The tropical momentum map: a classification of toric log symplectic manifolds

Abstract

We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentum map, we obtain a complete classification of such manifolds in terms of decorated log affine polytopes, hence extending the classification of symplectic toric manifolds achieved by Atiyah, Guillemin-Sternberg, Kostant, and Delzant.

Appendix

1.1 Local normal forms for log affine manifolds

7.1 Let $(X,D,\xi )$ be a log affine manifold with D of normal crossing type. If $x\in X$ is such that precisely k components $D_1,\ldots , D_k$ of D meet x, then we may choose coordinates $(y_1,\ldots , y_n)$ in a neighborhood U centered at x such that D is the vanishing locus of the monomial $y_1\cdots y_k$. The coordinates also provide a natural trivialization of the algebroid $TX(-\log D)$, i.e., the sections of $TX(-\log D)$ are generated by the vector fields

$$\begin{aligned} y_1\partial _{y_1}, \ldots , y_k\partial _{y_k}, \partial _{y_{k+1}},\ldots , \partial _{y_{n}}. \end{aligned}$$

Proposition 7.2

There exist coordinates $(x_1,\ldots , x_n)$ such that D is the vanishing locus of $x_1\ldots x_k$ and

$$\begin{aligned} \xi = \sum _{i=1}^k x_i^{-1}dx_i\otimes v_i + \sum _{i=k+1}^n dx_{i}\otimes v_{i}, \end{aligned}$$

(30)

for $\{v_1,\ldots v_n\}$ a constant basis of $\mathbb {R}^n$.

Proof

Begin with coordinates $(y_1,\ldots , y_n)$ in the neighborhood U as above, so that D is given by $y_1\ldots y_k = 0$ and

$$\begin{aligned} \xi = \sum _{i=1}^k y_i^{-1} dy_i\otimes \alpha _i + \sum _{i=k+1}^n dy_i \otimes \alpha _i, \end{aligned}$$

where $\alpha _i: U\rightarrow \mathbb {R}^n$ are smooth vector-valued functions. It follows from $d\xi =0$ that for each $i\le k$, $\alpha _i$ is constant along the hypersurface $y_i=0$. Denoting the basis $\{\alpha _1(0),\ldots , \alpha _n(0)\}$ by $\{v_1,\ldots , v_n\}$, we conclude that

$$\begin{aligned} \xi - \left( \sum _{i=1}^k y_i^{-1} dy_i\otimes v_i + \sum _{i=k+1}^n dy_i\otimes v_i \right) = dF, \end{aligned}$$

for $F:U\rightarrow \mathbb {R}^n$ smooth and $F(0) = dF(0) = 0$.

We now expand F in terms of the basis $\{v_1,\ldots , v_n\}$,

$$\begin{aligned} F = \sum _{i=1}^n f_i v_i, \end{aligned}$$

and we define new coordinates on a possibly smaller neighborhood given by

$$\begin{aligned} x_i = {\left\{ \begin{array}{ll} y_i e^{f_i} &{}\quad \text {for} \quad i\le k,\\ y_i + f_i &{}\quad \text {for} \quad i > k. \end{array}\right. } \end{aligned}$$

(31)

In these coordinates, we have the required expression (30). $\square$

7.3 In the coordinates provided by Proposition 7.2, the $\mathbb {R}^n$-action on X may be written as

$$\begin{aligned} u\cdot (x_1,\ldots x_n) = (e^{v_1^*(u)}x_1,\ldots e^{v_k^*(u)}x_k, x_{k+1} + v^*_{k+1}(u), \ldots , x_n + v^*_n(u)), \end{aligned}$$

(32)

where $\{v_1^*,\ldots v_n^*\}$ is the dual basis to $\{v_1,\ldots , v_n\}$.

7.4 We see from (30) that for each $i\le k$, $v_i$ is the residue of $\xi$ along $D_i$, and so is an invariant of the log affine manifold. We also see that these residues are linearly independent, since they are contained in the basis $\{v_1,\ldots , v_n\}$. The remaining basis elements $\{v_i \mid i>k\}$, however, are not invariant and may be adjusted by a change of coordinates, as follows.

If $\{v_1,\ldots , v_k, w_{k+1},\ldots , w_n\}$ is any other basis, define $\{r_{ij}, s_{ij}\}$ via

$$\begin{aligned} v_j = \sum _{i>k} r_{ij} w_i + \sum _{i\le k} s_{ij} v_i,\quad \text { for all } j>k. \end{aligned}$$

(33)

We then make the coordinate change

$$\begin{aligned} {\tilde{x}}_i = {\left\{ \begin{array}{ll} x_i \exp ({\sum _{j>k} s_{ij} x_j}) &{}\quad \text { for } i \le k,\\ \sum _{j>k} r_{ij} x_j &{} \quad \text { for } i>k. \end{array}\right. } \end{aligned}$$

(34)

As a consequence, we obtain

$$\begin{aligned} \sum _{i=1}^k x_i^{-1}dx_i\otimes v_i + \sum _{i=k+1}^n dx_{i}\otimes v_{i}= \sum _{i=1}^k {\tilde{x}}_i^{-1}d{\tilde{x}}_i\otimes v_i + \sum _{i=k+1}^n d{\tilde{x}}_{i}\otimes w_{i}. \end{aligned}$$

(35)

Proposition 7.5

Let $(X,D,\xi )$ be a log affine n-manifold and suppose that $x\in X$ lies in exactly k components $D_1,\ldots , D_k$ of the normal crossing divisor D. Then the residues $v_i\in \mathbb {R}^n$ of $\xi$ along $D_i$ are linearly independent. In addition, for any extension $\{v_1,\ldots , v_k, v_{k+1},\ldots v_n\}$ to a basis, there are coordinates near x such that

$$\begin{aligned} \xi = \sum _{i=1}^k x_i^{-1}dx_i\otimes v_i + \sum _{i=k+1}^n dx_{i}\otimes v_{i}. \end{aligned}$$

(36)

Theorem 7.6

Let $(X,D,\xi )$ and $p\in U\subset X$ be as above. If we choose a basepoint $q\in U{\setminus } D$ and map it to any point $\phi (q)\in \mathbb {R}^n$, then there is a unique extension of $\phi$ to a map

$$\begin{aligned} \phi : U\rightarrow X_A \end{aligned}$$

(37)

with values in the log affine manifold $X_A$ constructed in (5), where $A=\{v_1,\ldots , v_k\}$ is the set of residues of $\xi$ in U. This map is an isomorphism of log affine manifolds onto its image.

Proof

Choose local coordinates $(x_1,\ldots , x_n)$ in U so that $\xi$ is given by (30) and choose similar adapted coordinates $(y_1,\ldots , y_n)$ for $X_A$, so that its log 1-form $\xi _A$ has the form

$$\begin{aligned} \xi _A = \sum _{i=1}^k y_i^{-1}dy_i\otimes v_i + \sum _{i=k+1}^n dy_{i}\otimes w_{i}, \end{aligned}$$

(38)

for a basis $\{v_1,\ldots , v_k, w_{k+1},\ldots w_n\}$ of $\mathbb {R}^n$. Using Proposition 7.5, we change coordinates so that

$$\begin{aligned} \xi _A = \sum _{i=1}^k {\tilde{y}}_i^{-1}d{\tilde{y}}_i\otimes v_i + \sum _{i=k+1}^n d{\tilde{y}}_{i}\otimes v_{i}. \end{aligned}$$

(39)

Therefore, the map $\phi$, which must satisfy $\phi ^* \xi _A = \xi$, is given by

$$\begin{aligned} {\tilde{y}}_i = {\left\{ \begin{array}{ll} e^{c_i} x_i &{}\quad \text { for } i\le k,\\ x_i + c_i &{}\quad \text { for } i>k, \end{array}\right. } \end{aligned}$$

(40)

where the constants $\{c_i\in \mathbb {R}\}$ are fixed by $\phi (p) = q$, as required. $\square$

1.2 Log symplectic reduction

7.7 Let $(M,Z,\omega )$ be a log symplectic manifold with an $S^1$-action generated by the vector field $\partial _\theta$, and suppose it is Hamiltonian, in the sense that there is a function $\mu :M\rightarrow \mathbb {R}$ such that

$$\begin{aligned} i_{\partial _\theta } \omega = - d\mu . \end{aligned}$$

(41)

We impose the following transversality condition: along $\widetilde{M} = \mu ^{-1}(0)$, the composition $\widetilde{d\mu }$ of $d\mu$ with the anchor map $TM(-\log Z)\rightarrow TM$

(42)

must be surjective. In particular, this implies that 0 is a regular value of $\mu$ and so $\widetilde{M}\subset M$ is a smooth hypersurface. If Z is a normal crossing divisor, (42) is surjective if and only if 0 is a regular value, not only for $\mu$, but also for the restriction of $\mu$ to any stratum of Z. As a result, the intersection $\widetilde{Z} = Z\cap \widetilde{M}$ is a normal crossing divisor in $\widetilde{M}$, and the kernel of the morphism (42) is the Lie algebroid $T\widetilde{M}(-\log \widetilde{Z})$.

If we assume further that $S^1$ acts freely and properly on $\widetilde{M}$, we obtain a normal crossing divisor $Z_0 = \widetilde{Z}/S^1$ in the quotient manifold $M_0=\widetilde{M}/S^1$. The quotient map $\pi : \widetilde{M}\rightarrow M_0$ induces a Lie algebroid morphism and an exact sequence over $\widetilde{M}$

(43)

As in the usual Marsden-Weinstein symplectic reduction, the corank 1 subalgebroid $T\widetilde{M}(-\log \widetilde{Z})\subset TM(-\log Z)$ is coisotropic with respect to the log symplectic form, whose restriction has rank 1 kernel generated by $\partial _\theta$. Therefore, the pull back of $\omega$ to $\widetilde{M}$ is basic relative to $\pi$, and may be expressed as the pull back of a unique log symplectic form $\omega _0\in \Omega ^2(M_0,\log Z_0)$, defining the logarithmic symplectic reduction. We state a slight extension of this result for arbitrary free divisors.

Proposition 7.8

Let $(M,Z,\omega )$ be a log symplectic manifold, let $\mu$ be a Hamiltonian generating a circle action via (41), and assume that (42) is surjective, which implies that $\mu ^{-1}(0)$ is smooth, with free divisor $\widetilde{Z} = \mu ^{-1}(0) \cap Z$. Assume that the $S^1$-action on $\mu ^{-1}(0)$ is free and proper, with quotient $M_0$ containing the free divisor $Z_0 = \widetilde{Z}/S^1$. Then the pull back of $\omega$ to $\mu ^{-1}(0)$ is basic in the sense of (43), and defines a logarithmic symplectic structure $(M_0,Z_0,\omega _0)$, called the log symplectic quotient.

1.3 Symplectic uncut

Let $(M,Z,\omega )$ be a compact orientable log symplectic manifold with corners and with Z of normal crossing type, equipped with an effective $T^n$-action. We construct a log symplectic principal $T^n$-bundle $(\widetilde{M}, \widetilde{Z}, \widetilde{\omega })$ over the quotient $M / T^n$, called the symplectic uncut (Definition 7.21), by applying the compressed blow-up operation to M along its submaximal orbit type strata, following an idea of Meinrenken [21]. This operation is inverse to the symplectic cut introduced in Sect. 5.2, which may be applied to $\widetilde{M}$ along $\widetilde{Z}$ to recover M.

1.3.1 Compressed blow-up

To define the compressed blow-up operation (Definition 7.11), we first recall the blow-up operation and the boundary compression construction.

7.9 For M a manifold and K a submanifold of codimension 2, we denote the real oriented blow-up of M along K by $\mathsf {Bl}_{K}(M)$. We denote the exceptional divisor by E, which is a boundary component of $\mathsf {Bl}_{K}(M)$. The blow-down map

$$\begin{aligned} p: \mathsf {Bl}_{K}(M) \rightarrow M \end{aligned}$$

(44)

restricts to a diffeomorphism on $\mathsf {Bl}_{K}(M){\setminus } E$ and expresses E as an $S^1$-fiber bundle over K.

7.10 We now define a compression operation closely related to the unfolding defined in [3]. Let M be a smooth manifold with corners and let Z be a smooth component of the boundary. Let $U \cong Z \times [0, \epsilon )$ be a tubular neighborhood of Z, and let $V = M {\setminus } Z$. Then M is the fibered coproduct, or gluing, of U and V along $U \cap V$. Consider the open embedding

$$\begin{aligned} \iota : U \cap V \cong Z \times (0, \epsilon ) \rightarrow \widetilde{U} = Z \times \left[ 0, \tfrac{1}{2}\epsilon ^2\right) , \quad (p, r) \mapsto \left( p, \tfrac{1}{2}r^2 \right) . \end{aligned}$$

(45)

The fibered coproduct of $\widetilde{U}$ and V along $U \cap V$ is a smooth manifold with corners $\widetilde{M}$. The identity map on V extends to a smooth homeomorphism $q: M \rightarrow \widetilde{M}$ such that the restriction to Z is a diffeomorphism onto its image $\widetilde{Z} = q(Z)$. The triple $(\widetilde{M}, \widetilde{Z}, q)$ is called the compression of (M, Z).

We now compress the blow-up along its exceptional divisor to obtain the following.

Definition 7.11

Let M be a manifold with corners and let $K \subset M$ be a submanifold of codimension 2. Then the compressed blow-up of M along K is the compression $(\widetilde{M}_K, \widetilde{E}, q)$ of $(\mathsf {Bl}_{K}(M), E)$.

If $S \subset M$ is a submanifold that intersects K cleanly, the compressed proper transform of S is the submanifold

$$\begin{aligned} \widetilde{S} = \overline{(q\circ p^{-1})(S {\setminus } K)} \subset \widetilde{M}_K. \end{aligned}$$

Example 7.12

Let $M=\mathbb {R}^2$ and $K=\{0\}\subset M$. The real oriented blow-up is then $\mathsf {Bl}_{K}(M)=[0,\infty )\times S^1$; using standard coordinates $(r,\theta )$ for $\mathsf {Bl}_{K}(M)$ and the complex coordinate on M, the blow-down map is $p(r,\theta ) = r e^{i\theta }$. The compressed blow-up is also given by $\widetilde{M}_K = [0,\infty )\times S^1$, and receives a smooth homeomorphism from the blow-up $q(r,\theta ) = (\tfrac{1}{2}r^2,\theta )$. That is, $q: \mathsf {Bl}_{K}(M) \rightarrow \widetilde{M}_K$ is smooth with a continuous inverse.

7.13 We apply now the compressed blow-up operation to the submaximal strata of an effective $T^n$-action. Let M be a 2n-dimensional compact orientable manifold with corners equipped with an effective $T^n$-action. By [10, Corollary B.48], the fixed point set of a circle subgroup $H \subset T^n$ is a disjoint union of closed submanifolds of codimension at least 2.

Let $H_1, H_2, \ldots , H_m \subset T^n$ be the circle subgroups such that the fixed point set $M^{H_i}$ contains at least one codimension 2 component (there are finitely many $H_i$, because M is compact, and therefore has finitely many orbit type strata). Let $K_i$ be the union of the codimension 2 components of $M^{H_i}$.

Let $\widetilde{M}_{K_1}$ be the compressed blow-up of M along $K_1$. The compressed proper transform

$$\begin{aligned} \widetilde{K_2} \subset \widetilde{M}_{K_1} \end{aligned}$$

is a codimension 2 submanifold of $\widetilde{M}_{K_1}$, so we may apply the compressed blow-up operation to $\widetilde{M}_{K_1}$ along $\widetilde{K_2}$. Continuing in this way, we obtain the following.

Definition 7.14

Let M be a 2n-dimensional compact orientable manifold with corners with an effective $T^n$-action. The iterated compressed blow-up $\widetilde{M}$ is the manifold with corners obtained from M by successively applying the compressed blow-up operation to the fixed point sets $\{K_1, K_2, \ldots , K_m\}$.

The iterated compressed blow-up $\widetilde{M}$ receives the compression map

$$\begin{aligned} q: \mathsf {Bl}_{K}(M) \rightarrow \widetilde{M}, \end{aligned}$$

where $\mathsf {Bl}_{K}(M)$ is the iterated real oriented blow-up of M along $\{K_1, K_2, \ldots , K_m\}$.

Lemma 7.15

Let M be a 2n-dimensional compact orientable manifold with corners endowed with an effective $T^n$-action. The iterated compressed blow-up $\widetilde{M}$ in Definition 7.14 is a principal $T^n$-bundle over the quotient $\Delta = M / T^n$, and therefore $\Delta$ is a compact orientable manifold with corners. Furthermore, the following diagram commutes

(46)

and the blow-down and folding maps p, q are $T^n$-equivariant.

In addition, if Z is a $T^n$-invariant normal crossing divisor in M, then the compressed proper transform $\widetilde{Z}$ is a normal crossing divisor in $\widetilde{M}$, and $D = Z / T^n$ is a normal crossing divisor in $\Delta$.

Proof

Because M is orientable, $T^n$ acts freely on the principal orbit stratum, and every point with non-trivial stabilizer is contained in some $K_i$.

For any point $x \in K_{i_1} \cap K_{i_2} \cap \cdots \cap K_{i_k}$, there exists a $T^n$-invariant neighborhood $U_x$ that is equivariantly isomorphic to a neighborhood of

$$\begin{aligned} (x, 0)\in (U_x \cap K_{i_1} \cap K_{i_2} \cap \cdots \cap K_{i_k}) \times \mathbb {C}^k, \end{aligned}$$

where $\mathbb {C}^k$ is endowed with the standard $T^k$-action. It follows that $q ( p^{-1}(x))$ is equivariantly isomorphic to a k-torus $T^k$, and $q ( p^{-1}(U_x))$ is equivariantly isomorphic to a neighborhood of

$$\begin{aligned} q ( p^{-1}(x)) \cong \{(x, 0)\} \times T^k \in (U_x \cap K_{i_1} \cap K_{i_2} \cap \cdots \cap K_{i_k}) \times \mathbb {R}_+^k \times T^k, \end{aligned}$$

where $\mathbb {R}_+ = [0, \infty )$. Therefore, the induced $T^n$-action on $\widetilde{M}$ is free and proper, and diagram (46) commutes.

Since M is orientable, it follows that the iterated compressed blow-up $\widetilde{M}$ is orientable, and therefore $\Delta$ is orientable.

Let $Z_1, Z_2, \ldots , Z_l$ be components of Z. The intersection

$$\begin{aligned} L = \bigcap _{i =1, \ldots , l} Z_i \end{aligned}$$

is $T^n$-invariant. By induction on l, the $T^n$-action on L is effective.

The isotropy action of $H_i$ on $T_xM$ descends to the normal space $N_xK_i$. Up to a linear transformation, $H_i$ acts on $N_xK_i \cong \mathbb {C}$ by the standard $S^1$-action. Because the $T^n$-action on L is effective, it follows that the $T^n$-invariant subspace $T_xL \subset T_xM$ is transverse to $T_xK_i$. Therefore $\widetilde{Z}$ is a normal crossing divisor in $\widetilde{M}$ and D is a normal crossing divisor in $\Delta$. $\square$

1.3.2 Local normal form

Given a log symplectic manifold $(M, Z, \omega )$ equipped with an effective $T^n$-action, we show that intersections among the codimension 2 circle fixed point sets $\{K_1, K_2, \ldots , K_m\}$ are log symplectic submanifolds. We also show the existence of local normal forms for neighborhoods of points on such submanifolds. We begin with a convenient definition of log symplectic submanifold.

Definition 7.16

Let $(M,Z,\omega )$ be a log symplectic manifold. An embedded submanifold $i:N \hookrightarrow M$ is called a log symplectic submanifold if $Ti:TN\rightarrow TM$ is transverse to the anchor map $TM(-\log Z) \rightarrow TM$ and $\omega$ is non-degenerate on the resulting intersection.

If Z is of normal crossing type, then the above transversality condition is equivalent to the condition that N be transverse to all possible intersections among components of Z. This means that $Z_N = N\cap Z$ is a normal crossing divisor in N, and then N is log symplectic when $\omega$ is non-degenerate when pulled back to the subbundle

$$\begin{aligned} TN(-\log Z_N)\hookrightarrow TM(-\log Z). \end{aligned}$$

Lemma 7.17

Let $(M,Z,\omega )$ be a 2n-dimensional compact orientable log symplectic manifold with Z of normal crossing type, equipped with an effective $T^n$-action. Let K be a codimension 2 component of the fixed point set $M^H$ of a circle subgroup $H \subset T^n$.

Then K is a log symplectic submanifold. For every point $x \in K$, there exists a $T^n$-invariant neighborhood $U_x$ such that $U_x$ is isomorphic to a neighborhood of

$$\begin{aligned} (x, 0) \in M^H \times \mathbb {C}, \end{aligned}$$

(47)

where $\mathbb {C}$ is equipped with the symplectic structure

$$\begin{aligned} \frac{i}{2}dz\wedge d\overline{z} \end{aligned}$$

and H acts on $\mathbb {C}$ by the standard $S^1$-action.

Proof

Let $Z_1, Z_2, \ldots , Z_\ell$ be components of Z. As shown in the proof of Lemma 7.15, K is transverse to the intersection

$$\begin{aligned} \bigcap _{i =1, \ldots , \ell } Z_i. \end{aligned}$$

This implies that $K \cap Z$ is a normal crossing divisor in K. For every point $x\in K$, the log tangent space

$$\begin{aligned} T_x(M, - \log Z) \end{aligned}$$

is a symplectic vector space, and the H-invariant subspace

$$\begin{aligned} T_x(K, - \log (K \cap Z) ) \end{aligned}$$

is a symplectic subspace. This shows that K is a log symplectic submanifold.

Taking a smaller neighborhood, if necessary, we may assume that $U_x$ is contractible and

$$\begin{aligned} U_x \cong (U_x \cap K) \times D, \end{aligned}$$

where $D \subset \mathbb {C}$ is an open disk. We use the Moser method to show that $(U_x, \omega )$ is equivariantly symplectomorphic to $(U_x \cap K, \omega _K) \times (D, \sigma )$, where $\omega _K$ is the induced log symplectic structure on $U_x \cap K$ and $D \subset \mathbb {C}$ is equipped with the symplectic structure $\sigma = \frac{i}{2}dz\wedge d\overline{z}$ and the standard $S^1$-action. We write

$$\begin{aligned} \omega _t := (1-t)\omega + t (\omega _K \oplus \sigma ), \quad 0\le t \le 1. \end{aligned}$$

Because $K \pitchfork (Z_i \cap Z_j)$, for small enough $U_x$, we have

$$\begin{aligned} \bigoplus _{i, j} H^0(U_x \cap Z_i \cap Z_j) = \bigoplus _{i, j} H^0(K \cap U_x \cap Z_i \cap Z_j), \end{aligned}$$

and therefore

$$\begin{aligned} H^2(U_x, \log Z) = H^2(K \cap U_x, \log (K \cap Z)). \end{aligned}$$

This implies $[\omega _0] = [\omega _t] \in H^2(U_x, \log (U_x \cap Z))$, and so

$$\begin{aligned} \omega _0 - \omega _1 = t d \tau ' \end{aligned}$$

for some $\tau ' \in \Omega ^1(U_x, \log (U_x \cap Z))$.

Choose a chart centered at x such that

$$\begin{aligned} \tau ' |_x = \sum _{i = 1}^\ell c_i \frac{d x_i}{x_i} + \sum _{i = \ell +1}^{2n} c_i d x_i. \end{aligned}$$

(48)

We use the right hand side of (48) to define a closed logarithmic 1-form $\tau _0$ on $U_x$. Consequently,

$$\begin{aligned} \omega _0 - \omega _1 = t d \tau ' = t d (\tau ' - \tau _0) = t d \tau , \end{aligned}$$

where $\tau := \tau ' - \tau _0$ has the property that $\tau |_x = 0$. We average $\tau$ over $T^n$, getting a new map, also called $\tau$, which is $T^n$-equivariant. The usual Moser method yields the desired result. $\square$

By Lemma 7.17, we deduce that K is a compact orientable log symplectic submanifold of dimension $2n-2$ with an induced effective action by the torus $T^n / H$, so we obtain the following result by induction on dimension.

Proposition 7.18

Let $(M,Z,\omega )$ be a 2n-dimensional compact orientable log symplectic manifold with Z of normal crossing type, equipped with an effective $T^n$-action. Let $\{K_1,\ldots , K_m\}$ be the submaximal orbit type strata as described in 7.13. Then for $1 \le i_1, i_2, \ldots , i_k \le m$, where $i_j$’s are distinct,

$$\begin{aligned} N = K_{i_1} \cap K_{i_2} \cap \cdots \cap K_{i_k} \end{aligned}$$

is a log symplectic submanifold with the induced action by the torus

$$\begin{aligned} T^n/H_{i_1} H_{i_2} \ldots H_{i_k}. \end{aligned}$$

For a point $x \in N$, there exists a $T^n$-invariant neighborhood $U_x$ that is equivariantly isomorphic to a neighborhood of

$$\begin{aligned} (x, 0) \in (N \cap U_p) \times \mathbb {C}^k, \end{aligned}$$

(49)

where $\mathbb {C}^k$ is equipped with the symplectic structure

$$\begin{aligned} \frac{i}{2} \left( dz_1\wedge d\overline{z}_1 + dz_2\wedge d\overline{z}_2 + \cdots + dz_k\wedge d\overline{z}_k \right) \end{aligned}$$

and each $H_{i_p}$ acts only on the $p^{th}$ factor of $\mathbb {C}^k$ by the standard $S^1$-action.

Example 7.19

We continue with Example 7.12. Equip $M=\mathbb {R}^2$ with the standard symplectic form ($\sigma =\tfrac{i}{2}dz\wedge d\bar{z}$, in complex coordinates) which is invariant by rotations of the plane. The blow-up at the origin then has induced presymplectic structure $rdr\wedge d\theta$, while the compressed blow-up is endowed with the symplectic structure $\widetilde{\sigma } = {d} t \wedge {d} \theta$ in standard coordinates $(t,\theta )$ for $\widetilde{M}_K = [0,1)\times S^1$. The Hamiltonian function t generates the induced $S^1$-action on $\widetilde{M}_K$. Finally, the symplectic cut of $(\widetilde{M}_K, \widetilde{\sigma })$ with respect to the function t is equivariantly isomorphic to $(M, \sigma )$.

1.3.3 Symplectic uncut

We use Proposition 7.18 to show that there is an induced $T^n$-invariant log symplectic structure on the compressed blow-up $\widetilde{M}$. By Lemma 7.15, $\widetilde{M}$ defines a log symplectic principal $T^n$-bundle, which we call the symplectic uncut (see Definition 7.21 below).

Proposition 7.20

Let $(M,Z,\omega )$ be a 2n-dimensional compact orientable log symplectic manifold with corners, where Z is of normal crossing type, equipped with an effective $T^n$-action. Let $\Delta = M / T^n$ and $D = Z / T^n$.

Then the iterated compressed blow-up $(\widetilde{M}, \widetilde{Z})$ in Lemma 7.15, as a principal $T^n$-bundle over $(\Delta , D)$, is endowed with a unique $T^n$-invariant log symplectic structure $\widetilde{\omega }$ such that the pull back of $\widetilde{\omega }$ to $\mathsf {Bl}_{K}(M)$ coincides with the pull back of $\omega$ to $\mathsf {Bl}_{K}(M)$, i.e., we have $p^*\omega = q^*\widetilde{\omega }$ in diagram (46).

Proof

For a point

$$\begin{aligned} x \in N = K_{i_1} \cap K_{i_2} \cap \cdots \cap K_{i_k} \subset M, \end{aligned}$$

let $U_x \subset M$ be a $T^n$-invariant neighborhood that is isomorphic to a neighborhood of

$$\begin{aligned} (x, 0) \in (N \cap U_x) \times \mathbb {C}^k. \end{aligned}$$

as in Proposition 7.18. In the proof of Lemma 7.15, $q (p^{-1}(U_x)) \subset \widetilde{M}$ is equivariantly isomorphic to a neighborhood of

$$\begin{aligned} (x, 0) \times T^k \subset (N \cap U_x ) \times \left( \mathbb {R}_+ \times S^1 \right) ^k, \end{aligned}$$

where $\mathbb {R}_+ = [0, \infty )$. In these coordinates, the $T^n$-invariant symplectic form

$$\begin{aligned} \widetilde{\omega } = \omega _N \oplus \left( d t_1 \wedge d\theta _1 + d t_2 \wedge d\theta _2 + \cdots + d t_k\wedge d\theta _k \right) \end{aligned}$$

satisfies $p^*\omega = q^*\widetilde{\omega }$. $\square$

Definition 7.21

Let $(M,Z,\omega )$ be a compact orientable log symplectic manifold with corners with Z of normal crossing type, equipped with an effective $T^n$-action. The triple $(\widetilde{M}, \widetilde{Z}, \widetilde{\omega })$, as a log symplectic principal $T^n$-bundle over $(\Delta , D)$, is called the symplectic uncut of $(M, Z, \omega )$.

1.4 Logarithmic de Rham cohomology

7.22 Let M be a smooth manifold and Z a codimension 1 closed hypersurface of M. Let $TM(-\log Z)$ be the log tangent algebroid. Then the residue exact sequence of de Rham complexes

smoothly splits as a sequence of complexes, giving a natural decomposition of cohomology groups

$$\begin{aligned} H^k(M,\log Z)\cong H^k(M)\oplus H^{k-1}(Z), \end{aligned}$$

(50)

recovering the well-known isomorphism of Mazzeo-Melrose [19].

7.23 If Z is the degeneracy locus of a log symplectic structure, then Z has a foliation F by codimension 1 symplectic leaves, and the Poisson Lie algebroid is isomorphic to $TM(-\log Z,F)$, the sheaf of vector fields tangent to Z and to F. A local computation shows that the morphism of complexes dualizing the morphism of algebroids $TM(-\log Z, F)\rightarrow TM(-\log Z)$, namely,

$$\begin{aligned} \Omega ^\bullet (M, \log Z)\rightarrow \Omega ^{\bullet }(M, \log Z,F), \end{aligned}$$

is a quasi-isomorphism¹ and so the Poisson cohomology coincides with the logarithmic cohomology of the hypersurface Z.

7.24 If there are several smooth hypersurfaces $Z_1,\ldots , Z_k$ which intersect in a normal crossing fashion, for example the boundary of a manifold with corners, the Mazzeo-Melrose theorem easily generalizes to give (for the free divisor $Z = Z_1+\cdots + Z_k$)

$$\begin{aligned} H^k(M,\log Z) \cong H^k(M) \oplus \sum _i H^{k-1}(Z_i) \oplus \sum _{i<j} H^{k-2}(Z_i\cap Z_j)\oplus \cdots . \end{aligned}$$

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Furthermore, if Z is the degeneracy locus of a log symplectic structure $\omega \in \Omega ^2(M, \log Z)$, then the corresponding Poisson structure $\pi = \omega ^{-1} \in C^\infty (M, \wedge ^2 TM(-\log Z))$ defines a morphism of Lie algebroids

$$\begin{aligned} \pi : T^* M \rightarrow TM(-\log Z), \qquad \alpha \mapsto i_\omega \pi , \end{aligned}$$

which dualizes to a quasi-isomorphism of cochain complexes:

$$\begin{aligned} (\Omega ^\bullet (M, \log Z), d)\rightarrow (\mathfrak {X}^{\bullet }(M), [\pi , \cdot ]). \end{aligned}$$

It follows that the Poisson cohomology is isomorphic to the logarithmic de Rham cohomology of the hypersurface arrangement. This allows a computation of the Poisson cohomology by way of the Mazzeo-Melrose isomorphism.

1.5 Poisson toric variety and the tropicalization map

We outline here the relationship between the tropical momentum map introduced in this paper and the notion of extended tropicalization which exists in tropical geometry. To see the connection, we use a canonical real Poisson structure on any smooth complex toric variety introduced by Caine [2].

7.25 First, we recall the construction of Caine’s Poisson toric variety. Let $\mathfrak {t}^n$ be the Lie algebra of the real n-torus $T^n$. Let $\Sigma$ be a complete simplicial fan with d 1-dimensional cones. We assume that $\Sigma$ is dual to a Delzant polytope, so that the resulting toric variety is smooth. We label each 1-dimensional cone with its integral generating vector $v_i \in \mathfrak {t}^n$, $i=1, \ldots , d$. The linear projection defined by $\mathfrak {t}^d\ni e_i \mapsto v_i \in \mathfrak {t}^n$ for $i = 1, \ldots , d$, where $\{e_1, \ldots , e_d\}$ is the standard basis in $\mathfrak {t}^d \cong \mathbb {R}^d$, induces the short exact sequence of Lie algebras

$$\begin{aligned} 0 \longrightarrow \mathfrak {n} \longrightarrow \mathfrak {t}^d \longrightarrow \mathfrak {t}^n \longrightarrow 0. \end{aligned}$$

Now let $T_\mathbb {C}^d$ be the complex torus with the standard action on $\mathbb {C}^d$, and let N be the $(d-n)$-dimensional torus integrating the Lie algebra $\mathfrak {n}$. The complex subtorus $N_\mathbb {C}\subset T_\mathbb {C}^d$ acts effectively on $\mathbb {C}^d$, and there is an open dense orbit stratum $U \subset \mathbb {C}^d$ such that $N_\mathbb {C}$ acts freely. The compact holomorphic toric variety $\mathcal {X}$ of the simplicial fan $\Sigma$ is the quotient

$$\begin{aligned} \mathcal {X} = U / N_\mathbb {C}, \end{aligned}$$

which is equipped with the residual $T_\mathbb {C}^n$-action.

Now, if we consider the real Poisson structure

$$\begin{aligned} \Pi = \sum _{k=1}^d i \overline{z}_k \frac{\partial }{\partial \overline{z}_k} \wedge z_k \frac{\partial }{\partial z_k} \end{aligned}$$

on $\mathbb {C}^d$, which is invariant under the $N_\mathbb {C}$-action, then $\Pi$ descends to a Poisson structure $\pi$ on $\mathcal {X}$. The Poisson manifold $(\mathcal {X}, \pi )$ is what Caine called Poisson toric variety. Caine proved that $\pi$ is generically non-degenerate and has quadratic degeneracy along the submaximal orbits.

Because the simplicial fan $\Sigma$ is due to a Delzant polytope, it means that each n-dimensional cone of $\Sigma$ can be mapped to the standard positive orthant of $\mathfrak {t}^n$ by an automorphism of the integral lattice $\mathbb {Z}^n \subset \mathfrak {t}^n$. If the standard positive orthant of $\mathfrak {t}^n$ is indeed a cone of $\Sigma$, then the Caine Poisson structure $\pi$ takes the form

$$\begin{aligned} \pi = \sum _{k=1}^n i \overline{Z}_k \wedge Z_k, \end{aligned}$$

where $\left\{ Z_1, Z_2, \ldots , Z_n \right\}$ are the holomorphic vector fields on $\mathcal {X}$ that generate the action of $T^n_\mathbb {C}\cong \left( \mathbb {C}^* \right) ^n$.

7.26 Our key observation is that if we carry out a real oriented blow-up along the submaximal orbits of $\mathcal {X}$, then the blow-up space is a smooth manifold with corners $\widetilde{\mathcal {X}}$, and $\pi$ lifts to a Poisson structure $\widetilde{\pi }$ on $\widetilde{\mathcal {X}}$ such that $\widetilde{\omega } = \widetilde{\pi }^{-1}$ is log symplectic and degenerates along the boundary $\partial \widetilde{\mathcal {X}}$. Furthermore, the induced $T^n$-action on $\widetilde{\mathcal {X}}$ is free, proper, and Lagrangian, so by Theorem 6.3, the quotient $X = \widetilde{\mathcal {X}} / T^n$ is equipped with a log affine structure $\xi$ which degenerates along the boundary $\partial X$, and the quotient map

$$\begin{aligned} \mu : \left( \widetilde{\mathcal {X}}, \partial \widetilde{\mathcal {X}}, \widetilde{\omega }\right) \rightarrow (X, \partial X, \xi ). \end{aligned}$$

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is a tropical momentum map as in Definition 4.5. By Proposition 3.11, the log affine manifold $(X, \partial X, \xi )$ defines a labeled simplicial fan, which is simply the labeled simplicial fan $\Sigma$ of the toric variety $\mathcal {X}$.

7.27 Now let $U_\mathcal {X} \cong T_\mathbb {C}^n$ be the open dense orbit of $\mathcal {X}$. We have the natural tropicalization map $\nu : U_\mathcal {X} \rightarrow \mathfrak {t}^n$. Upon choosing a basis of $\mathfrak {t}^n$, the tropicalization map, see, e.g., Definition 1.1 of [22], takes the form

$$\begin{aligned} \nu : U_\mathcal {X} \cong (\mathbb {C}^*)^n \rightarrow \mathfrak {t}^n \cong \mathbb {R}^n, \qquad (z_1, \ldots , z_n) \mapsto (\log |z_1|, \ldots , \log |z_n|). \end{aligned}$$

The extended tropicalization map introduced by Kajiwara [15] and Payne [23] is the natural extension:

$$\begin{aligned} \nu : \mathcal {X} \rightarrow \mathbf {Trop}(\mathcal {X}), \end{aligned}$$

where the codomain $\mathbf {Trop}(\mathcal {X})$, called the tropicalization of $\mathcal {X}$, is a natural compactification of $\mathfrak {t}^n$. From our perspective, if we regard the simplicial fan $\Sigma$ of $\mathcal {X}$ as an admissible decorated cubic complex, then $\mathbf {Trop}(\mathcal {X})$ is nothing but the log affine manifold X, which is constructed from $\Sigma$ by the welding procedure in Proposition 3.29.

We then have the following commutative diagram:

(53)

where $p: \widetilde{\mathcal {X}} \rightarrow \mathcal {X}$ is the blow-down map.

Example 7.28

The complex projective space $\mathbb {C}P^1$ may be thought as the quotient of $\mathbb {C}^2$ by the diagonal $\mathbb {C}^*$-action. The residual anti-diagonal $\mathbb {C}^*$-action renders $\mathbb {C}P^1$ into a toric variety. If we use the homogeneous coordinates $z_1$ and $z_2$ on $\mathbb {C}P^1$ with $z_1 = \frac{1}{z_2}$, then

$$\begin{aligned} \pi = i \overline{z}_1 \frac{\partial }{\partial \overline{z}_1} \wedge z_1 \frac{\partial }{\partial z_1} = i \overline{z}_2 \frac{\partial }{\partial \overline{z}_2} \wedge z_2 \frac{\partial }{\partial z_2}. \end{aligned}$$

Let $\widetilde{\mathcal {X}}$ be the blow-up of $\mathbb {C}P^1$ along the two points $z_1 = 0$ and $z_2 = 0$. We cover $\widetilde{\mathcal {X}}$ by the charts

$$\begin{aligned} (r_k = |z_k|, \theta _k = \arg z_k), \quad r_k \ge 0, \quad \theta _k \in \mathbb {R}/ 2\pi \mathbb {Z}, \quad k = 1, 2, \end{aligned}$$

the maps in the commutative diagram (53) are as follows:

$$\begin{aligned} \begin{array}{lll} &{} p: \widetilde{\mathcal {X}} \rightarrow \mathbb {C}P^1, \qquad (r_k, \theta _k) \mapsto z_k = r_k e^{i\theta _k}, \quad k = 1, 2; \\ &{} \nu : \mathbb {C}P^1 \rightarrow X, \qquad z_k \mapsto r_k = |z_k|, \quad k = 1, 2; \\ &{} \mu : \widetilde{\mathcal {X}} \rightarrow X, \qquad (r_k, \theta _k) \mapsto r_k, \quad k = 1, 2. \end{array} \end{aligned}$$