorphism of Poisson algebras. We refer to the data of the variety $X\sslash K$ and the Poisson algebra $\mathbb{C}[X\sslash K]$ as the Hamiltonian reduction of $(X,P,\nu)$ if (2.5) exists and $K$ is connected and reductive.

The Hamiltonian reduction process will yield a richer geometric object in the presence of certain assumptions about the $K$-action on $X$. To this end, let $K$ be a linear algebraic group and suppose that $(X,P,\nu)$ is a Hamiltonian $K$-variety. Assume that the geometric quotient (2.5) exists, and that $K$ acts freely on $\nu^{-1}(0)$. The closed subvariety $\nu^{-1}(0)\subseteq X$ is then smooth, and one deduces that $X\sslash K$ is also smooth. One may also define a Poisson bivector $P_{X\sslash K}$ on $X\sslash K$ as follows. Suppose that $x\in\nu^{-1}(0)$ and let

be the dual of the differential $d\pi_{x}:T_{x}(\nu^{-1}(0))\longrightarrow T_{\pi(x)}(X\sslash K)$. Set

for all $\alpha\in T_{\pi(x)}^{*}(X\sslash K)$, where $\tilde{\alpha}\in T_{x}^{*}X$ is any element that annihilates $T_{x}(Kx)$ and coincides with $d\pi_{x}^{*}(\alpha)$ on $T_{x}(\nu^{-1}(0))$. The bivector $P_{X\sslash K}$ renders $\mathcal{O}_{X\sslash K}$ a sheaf of Poisson algebras, recovering the above-described Poisson bracket on $\mathbb{C}[X\sslash K]$. We call the Poisson variety $(X\sslash_{\zeta}K,P_{X\sslash_{\zeta}K})$ the Hamiltonian reduction of $(X,P,\nu)$ at level $\zeta$, provided that (2.5) exists and $K$ is known to act freely on $\nu^{-1}(\zeta)$.

The preceding construction generalizes to allow for Hamiltonian reduction at an arbitrary level $\zeta\in\mathfrak{k}^{*}$. To this end, let $K_{\zeta}$ denote the $K$-stabilizer of $\zeta$ with respect to the coadjoint action. One simply sets

if the right-hand side exists as a geometric quotient. The definitions of the Poisson bracket on $\mathbb{C}[X\sslash_{\zeta}K]$ and Poisson bivector $P_{X\sslash_{\zeta}K}$ are analogous to their counterparts above.

Let $G$ be a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$. Note that $\mathfrak{g}$ is a $G$-module via the adjoint representation

and a $\mathfrak{g}$-module via the other adjoint representation

One obtains an induced action of $G$ on the coordinate ring $\mathbb{C}[\mathfrak{g}]=\mathrm{Sym}(\mathfrak{g}^{*})$, and we write $\mathbb{C}[\mathfrak{g}]^{G}\subseteq\mathbb{C}[\mathfrak{g}]$ for the subalgebra of all functions fixed by $G$. The inclusion $\mathbb{C}[\mathfrak{g}]^{G}\subseteq\mathbb{C}[\mathfrak{g}]$ then determines a morphism of affine varieties

often called the adjoint quotient.

Define the centralizer subalgebra

for each $y\in\mathfrak{g}$. An element $y\in\mathfrak{g}$ is called regular if the dimension of $\mathfrak{g}_{y}$ coincides with the rank of $\mathfrak{g}$. The set of all regular elements is a $G$-invariant open dense subvariety of $\mathfrak{g}$ that we denote by $\mathfrak{g}^{\text{r}}$.

Recall that $(\xi,h,\eta)\in\mathfrak{g}^{\oplus 3}$ is an $\mathfrak{sl}_{2}$-triple if the identities

hold in $\mathfrak{g}$, and that the associated Slodowy slice is defined by

Now assume that $\tau$ is a principal $\mathfrak{sl}_{2}$-triple, i.e. an $\mathfrak{sl}_{2}$-triple for which $\xi,h,\eta\in\mathfrak{g}^{\text{r}}$. The slice $\mathcal{S}_{\tau}$ then lies in $\mathfrak{g}^{\text{r}}$ and is a fundamental domain for the $G$-action on $\mathfrak{g}^{\text{r}}$ [35, Theorem 8]. This slice is also known to be a section of the adjoint quotient, meaning that the restriction

is a variety isomorphism [35, Theorem 7]. Let us write

for each $y\in\mathfrak{g}$. In other words, $y_{\tau}$ is the unique point at which $\mathcal{S}_{\tau}$ meets $\chi^{-1}(\chi(y))$.

Let $\langle\cdot,\cdot\rangle:\mathfrak{g}\otimes_{\mathbb{C}}\mathfrak{g}\longrightarrow\mathbb{C}$ denote the Killing form on $\mathfrak{g}$. This bilinear form is non-degenerate and $G$-invariant, i.e. the map

is a $G$-module isomorphism. The canonical Poisson structure on $\mathfrak{g}^{*}$ thereby corresponds to a Poisson structure on $\mathfrak{g}$, determined by the following condition:

for all $f_{1},f_{2}\in\mathbb{C}[\mathfrak{g}]$ and $y\in\mathfrak{g}$, where the right-hand side uses (2.8) to regard $(df_{1})_{y},(df_{2})_{y}\in\mathfrak{g}^{*}$ as elements of $\mathfrak{g}$. By means of (2.8), we shall make no further distinction between $\mathfrak{g}$ and $\mathfrak{g}^{*}$. One also has the ($G\times G$)-module isomorphism

through which we shall identify $\mathfrak{g}\oplus\mathfrak{g}$ with $(\mathfrak{g}\oplus\mathfrak{g})^{*}$.

In this subsection, we assume that $G$ is the adjoint group of $\mathfrak{g}$. Let $n=\dim\mathfrak{g}$ and write $\mathrm{Gr}(n,\mathfrak{g}\oplus\mathfrak{g})$ for the Grassmannian of all $n$-dimensional subspaces in $\mathfrak{g}\oplus\mathfrak{g}$. Note that $G\times G$ acts on $\mathrm{Gr}(n,\mathfrak{g}\oplus\mathfrak{g})$ by

and on $G$ itself by

Let $\mathfrak{g}_{\Delta}\subseteq\mathfrak{g}\oplus\mathfrak{g}$ denote the diagonally embedded copy of $\mathfrak{g}$ in $\mathfrak{g}\oplus\mathfrak{g}$, and consider the ($G\times G$)-equivariant locally closed immersion

We thereby view $G$ as a subvariety of $\mathrm{Gr}(n,\mathfrak{g}\oplus\mathfrak{g})$ and write $\overline{G}$ for its closure in $\mathrm{Gr}(n,\mathfrak{g}\oplus\mathfrak{g})$. The closed subvariety $\overline{G}$ is ($G\times G$)-invariant, smooth, and called the wonderful compactification of $G$ [18]. The complement $D:=\overline{G}\setminus G$ is known to be a normal crossing divisor in $\overline{G}$.

The pair $(G,D)$ determines a so-called log cotangent bundle $T^{*}\overline{G}(\log D)\longrightarrow\overline{G}$. One may realize this vector bundle as the pullback of the tautological bundle $\mathcal{T}\longrightarrow\mathrm{Gr}(n,\mathfrak{g}\oplus\mathfrak{g})$ along the inclusion $\overline{G}\hookrightarrow\mathrm{Gr}(n,\mathfrak{g}\oplus\mathfrak{g})$. This amounts to setting

and defining the bundle projection to be

The action of ($G\times G$) on $\overline{G}$ then lifts to the following ($G\times G$)-action on $T^{*}\overline{G}(\log D)$:

Let all objects and notation be as set in 2.5. Note that the left trivialization and Killing form combine to yield a variety isomorphism

We shall thereby make no further distinction between $T^{*}G$ and $G\times\mathfrak{g}$. The canonical symplectic form $\omega$ on $T^{*}G$ is then defined as follows on each tangent space $T_{(g,x)}(G\times\mathfrak{g})=T_{g}G\oplus\mathfrak{g}$: