On Toric Poisson Structures of Type $(1,1)$ and their Cohomology

On Toric Poisson Structures of Type $\boldsymbol{(1,1)}$
and their Cohomology††This paper is a contribution to the Special Issue “Gone
Fishing”. The full collection is available at
http://www.emis.de/journals/SIGMA/gone-fishing2016.html

Arlo CAINE and Berit Nilsen GIVENS A. Caine and B.N. Givens California State
Polytechnic University Pomona,
3801 W. Temple Ave., Pomona, CA, 91768, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received October 29, 2016, in final form March 28, 2017; Published online
April 06, 2017

We classify real Poisson structures on complex toric manifolds of type $(1,1)$
and initiate an investigation of their Poisson cohomology. For smooth toric
varieties, such structures are necessarily algebraic and are homogeneous
quadratic in each of the distinguished holomorphic coordinate charts
determined by the open cones of the associated simplicial fan. As an
approximation to the smooth cohomology problem in each ${\mathbb{C}}^{n}$
chart, we consider the Poisson differential on the complex of polynomial
multi-vector fields. For the algebraic problem, we compute $H^{0}$ and $H^{1}$
under the assumption that the Poisson structure is generically non-degenerate.
The paper concludes with numerical investigations of the higher degree
cohomology groups of $({\mathbb{C}}^{2},\pi_{B})$ for various $B$.

toric; Poisson structures; group-valued momentum map; Poisson cohomology

53D17; 37J15

##  1 Introduction

An almost complex structure $J$ on a smooth manifold $M$ determines a
splitting of the complexified tangent bundle
$TM\otimes_{\mathbb{R}}{\mathbb{C}}=T^{1,0}\oplus T^{0,1}$ into sub-bundles on
which $J$ acts by $i$ and $-i$, respectively. This induces a decomposition

$\displaystyle\textstyle{\bigwedge^{2}}TM\otimes_{\mathbb{R}}{\mathbb{C}}=\big{(}T^{1,0}\wedge
T^{1,0}\big{)}\oplus\big{(}T^{1,0}\wedge
T^{0,1}\big{)}\oplus\big{(}T^{0,1}\wedge T^{0,1}\big{)}$

and the smooth sections of $T^{1,0}\wedge T^{0,1}$ are termed bi-vector fields
on $M$ of type $(1,1)$. Observe that the canonical complex conjugation in the
fibers $TM\otimes_{\mathbb{R}}{\mathbb{C}}$ interchanges $T^{1,0}$ and
$T^{0,1}$. Thus, $T^{1,0}\wedge T^{0,1}$ is stable under the complex
conjugation on $\bigwedge^{2}TM\otimes_{\mathbb{R}}{\mathbb{C}}$ determined by
$X\wedge Y\mapsto\overline{X}\wedge\overline{Y}$ for each $X$, $Y$ in a fiber
of $TM\otimes_{\mathbb{R}}{\mathbb{C}}$. However conjugation maps
$T^{1,0}\wedge T^{1,0}$ to $T^{0,1}\wedge T^{0,1}$ and vice versa. A bi-vector
field on $M$ is real if and only if it is fixed by conjugation. So every real
bi-vector field on $M$ is a sum of a bi-vector field of type $(1,1)$ together
with a section of $T^{1,0}\wedge T^{1,0}$ plus its conjugate. In the context
of this paper, we will be concerned with complex manifolds but focus on real
bi-vector fields of type $(1,1)$, i.e., smooth sections $\pi$ of
$T^{1,0}\wedge T^{0,1}$ such that $\overline{\pi}=\pi$.

Suppose that $M$ is a complex toric manifold, i.e., a smooth complex manifold
on which a complex torus $T_{\mathbb{C}}$ acts holomorphically and effectively
with an open dense orbit $\mathcal{O}$. We write $T_{\mathbb{C}}$ for the
torus, regarding it as the complexification of a real compact torus $T$ in
analogy with ${\mathbb{T}}_{\mathbb{C}}$ representing the non-zero complex
numbers which is the complexification of the group ${\mathbb{T}}$ of complex
numbers of modulus $1$. By a toric Poisson structure on $M$ we mean a Poisson
structure which is invariant under the action of $T_{\mathbb{C}}$. Such
structures could be considered in the smooth or holomorphic category. For
example, ${\mathbb{C}}^{2}$ with complex coordinates $(z,w)$ is a complex
toric manifold acted on by ${\mathbb{T}}_{\mathbb{C}}^{2}$ via independent
dilation of each complex coordinate and one can check that
$\pi=-2izw\partial_{z}\wedge\partial_{w}$ is a holomorphic toric Poisson
structure. Holomorphic Poisson structures are necessarily bi-vector fields of
type $(2,0)$. No bi-vector field of type $(1,1)$ can be holomorphic, nor can
such a field occur as the real projection of holomorphic bi-vector field.

In this paper we will focus on the smooth category and consider structures of
type $(1,1)$ which are generically non-degenerate. For example, ${\mathbb{C}}$
with coordinates $(z,\bar{z})$ is a complex toric manifold acted on by
${\mathbb{T}}_{\mathbb{C}}$ via $\zeta.(z,\bar{z})=(\zeta z,\overline{\zeta
z})$ and $\pi=-2iz\bar{z}\partial_{z}\wedge\partial_{\bar{z}}$ is a smooth
toric Poisson structure. Note that
$\pi=-2iz\bar{z}\partial_{z}\wedge\partial_{\bar{z}}$ is real (fixed by
conjugation) and of type $(1,1)$. In terms of real variables, $z=x+iy$ and
$\partial_{z}=\frac{1}{2}(\partial_{x}-i\partial_{y})$ while
$\partial_{\bar{z}}=\frac{1}{2}(\partial_{x}+i\partial_{y})$ and one can check
that

$\displaystyle\pi=-2iz\bar{z}\partial_{z}\wedge\partial_{\bar{z}}=\big{(}x^{2}+y^{2}\big{)}\partial_{x}\wedge\partial_{y},$
(1.1)

which means that $\pi$ is also a quadratic Poisson structure of elliptic type
on ${\mathbb{R}}^{2}$ in the terminology of [3].

In [1], examples of toric Poisson structures of type $(1,1)$ on smooth toric
varieties generalizing this example were constructed via a quotient
construction. A smooth toric variety $M$ determines and is determined by a
simplicial fan $\Sigma$ in ${\mathfrak{t}}$, the real Lie algebra of the torus
$T$, which is simplicial with respect to the coweight lattice
$\Lambda\subset{\mathfrak{t}}$ of $T_{\mathbb{C}}$. The variety can be
recovered by a quotient ${\mathbb{C}}^{d}//N_{\mathbb{C}}\simeq M$ where the
dimension $d$ and the sub-torus $N_{\mathbb{C}}$ of
${\mathbb{T}}_{\mathbb{C}}^{d}$ are determined from $\Sigma$. The double slash
indicates that one does not divide ${\mathbb{C}}^{d}$ by the action of
$N_{\mathbb{C}}$ but rather a dense open
${\mathbb{T}}_{\mathbb{C}}^{d}$-stable subset $\mathcal{U}_{\Sigma}$,
determined from $\Sigma$, on which $N_{\mathbb{C}}$ acts freely. The product
Poisson structure $\pi\oplus\cdots\oplus\pi$ on ${\mathbb{C}}^{d}$, using
$\pi$ from (1.1), restricts to a real toric Poisson structure on
$\mathcal{U}_{\Sigma}$ of type $(1,1)$ which is generically non-degenerate
and, via the quotient map
$\mathcal{U}_{\Sigma}\to{\mathbb{C}}^{d}//N_{\mathbb{C}}$, coinduces a real
toric Poisson structure of type $(1,1)$ on $M$ which is generically non-
degenerate. Each open cone in $\Sigma$ determines a distinguished holomorphic
coordinate chart ${\mathbb{C}}^{n}$ on $M$ in which the action of
$T_{\mathbb{C}}\simeq{\mathbb{T}}_{\mathbb{C}}^{d}/N_{\mathbb{C}}$ is
equivalent to the action of ${\mathbb{T}}_{\mathbb{C}}^{n}$ on
${\mathbb{C}}^{n}$. In terms of such coordinates $(z_{1},\dots,z_{n})$, the
Poisson structure on $M$ has the form

$\displaystyle-2i\sum_{p,q=1}^{n}B_{pq}z_{p}\bar{z}_{q}\partial_{z_{p}}\wedge\partial_{\bar{z_{q}}}$
(1.2)

for some symmetric positive definite integral matrix $[B_{pq}]$ determined by
the open cone in $\Sigma$.

Let $Z_{1}=z_{1}\partial_{z_{1}},\dots,Z_{n}=z_{n}\partial_{z_{n}}$. Then we
can write (1.2) in the form

$\displaystyle-2i\sum_{p,q=1}^{n}B_{pq}Z_{p}\wedge\overline{Z_{q}}$

making it clear that it is the image of an element of
${\mathfrak{t}}_{\mathbb{C}}\wedge\overline{{\mathfrak{t}}_{\mathbb{C}}}$
under the natural map induced infinitesimally by the holomorphic action of
$T_{\mathbb{C}}$ on $M$. In fact, since ${\mathfrak{t}}_{\mathbb{C}}$ is
abelian and the map from ${\mathfrak{t}}_{\mathbb{C}}$ into sections of
$T^{1,0}$ is a homomorphism of Lie algebras, it is clear that any element of
${\mathfrak{t}}_{\mathbb{C}}\wedge\overline{{\mathfrak{t}}_{\mathbb{C}}}$ maps
to a toric Poisson structure on $M$ of type $(1,1)$. In terms of the
coordinates $(z_{1},\dots,z_{n})$, such a structure has an expression of the
form (1.2) for some matrix of complex numbers $[B_{pq}]$. In order for such a
Poisson structure to be real, it turns out that $[B_{pq}]$ must be Hermitian
and for it to be generically non-degenerate, it must be invertible. Given a
formula such as (1.2) on ${\mathbb{C}}^{d}$, one could repeat the construction
and coinduce a different real Poisson structure on a given toric variety $M$.
From this slightly more general perspective, one can think of the construction
in [1] as simply that which begins with the toric Poisson structure on
${\mathbb{C}}^{d}$ corresponding to the identity matrix.

The purpose of this paper is to address the following natural questions
regarding toric Poisson structures of type $(1,1)$.

  1. 1.

To what extent does this slight modification of the construction in [1]
classify all toric Poisson structures of type $(1,1)$?

  2. 2.

What is the significance, if any, of the rationality of the coefficients
$[B_{pq}]$ of a toric Poisson structure as in (1.2)?

  3. 3.

To what extent does the cohomology of a toric Poisson structure of type
$(1,1)$ depend on the coefficient matrix $[B_{pq}]$ in each chart and the
topology of the toric variety $M$?

In the first section of the paper we classify real toric Poisson structures of
type $(1,1)$ establishing the following theorem.

######  Theorem 1.1.

Suppose that $(M,T_{\mathbb{C}})$ is a complex toric manifold. Then the real
toric Poisson structures of type $(1,1)$ on $M$ are in one-to-one
correspondence with the Hermitian forms on ${\mathfrak{t}}_{\mathbb{C}}^{*}$.
We denote this correspondence $\pi_{B}\leftrightarrow B$. Furthermore, if $M$
is a smooth toric variety with corresponding simplicial fan $\Sigma$ in
${\mathfrak{t}}$, then in each distinguished holomorphic coordinate chart on
$M$ associated to an open cone of the fan for $M$, $\pi_{B}$ has the form
(1.2) where $[B_{pq}]$ is the matrix representing $B$ with respect to the
integral basis spanning the cone.

The formulas for $\pi_{B}$ in each of the distinguished holomorphic coordinate
charts are then given by (1.2) for different matrices $[B_{pq}]$ which are
congruent by elements of $GL(n,{\mathbb{Z}})$. Such structures are generically
non-degenerate if and only if the Hermitian form $B$ is non-degenerate. We
show in the following section that those forms which are rational with respect
to the weight lattice $\Lambda^{*}$ in ${\mathfrak{t}}_{\mathbb{C}}^{*}$ then
have an additional symmetry.

######  Theorem 1.2.

Suppose that $(M,T_{\mathbb{C}})$ is a complex toric manifold and that
$\pi_{B}$ is a generically non-degenerate real toric Poisson structure of type
$(1,1)$ on $M$ associated to a non-degenerate Hermitian form $B$ on
${\mathfrak{t}}_{\mathbb{C}}^{*}$. Then the action of $T_{\mathbb{C}}$ on the
open orbit $\mathcal{O}$ is Hamiltonian with respect to $\pi_{B}$ if and only
if $B$ is rational with respect to the weight lattice
$\Lambda^{*}\subset{\mathfrak{t}}_{\mathbb{C}}^{*}$.

In the remaining sections of the paper, we initiate an investigation of the
cohomology of such structures, focusing on $M={\mathbb{C}}^{2}$ and various
example forms $B$ in an effort to begin addressing the much harder third
question. These examples remove topology from consideration because
${\mathbb{C}}^{2}$ is contractible and focus only on contributions to the
cohomology from the degeneracies of the Poisson tensor. Since toric Poisson
structures are algebraic in the distinguished coordinate charts, we consider
the cohomology problem first in the algebraic category as an approximation to
the smooth problem. The formal cohomology, in this sense, of quadratic Poisson
structures was computed generally in [5] although it takes some work to
interpret and apply the results in this setting. We describe how to use
symbolic computation with Mathematica to compute the cohomology for
multivector fields of low monomial degree. These numerical results confirm the
results of [5] in these cases and suggest that the cohomology of
$({\mathbb{C}}^{2},\pi_{B})$ is quite small for generic $B$ and, in
particular, is finite dimensional. Rather surprisingly, we found an example of
a $({\mathbb{C}}^{2},\pi_{B})$ where $B$ is symmetric positive definite and
integral, yet the cohomology is infinite dimensional, having infinitely many
generators in wedge degrees 3 and 4. This example is the local form of the
toric Poisson structure constructed in [1] on one of the Hirzebruch surfaces.
In general, classes in $H^{0}(M,\pi)$, $H^{1}(M,\pi)$, and $H^{2}(M,\pi)$ have
well-known interpretations, but the interpretation of non-trivial classes of
higher wedge degree is unclear. It is our hope that our examples will be
studied further so that interpretations of higher degree classes can be found.

If $\pi$ is a generically non-degenerate real toric Poisson structure of type
$(1,1)$ on a complex toric manifold $X$, it is easy to see that
$H^{0}(M,\pi)={\mathbb{C}}\langle 1\rangle$ if $X$ is connected. It is a more
subtle fact that ${\mathfrak{t}}_{\mathbb{C}}$ includes in $H^{1}(M,\pi)$ as a
real subspace if the fan for $M$ is closed in ${\mathfrak{t}}$. The main
result of the later sections is that there is nothing more in
$H^{1}({\mathbb{C}}^{n},\pi_{B})$ when $B$ is non-degenerate, at least in the
algebraic category.

######  Theorem 1.3.

If $[B_{pq}]$ is an invertible $n\times n$ Hermitian matrix, then for the
algebraic Poisson cohomology, the inclusion of ${\mathfrak{t}}_{\mathbb{C}}$
into $H^{1}({\mathbb{C}}^{n},\pi_{B})$ is an isomorphism

$\displaystyle
H^{1}({\mathbb{C}}^{n},\pi_{B})\simeq{\mathfrak{t}}_{\mathbb{C}}$

of real vector spaces.

In a forthcoming paper, we will use representation theory to completely
determine the Poisson cohomology of $({\mathbb{C}}^{n},\pi_{B})$, reducing the
determination of generators for the cohomology to the solution of certain
Diophantine inequalities depending on the matrix $B$. This work will
reinterpret the computations of [5].

### Conventions

Throughout the paper ${\mathbb{T}}$ denotes the Lie group of complex numbers
of modulus one. The Lie algebra of ${\mathbb{T}}$ will be identified with
$i{\mathbb{R}}$ via the map $i{\mathbb{R}}\to{\mathbb{T}}$ given by $s\mapsto
e^{s}$. The kernel of this group homomorphism is the lattice $\tau
i{\mathbb{Z}}$ in $i{\mathbb{R}}$ where $\tau$ denotes the circumference of
the unit circle. We use $\tau$ instead of $2\pi$ to avoid a clash of notation
when this number and a Poisson structure labeled $\pi$ need to appear in the
same formula. By taking products, this induces an identification of the Lie
algebra of ${\mathbb{T}}^{d}$ with $(i{\mathbb{R}})^{d}$. The kernel of the
product map $(i{\mathbb{R}})^{d}\to{\mathbb{T}}^{d}$ is the lattice $(\tau
i{\mathbb{Z}})^{d}\subset(i{\mathbb{R}})^{d}$. We will write
$e_{1},e_{2},\dots,e_{d}$ for the standard basis of ${\mathbb{Z}}^{d}$ so that
$\tau ie_{1},\dots,\tau ie_{d}$ generate $(\tau i{\mathbb{Z}})^{d}$. Writing
${\mathbb{T}}_{\mathbb{C}}$ for the complex Lie group of non-zero complex
numbers and similarly identifying the Lie algebra of
${\mathbb{T}}_{\mathbb{C}}$ with ${\mathbb{C}}$ via the map
${\mathbb{C}}\to{\mathbb{T}}_{\mathbb{C}}$ given by $s\mapsto e^{s}$, we thus
extend the identification of the real Lie subalgebra of ${\mathbb{T}}$ with
the real subspace $i{\mathbb{R}}$ of ${\mathbb{C}}$.

Let $T$ denote a compact connected abelian Lie group of real dimension $n$,
i.e., a compact torus, and let $T_{\mathbb{C}}$ denote its complexification.
The kernel of the group homomorphism ${\mathfrak{t}}_{\mathbb{C}}\to
T_{\mathbb{C}}$ defined by $\xi\mapsto\exp(\xi)$ is the coweight lattice
$\Lambda\subset{\mathfrak{t}}\subset{\mathfrak{t}}_{\mathbb{C}}$. If we choose
a basis $\xi_{1},\dots,\xi_{n}$ for $\Lambda$ and define a
${\mathbb{Z}}$-linear map $(\tau i{\mathbb{Z}})^{n}\to\Lambda$ by $\tau
ie_{k}\mapsto\xi_{k}$, then by extending to a ${\mathbb{C}}$-linear map
${\mathbb{C}}^{n}\to{\mathfrak{t}}_{\mathbb{C}}$ we obtain a commutative
diagram

$\displaystyle\begin{matrix}0&\longrightarrow&(\tau
i{\mathbb{Z}})^{n}&\longrightarrow&{\mathbb{C}}^{n}&\longrightarrow&({\mathbb{T}}_{\mathbb{C}})^{n}&\longrightarrow&1\\\
&&\big{\downarrow}&&\big{\downarrow}&&&&\\\
0&\longrightarrow&\Lambda&\longrightarrow&{\mathfrak{t}}_{\mathbb{C}}&\longrightarrow&T_{\mathbb{C}}&\longrightarrow&1\end{matrix}$

with exact rows and in which the two vertical arrows are isomorphisms. This
induces an isomorphism $({\mathbb{T}}_{\mathbb{C}})^{n}\to T_{\mathbb{C}}$
making the diagram commute. If
$(z_{1},\dots,z_{n})=(e^{s_{1}},\dots,e^{s_{n}})$ in
$({\mathbb{T}}_{\mathbb{C}})^{n}$ then $(z_{1},\dots,z_{n})$ corresponds to

$\displaystyle\exp\left(\frac{s_{1}}{\tau i}\xi_{1}+\dots+\frac{s_{n}}{\tau
i}\xi_{n}\right)=\exp\left(\frac{s_{1}}{\tau
i}\xi_{1}\right)\cdots\exp\left(\frac{s_{n}}{\tau i}\xi_{n}\right)$

in $T_{\mathbb{C}}$.

######  Note 1.4.

For $z\not=0$ in ${\mathbb{C}}$ and $\xi\in\Lambda$, we write $z^{\xi}$ for
$\exp(\frac{1}{\tau i}\log(z)\xi)\in T_{\mathbb{C}}$. This is well defined
because $\exp(k\xi)=1$ for each $k\in{\mathbb{Z}}$. Thus, for a choice of
basis $\xi_{1},\dots,\xi_{n}$ of $\Lambda$, we obtain an isomorphism
$(T_{\mathbb{C}})^{n}\to T_{\mathbb{C}}$ given by $(z_{1},\dots,z_{n})\mapsto
z_{1}^{\xi_{1}}\cdots z_{n}^{\xi_{n}}$.

##  2 Real toric Poisson structures of type $\boldsymbol{(1,1)}$

Let $(M,T_{\mathbb{C}})$ be a complex toric manifold and choose a
${\mathbb{C}}$-basis $\xi_{1},\dots,\xi_{n}$ for
${\mathfrak{t}}_{\mathbb{C}}$. Then through the holomorphic action of
$T_{\mathbb{C}}$ we obtain corresponding holomorphic
$T_{\mathbb{C}}$-invariant vector fields $X_{1},\dots,X_{n}$ on $M$, i.e.,
holomorphic sections of $T^{1,0}$, which are $T_{\mathbb{C}}$-invariant. Given
a Hermitian matrix of complex numbers, one can use this data to construct a
real toric Poisson structure on $M$ of type $(1,1)$.

######  Lemma 2.1.

If $B=[B_{pq}]$ is a Hermitian matrix of complex numbers, then

$\displaystyle\pi_{B}=2i\sum_{p,q=1}^{n}B_{pq}X_{p}\wedge\overline{X_{q}}$
(2.1)

is a real toric Poisson structure on $M$ of type $(1,1)$.

###### Proof.

By construction, $\pi_{B}$ is a $T_{\mathbb{C}}$-invariant bi-vector field on
$M$ on type $(1,1)$. Since

$\displaystyle\overline{\pi_{B}}=\overline{2i\sum_{p,q=1}^{n}B_{pq}X_{p}\wedge\overline{X_{q}}}=-2i\sum_{p,q=1}^{n}\overline{B_{pq}}\,\overline{X_{p}}\wedge
X_{q}$
$\displaystyle\hphantom{\overline{\pi_{B}}}{}=2i\sum_{p,q=1}^{n}\overline{B_{pq}}\,X_{q}\wedge\overline{X_{p}}=2i\sum_{p,q=1}^{n}\overline{B_{qp}}\,X_{p}\wedge\overline{X_{q}}$

we see that $\pi_{B}$ is real ($\overline{\pi_{B}}=\pi_{B}$) because
$\overline{B_{qp}}=B_{pq}$ for each $p,q=1,\dots,n$. The holomorphic action of
$T_{\mathbb{C}}$ on $M$ induces a homomorphism of Lie algebras
${\mathfrak{t}}_{\mathbb{C}}\to T^{1,0}$ and this carries $\xi_{p}\to X_{p}$
for each $p$ by definition. Thus, $[X_{p},X_{p^{\prime}}]=0$ for each
$p,p^{\prime}=1,2,\dots,n$. Since conjugation preserves brackets, we know that
$[\overline{X_{q}},\overline{X_{q^{\prime}}}]=0$ for all
$q,q^{\prime}=1,2,\dots,n$. Recalling that holomorphic derivations and anti-
holomorphic derivations commute, we find that $[X_{p},\overline{X_{q}}]=0$ for
all $p,q=1,2,\dots,n$. Thus $\pi_{B}$ is a linear combination of wedge
products of commuting vector fields and hence $[\pi_{B},\pi_{B}]=0$ by
linearity and the graded Leibniz rule for the Schouten bracket. ∎

The next result shows that every real toric Poisson structure of type $(1,1)$
on $M$ is of the form $\pi_{B}$. The representation (2.1) depends on the basis
$\xi_{1},\dots,\xi_{n}$ which determines both the holomorphic vector fields
$X_{1},\dots,X_{n}$ and, as we will see, the matrix $[B_{pq}]$.

######  Theorem 2.2.

Suppose that $\pi$ is a real toric Poisson structure on $M$ of type $(1,1)$.
For each ${\mathbb{C}}$-basis $\xi_{1},\dots,\xi_{n}$ of
${\mathfrak{t}}_{\mathbb{C}}$ there exists an $n\times n$ Hermitian matrix
$[B_{pq}]$ such that $\pi=\pi_{B}$ as in (2.1).

###### Proof.

The key point is that a ${\mathbb{C}}$-basis $\xi_{1},\dots,\xi_{n}$ for
${\mathfrak{t}}_{\mathbb{C}}$ determines a frame $X_{1},\dots,X_{n}$ for
$T^{1,0}$ over the open dense $T_{\mathbb{C}}$-orbit $\mathcal{O}$. This is
because the action of $T_{\mathbb{C}}$ is free and transitive on
$\mathcal{O}$. By conjugation, $\overline{X_{1}},\dots,\overline{X_{n}}$ is
then a frame for $T^{0,1}$ over $\mathcal{O}$. Consequently, the bi-vector
fields $X_{p}\wedge\overline{X_{q}}$ for $p,q=1,2,\dots,n$ form a frame for
$T^{1,0}\wedge T^{0,1}$ over $\mathcal{O}$. Thus, there exist smooth complex
valued functions $B_{pq}$ on $\mathcal{O}$ such that

$\displaystyle\pi=2i\sum_{p,q=1}^{n}B_{pq}X_{p}\wedge\overline{X_{q}}$ (2.2)

on $\mathcal{O}$. As we saw in the proof of Lemma 2.1, (2.2) implies that

$\displaystyle\overline{\pi}=2i\sum_{p,q=1}^{n}\overline{B_{qp}}\,X_{p}\wedge\overline{X_{q}}$

on $\mathcal{O}$. Since $\pi$ is real, we can match coefficients to conclude
that $B_{pq}=\overline{B_{qp}}$ for each $p,q=1,\dots,n$, so that $[B_{pq}]$
is a Hermitian matrix of functions. It remains to show that these functions
are constant.

If $\pi$ is also $T_{\mathbb{C}}$-invariant, then the Schouten bracket
$[X_{r},\pi]=0$ for each $r=1,\dots,n$. Observe that

$\displaystyle[X_{r},B_{pq}X_{p}\wedge\overline{X_{q}}]=X_{r}(B_{pq})X_{p}\wedge\overline{X_{q}}+B_{pq}[X_{r},X_{p}]\wedge\overline{X_{q}}+B_{pq}X_{p}\wedge[X_{r},\overline{X_{q}}]$
$\displaystyle\hphantom{[X_{r},B_{pq}X_{p}\wedge\overline{X_{q}}]}{}=X_{r}(B_{pq})X_{p}\wedge\overline{X_{q}}$

because the Schouten bracket is a graded derivation. Thus, $[X_{r},\pi]=0$ on
${\mathbb{T}}_{\mathbb{C}}^{n}$ for each $r=1,\dots,n$ if and only if
$X_{r}(B_{pq})=0$ on $\mathcal{O}$ for each $p,q,r=1,\dots,n$. Hence, each
$B_{pq}$ is anti-holomorphic on $\mathcal{O}$. Likewise, the fact that
$[\overline{X_{r}},\pi]=0$ for all $r=1,\dots,n$ can be used to argue that
$B_{pq}$ is also holomorphic on $\mathcal{O}$. Therefore, each $B_{pq}$ is
constant on $\mathcal{O}$. But since the vector fields $X_{p}$ and
$\overline{X_{q}}$ are smooth on $M$ and the functions $B_{pq}$ are constant
on the open dense set $\mathcal{O}$, the representation (2.2) is actually
valid on all of $M$. Hence $\pi=\pi_{B}$ on $M$ as desired. ∎

######  Proposition 2.3.

If $\pi=\pi_{B}$ is a real toric Poisson structure of type $(1,1)$ on $M$
which is generically non-degenerate, then $[B_{pq}]$ is invertible.

###### Proof.

Since $\pi_{B}$ is $T_{\mathbb{C}}$-invariant,

$\displaystyle\pi_{B}^{n}=(2i)^{n}\det([B_{pq}])X_{1}\wedge\dots\wedge
X_{n}\wedge\overline{X_{1}}\wedge\dots\wedge\overline{X_{n}}$

is $T_{\mathbb{C}}$-invariant. Since $X_{1}\wedge\dots\wedge
X_{n}\wedge\overline{X_{1}}\wedge\dots\wedge\overline{X_{n}}$ is non-zero on
$\mathcal{O}$ we conclude that $\det([B_{pq}])\not=0$ on $\mathcal{O}$ if
$\pi$ is generically non-degenerate. Hence $[B_{pq}]$ is invertible. ∎

######  Remark 2.4.

By applying Theorem 2.2 for a different choice of basis
$\xi_{1}^{\prime},\dots,\xi_{n}^{\prime}$ of ${\mathfrak{t}}_{\mathbb{C}}$,
one obtains another Hermitian matrix $[B_{pq}^{\prime}]$ such that
$\pi=\pi_{B^{\prime}}$. One can check that the matrices $[B_{pq}]$ and
$[B_{pq}^{\prime}]$ are congruent via the change of basis matrix from
$\xi_{1},\dots,\xi_{n}$ to $\xi_{1}^{\prime},\dots,\xi_{n}^{\prime}$. Thus,
these matrices represent the same Hermitian form on
${\mathfrak{t}}_{\mathbb{C}}^{*}$. The first half of Theorem 1.1 from the
introduction has now been established. The real toric Poisson structures on
$M$ of type $(1,1)$ are in one-to-one correspondence with the Hermitian forms
on ${\mathfrak{t}}_{\mathbb{C}}^{*}$.

If $M$ is a smooth compact toric variety with associated simplicial fan
$\Sigma$ in ${\mathfrak{t}}$, then we obtain a distinguished family of
holomorphic coordinate charts on $M$, one for each open cone in $\Sigma$. Such
a cone is spanned by a ${\mathbb{Z}}$-basis $\xi_{1},\dots,\xi_{n}$ of the
coweight lattice $\Lambda$ in ${\mathfrak{t}}$ and with this basis we obtain
an isomorphism ${\mathbb{T}}_{\mathbb{C}}^{n}\to T_{\mathbb{C}}$ given by
$(z_{1},\dots,z_{n})\mapsto z_{1}^{\xi_{1}}\dots z_{n}^{\xi_{n}}$ as in Note
1.4. By choosing a point $x\in\mathcal{O}$, we obtain a map
${\mathbb{T}}_{\mathbb{C}}^{n}\to T_{\mathbb{C}}.x=\mathcal{O}$ and
holomorphic coordinates on $\mathcal{O}$. In terms of these coordinates, the
holomorphic vector fields $X_{1},\dots,X_{n}$ associated to
$\xi_{1},\dots,\xi_{n}$ have the form $X_{p}=z_{p}\partial_{z_{p}}$. Thus

$\displaystyle\pi=i\sum_{p,q=1}^{n}B_{pq}z_{p}\bar{z}_{q}\partial_{z_{p}}\wedge\partial_{\bar{z}_{q}}$
(2.3)

in these coordinates.

In [1], local forms such as these were obtained for Poisson structures on
smooth toric varieties which were coinduced from a product Poisson structure
on ${\mathbb{C}}^{d}$ via a quotient construction. There, however, the
matrices $[B_{pq}]$ were also integral and thus associated to integral
Hermitian forms on ${\mathfrak{t}}_{\mathbb{C}}^{*}$. We will next show that
toric Poisson structures of type $(1,1)$ associated to non-degenerate
Hermitian forms which are rational with respect to $\Lambda$ are precisely
those for which the action of $T_{\mathbb{C}}$ on $\mathcal{O}$ is
Hamiltonian.

The sense in which we will consider the action to be Hamiltonian is the
following from [4]. Regarding $T_{\mathbb{C}}$ as a Poisson Lie group with the
trivial Poisson Lie group structure, we can think of the action
$(\mathcal{O},\pi_{B})\times(T_{\mathbb{C}},0)\to(\mathcal{O},\pi_{B})$ as a
Poisson action and search for a momentum mapping for the action taking values
in the dual Poisson Lie group. However, we must specify which model of the
dual group we are considering. Since $T_{\mathbb{C}}$ is abelian, the simply
connected model for the dual group is simply the vector space
${\mathfrak{t}}_{\mathbb{C}}^{*}$. But an alternative is the quotient
${\mathfrak{t}}_{\mathbb{C}}^{*}/\Lambda^{*}$ which is isomorphic to
$T_{\mathbb{C}}$. With this model, we can regard $(T_{\mathbb{C}},0)$ as a
self-dual Poisson Lie group. Our final goal in this section is to classify
which $\pi_{B}$ admit such a momentum map with values in $T_{\mathbb{C}}$ or
some finite quotient of $T_{\mathbb{C}}$.

Suppose that $(N,\pi)\times(G,\pi_{G})\to(N,\pi)$ is a Poisson action of a
Poisson Lie group $(G,\pi_{G})$ on a Poisson manifold $(N,\pi)$. Let
$\mu\colon N\to G^{*}$ be a smooth map from $(N,\pi)$ to a dual group $G^{*}$
of $(G,\pi_{G})$. Recall that $\mu$ is a $G^{*}$-valued _momentum map_ [4] if
for each $\xi\in{\mathfrak{g}}$, the Lie algebra of $G$,

$\displaystyle X_{\xi}=\pi^{\\#}(\mu^{*}(\theta_{\xi})),$ (2.4)

where $X_{\xi}$ is the vector field on $N$ generated by the infinitesimal
action of $\xi$, $\theta_{\xi}$ is the right-invariant 1-form on $G^{*}$
generated by $\xi\in{\mathfrak{g}}=(T_{e}G^{*})^{*}$, and $\mu^{*}\colon
T^{*}G^{*}\to T^{*}N$ is the cotangent lift of $\mu\colon N\to G^{*}$. We will
say that a Poisson action of $(G,\pi_{G})$ on $(N,\pi)$ is $G^{*}$-Hamiltonian
if there exists a $G^{*}$-valued momentum map for the action.

######  Theorem 2.5.

Suppose $\pi_{B}$ is a real toric Poisson structure on $M$ of type $(1,1)$
associated to a non-degenerate Hermitian form $B$ on
${\mathfrak{t}}_{\mathbb{C}}^{*}$. Then the action of $T_{\mathbb{C}}$ on
$\mathcal{O}$ is $T_{\mathbb{C}}$-Hamiltonian if and only if $B$ is rational
with respect to $\Lambda^{*}$ in ${\mathfrak{t}}_{\mathbb{C}}^{*}$.

###### Proof.

We will compute in the global coordinates $(z_{1},\dots,z_{n})$ for
$\mathcal{O}$ associated to an integral basis $\xi_{1},\dots,\xi_{n}$ for
$\Lambda$. This identifies $\mathcal{O}$ with ${\mathbb{T}}_{\mathbb{C}}^{n}$
and ensures that $\pi_{B}$ has the form (2.3) where $[B_{pq}]$ is the matrix
representing $B$ in terms of that basis. Furthermore, we can regard
$T_{\mathbb{C}}$ as ${\mathbb{T}}_{\mathbb{C}}^{n}$ acting on
${\mathbb{T}}_{\mathbb{C}}^{n}$ by multiplication. So, to specify a
$G^{*}=T_{\mathbb{C}}$-valued momentum map on $\mathcal{O}$ is the same as
specifying a map
$\mu\colon{\mathbb{T}}_{\mathbb{C}}^{n}\to{\mathbb{T}}_{\mathbb{C}}^{n}$
satisfying the differential condition (2.4). To proceed, we will classify the
local solutions $\mu$ to (2.4) and show that these local solutions can be
extended globally to ${\mathbb{T}}_{\mathbb{C}}^{n}$ if and only if $[B_{pq}]$
is a rational matrix.

If $\xi=x_{1}e_{1}+\dots+x_{n}e_{n}$ and $\eta=y_{1}e_{1}+\dots+y_{n}e_{n}$ in
${\mathbb{C}}^{n}$, let
$\langle\xi,\eta\rangle=\sum\limits_{k=1}^{n}\bar{x}_{k}y_{k}$ so that
$\langle\cdot,\cdot\rangle$ is the standard Hermitian inner product on
${\mathbb{C}}^{n}$ which is conjugate linear in its first argument. We will
identify the real dual of ${\mathbb{C}}^{n}$ with ${\mathbb{C}}^{n}$ using
$\mathrm{Im}\langle\cdot,\cdot\rangle$. Then $\xi\in{\mathbb{C}}^{n}$ with
$\xi=\sum\limits_{k=1}^{n}x_{k}e_{k}$ generates a holomorphic vector field
$X_{\xi}$ on $\mathcal{O}$ and the invariant 1-form

$\displaystyle\theta_{\xi}=\operatorname{Re}\left(\sum_{k=1}^{n}\overline{ix}_{k}\frac{dw_{k}}{w_{k}}\right)$

on the dual group $G^{*}={\mathbb{T}}_{\mathbb{C}}^{n}$ with coordinates
$(w_{1},\dots,w_{n})$. To verify that this formula for $\theta_{\xi}$ is
correct, observe that $\theta_{\xi}$ is invariant under multiplication by
$(\zeta_{1},\dots,\zeta_{n})\in{\mathbb{T}}_{\mathbb{C}}^{n}$ and restricts to
the form $\operatorname{Re}\langle
i\xi,\cdot\rangle=\mathrm{Im}\langle\xi,\cdot\rangle$ at the identity.

Let $[B^{pq}]$ denote the matrix inverse to $[B_{pq}]$, so that
$[\overline{B^{pq}}]$ is the matrix representing the conjugate Hermitian form
dual to $B$ on ${\mathfrak{t}}_{\mathbb{C}}^{*}$. We set

$\displaystyle\mu(z_{1},\dots,z_{n})=\left(z_{1}^{\overline{B^{11}}}\cdots
z_{n}^{\overline{B^{n1}}},\dots,z_{1}^{\overline{B^{1n}}}\cdots
z_{n}^{\overline{B^{nn}}}\right)=(w_{1},\dots,w_{n})$

and obtain a multi-valued function on $M={\mathbb{T}}_{\mathbb{C}}^{n}$ with
values in ${\mathbb{T}}_{\mathbb{C}}^{n}$. We will show that $\mu$ is the
unique solution to (2.4) satisfying $\mu(1,\dots,1)=(1,\dots,1)$. By
inspection, we see that $\mu$ can be interpreted as a single-valued function
on $M={\mathbb{T}}_{\mathbb{C}}^{n}$ with values in a finite quotient of
$T_{\mathbb{C}}$ if and only if each $\overline{B^{pq}}$ is rational. But this
of course requires that each $B_{pq}$ be rational as well.

Observe that

$\displaystyle\mu^{*}\left(\frac{dw_{k}}{w_{k}}\right)=\frac{d\big{(}z_{1}^{\overline{B^{1k}}}\cdots
z_{n}^{\overline{B^{nk}}}\big{)}}{z_{1}^{\overline{B^{k1}}}\cdots
z_{n}^{\overline{B^{kn}}}}=\sum_{j=1}^{n}\overline{B^{jk}}\frac{dz_{j}}{z_{j}}$

for each $k=1,2,\dots,n$. Thus,

$\displaystyle\mu^{*}(\theta_{\xi})=\operatorname{Re}\left(\sum_{k=1}^{n}\overline{ix_{k}}\left(\sum_{j=1}^{n}\overline{B^{jk}}\frac{dz_{j}}{z_{j}}\right)\right)=\operatorname{Re}\left(\sum_{j,k=1}^{n}\overline{ix_{k}}B^{kj}\frac{dz_{j}}{z_{j}}\right)$

because $[B^{jk}]$ is Hermitian. Applying $\pi_{B}^{\\#}$ from (2.3), we
obtain

$\displaystyle\pi_{B}^{\\#}(\mu^{*}(\theta_{\xi}))=\pi_{B}^{\\#}\left(\operatorname{Re}\left(\sum_{j,k=1}^{n}\overline{ix_{k}}B^{kj}\frac{dz_{j}}{z_{j}}\right)\right)=\operatorname{Re}\left(\pi_{B}^{\\#}\left(\sum_{j,k=1}^{n}\overline{ix_{k}}B^{kj}\frac{dz_{j}}{z_{j}}\right)\right)$
(2.5)

since $\pi_{B}^{\\#}$ is real. Now,

$\displaystyle\pi_{B}^{\\#}\left(\sum_{j,k=1}^{n}\overline{ix_{k}}B^{kj}\frac{dz_{j}}{z_{j}}\right)=2\sum_{p,q=1}^{n}\sum_{j,k=1}^{n}B_{pq}B^{kj}\frac{\bar{x}_{k}z_{p}\bar{z}_{q}}{z_{j}}(\partial_{z_{p}}\wedge\partial_{\bar{z}_{q}})^{\\#}(dz_{j})$
$\displaystyle\hphantom{\pi_{B}^{\\#}\left(\sum_{j,k=1}^{n}\overline{ix_{k}}B^{kj}\frac{dz_{j}}{z_{j}}\right)}{}=2\sum_{p,q=1}^{n}\sum_{j,k=1}^{n}B_{pq}B^{kj}\frac{\bar{x}_{k}z_{p}\bar{z}_{q}}{z_{j}}\delta_{pj}\partial_{\bar{z}_{q}}$
$\displaystyle\hphantom{\pi_{B}^{\\#}\left(\sum_{j,k=1}^{n}\overline{ix_{k}}B^{kj}\frac{dz_{j}}{z_{j}}\right)}{}=2\sum_{p,q=1}^{n}\sum_{k=1}^{n}B_{pq}B^{kp}\bar{x}_{k}\bar{z}_{q}\partial_{\bar{z}_{q}}=2\sum_{k=1}^{n}\bar{x}_{k}\bar{z}_{k}\partial_{\bar{z}_{k}}$
(2.6)

since $[B^{kp}][B_{pq}]=[\delta_{p}^{k}]$. Combining (2.5) and (2.6), we have

$\displaystyle\pi_{B}^{\\#}(\mu^{*}(\theta_{\xi}))=\operatorname{Re}\left(2\sum_{k=1}^{n}\bar{x}_{k}\bar{z}_{k}\partial_{\bar{z}_{k}}\right)=\sum_{k=1}^{n}x_{k}z_{k}\partial_{z_{k}}+\text{c.c.}=2\operatorname{Re}(X_{\xi})$

as required.

Thus, $\mu$ is locally a solution to (2.4). All other solutions differ from
$\mu$ by post-multiplication by an element $(\zeta_{1},\dots,\zeta_{n})$ of
${\mathbb{T}}_{\mathbb{C}}^{n}$ which ensures that
$(1,\dots,1)\mapsto(\zeta_{1},\dots,\zeta_{n})$. This completes the proof. ∎

######  Example 2.6.

For $\pi_{b}=-2ibz\bar{z}\partial_{z}\wedge\partial_{\bar{z}}$ on
${\mathbb{T}}_{\mathbb{C}}$, $b\in{\mathbb{R}}\setminus\\{0\\}$, the map
$\mu(z)=z^{\frac{1}{b}}$ satisfies the differential condition (2.4) with
respect to the infinitesimal action of ${\mathbb{T}}_{\mathbb{C}}$ on itself.
However, $\mu$ is single-valued only if $\frac{1}{b}\in{\mathbb{Z}}$. For
general $b=\frac{m}{n}\in\mathbb{Q}$, this can be remedied by replacing the
codomain ${\mathbb{T}}_{\mathbb{C}}$ of $\mu$ with a finite quotient of
${\mathbb{T}}_{\mathbb{C}}$. Since each of those quotients are isomorphic to
${\mathbb{T}}_{\mathbb{C}}$, this does not substantially change the model for
the dual group.

##  3 Algebraic Poisson cohomology

In complex dimension $n=1$, there are three examples of complex toric
manifolds, namely ${\mathbb{T}}_{\mathbb{C}}$,
${\mathbb{C}}={\mathbb{T}}_{\mathbb{C}}\cup\\{0\\}$, or
${\mathbb{C}}P^{1}={\mathbb{C}}\cup\\{\infty\\}$. Applying Theorem 1.1, the
real toric Poisson structures of type $(1,1)$ have the form

$\displaystyle\pi_{b}=-2ib\,z\bar{z}\,\partial_{z}\wedge\partial_{\bar{z}}$
(3.1)

for some $b\in{\mathbb{R}}$ with $b\not=0$ in terms of the holomorphic
coordinate on ${\mathbb{T}}_{\mathbb{C}}$. In terms of the underlying real
variables $z=x+iy$, this Poisson structure has the expression

$\displaystyle\pi_{b}=b\big{(}x^{2}+y^{2}\big{)}\partial_{x}\wedge\partial_{y}.$

In the example $M={\mathbb{C}}$, it is therefore a quadratic plane Poisson
structure of elliptic type. On
${\mathbb{R}}^{2}\setminus\\{(0,0)\\}={\mathbb{T}}_{\mathbb{C}}$ it is non-
degenerate and so its Poisson cohomology is isomorphic to the de Rham
cohomology of ${\mathbb{R}}^{2}\setminus\\{(0,0)\\}$. In [7], Nakanishi
performed a careful analysis of the partial differential equations encoded in
the condition $\sigma(Y)=0$, where $\sigma$ is the Poisson differential for
$\pi_{b}$ and $Y$ is a smooth multivector field on ${\mathbb{R}}^{2}$, and
through that analysis determined the Poisson cohomology of $\pi_{b}$ on
${\mathbb{C}}$. Since the expression (3.1) is invariant under the change of
variables $z\mapsto 1/z$, a Mayer–Vietoris argument can then be applied to
compute the Poisson cohomology of $\pi_{b}$ on ${\mathbb{C}}P^{1}$ using
Nakanishi’s result on ${\mathbb{T}}_{\mathbb{C}}\cup\\{0\\}={\mathbb{C}}$ and
${\mathbb{T}}_{\mathbb{C}}\cup\\{\infty\\}\simeq{\mathbb{C}}$. This was
observed in [1]. In each case, the result is independent of $b$, but is very
different depending on whether the domain is ${\mathbb{T}}_{\mathbb{C}}$,
${\mathbb{C}}$, or ${\mathbb{C}}P^{1}$. This is because Poisson cohomology is
affected both by the topology of $M$ and the degeneracy of the Poisson tensor.
Rather surprisingly, in each of these cases the Poisson cohomology was found
to be finite dimensional and, perhaps more surprisingly, it was possible to
find algebraic representatives for the Poisson cohomology classes.

As an approximation to the smooth cohomology problem, we consider the
algebraic cohomology of $\pi_{B}$ on ${\mathbb{C}}^{n}$ for more general $n$.
With the algebraic cohomology in hand, one might hope to compute the smooth
cohomology by taking some sort of limit. Then for smooth toric varieties,
which have a distinguished finite atlas of affine charts in which any toric
Poisson structure of type $(1,1)$ has the form (2.3), one could hope to
assemble the global smooth cohomology from the local results using a
Mayer–Vietoris argument.

The algebraic Poisson cohomology of $({\mathbb{C}}^{n},\pi_{B})$ is the
cohomology of the differential graded algebra
$\mathcal{V}^{\mathbb{R}}={\mathbb{R}}[x_{1},y_{1},\dots,x_{n},y_{n}]\otimes\bigwedge{\mathbb{R}}\langle\partial_{x_{1}},\partial_{y_{1}},\dots,\partial_{x_{n}},\partial_{y_{n}}\rangle$,
the exterior algebra of multi-vector fields on ${\mathbb{C}}^{n}$ (viewed as
${\mathbb{R}}^{2n}$) with polynomial coefficients. The differential
$\sigma\colon\mathcal{V}^{\mathbb{R}}\to\mathcal{V}^{\mathbb{R}}$ is given by
$\sigma(Y)=[Y,\pi_{B}]$, where $[\cdot,\cdot]$ is the Schouten bracket ([8]).
Since $\pi_{B}$ is homogeneous quadratic, $\sigma$ on the space of smooth
multi-vector fields leaves invariant the subspace of multi-vector fields with
polynomial coefficients. It will be convenient to complexify the problem so
that we can work with complex coordinates $z_{k}$ and $\bar{z}_{k}$ instead of
the real coordinates $x_{k}$ and $y_{k}$. Let

$\displaystyle\mathcal{V}=\mathcal{V}^{\mathbb{R}}\otimes_{\mathbb{R}}{\mathbb{C}}\simeq{\mathbb{C}}[x_{1},y_{1},\dots,x_{n},y_{n}]\otimes\bigwedge{\mathbb{C}}\langle\partial_{x_{1}},\partial_{y_{1}},\dots,\partial_{x_{n}},\partial_{y_{n}}\rangle.$

Then the assignments $z_{k}\mapsto x_{k}+iy_{k}$, $\bar{z}_{k}\mapsto
x_{k}-iy_{k}$,
$\partial_{z_{k}}\mapsto\frac{1}{2}(\partial_{x_{k}}-i\partial_{y_{k}})$, and
$\partial_{\bar{z}_{k}}\mapsto\frac{1}{2}(\partial_{x_{k}}+i\partial_{y_{k}})$
determine an isomorphism

$\displaystyle\mathcal{V}\simeq{\mathbb{C}}[z_{1},\dots,z_{n},\bar{z}_{1},\dots,\bar{z}_{n}]\otimes\bigwedge{\mathbb{C}}\langle\partial_{z_{1}},\dots,\partial_{z_{n}},\partial_{\bar{z}_{1}},\dots,\partial_{\bar{z}_{n}}\rangle,$

which carries the real coordinate expression for $\pi_{B}$ to the complex
coordinate expression for $\pi_{B}$. But then, at this algebraic level,
computing the real algebraic cohomology of $({\mathbb{C}}^{n},\pi_{B})$ with
complex coefficients is the same computing the complex algebraic cohomology of
$({\mathbb{C}}^{2n},\pi_{B})$ where

$\displaystyle\pi_{B}=-2i\sum_{p,q=1}^{n}B_{pq}z_{p}w_{q}\partial_{z_{p}}\wedge\partial_{w_{q}}$
(3.2)

in terms of the complex linear coordinates
$(z_{1},\dots,z_{n},w_{1},\dots,w_{n})$. Indeed, the assignments
$w_{k}\mapsto\bar{z}_{k}$ determine an isomorphism of DGAs from the complex
algebraic complex for $({\mathbb{C}}^{2n},\pi_{B})$ to the real algebraic
complex for $({\mathbb{C}}^{n},\pi_{B})$ but with complex coefficients.

Since the Poisson differential is linear in its dependence on the Poisson
structure, modifying the Poisson structure by an overall non-zero scalar
factor does not affect the cohomology. So the factor of $-2i$ in (3.2) may be
ignored. Thus, we consider the general problem of computing the cohomology of
the DGA whose algebra is the exterior algebra $R\otimes\bigwedge V$ over
$R={\mathbb{C}}[z_{1},\dots,z_{n},w_{1},\dots,w_{n}]$, where
$V={\mathbb{C}}\langle\partial_{z_{1}},\dots,\partial_{z_{n}},\partial_{w_{1}},\dots,\partial_{w_{n}}\rangle$,
equipped with the differential $\sigma$ over ${\mathbb{C}}$ defined by
$\sigma(Y)=[Y,\pi_{B}]$ for each $Y\in R\otimes\bigwedge V$ and

$\displaystyle\pi_{B}=\sum_{p,q=1}^{n}B_{pq}z_{p}w_{q}\partial_{z_{p}}\wedge\partial_{w_{q}}$
(3.3)

for some Hermitian matrix $[B_{pq}]$. We will be concerned primarily with
examples where $[B_{pq}]$ is invertible, so that $\pi$ is generically non-
degenerate and the cohomology has a chance at being finite dimensional.

###  3.1 General results for $\boldsymbol{H^{0}}$ and $\boldsymbol{H^{1}}$

The first general result on the cohomology concerns $H^{0}$, which we recall
is canonically isomorphic to the subspace elements of $\ker\sigma$ of wedge
degree $0$.

######  Proposition 3.1.

If $[B_{pq}]$ in (3.3) is invertible, so that $\pi_{B}$ is generically non-
degenerate, then $H^{0}({\mathbb{C}}^{2n},\pi_{B})={\mathbb{C}}\langle
1\rangle$.

###### Proof.

If $[B_{pq}]$ is invertible, then $\pi_{B}$ is non-degenerate on the open
dense set ${\mathbb{T}}_{\mathbb{C}}^{2n}\subset{\mathbb{C}}^{2n}$ by a
computation similar to that in the proof of Proposition 2.3. The complex
algebraic functions $f$ on ${\mathbb{C}}^{2n}$ with $[f,\pi_{B}]=0$ are those
for which $\pi_{B}^{\\#}(\partial f)=0$ on ${\mathbb{C}}^{2n}$, where
$\partial$ is the Dolbeault operator. Such functions satisfy
$\pi_{B}^{\\#}(\partial f)=0$ on ${\mathbb{T}}_{\mathbb{C}}^{2n}$ in
particular. Since ${\mathbb{T}}_{\mathbb{C}}^{2n}$ is connected and $\pi_{B}$
is non-degenerate there, we see that this requires $f$ to be constant. But an
algebraic function which is constant on ${\mathbb{T}}_{\mathbb{C}}^{2n}$ is
constant on ${\mathbb{C}}^{2n}$. Hence
$H^{0}({\mathbb{C}}^{2n},\pi_{B})={\mathbb{C}}\langle 1\rangle$ for the
algebraic cohomology. ∎

This argument certainly applies to holomorphic functions, so the same result
is valid in the holomorphic category. By setting $w_{k}=\bar{z}_{k}$, re-
inserting the factor of $-2i$ in $\pi_{B}$, and considering not the Dolbeault
operator $\partial$ but the full de Rham differential
$d=\partial+\bar{\partial}$, the above argument applies to the smooth real
problem as well. Furthermore, it applies to any complex toric manifold $M$
equipped with a generically non-degenerate real toric Poisson structure $\pi$
of type $(1,1)$. Since $\pi$ is non-degenerate on the open dense
$T_{\mathbb{C}}$-orbit $\mathcal{O}$, which is diffeomorphic to
$T_{\mathbb{C}}$, only the constant functions on $M$ can be constant on the
symplectic leaves of $(M,\pi)$ by continuity.

The classes in $H^{1}(M,\pi)$ are infinitesimal outer automorphisms of
$(M,\pi)$, i.e., classes of vector fields which preserve $\pi$ ($[X,\pi]=0$)
modulo those which do so because they are Hamiltonian ($X=\pi^{\\#}(df)$). The
standard basis for ${\mathbb{C}}^{2n}$, thought of as the Lie algebra of
${\mathbb{T}}_{\mathbb{C}}^{2n}$, determines the holomorphic vector fields
$z_{1}\partial_{z_{1}},\dots,z_{n}\partial_{z_{n}},w_{1}\partial_{w_{1}},\dots,w_{n}\partial_{w_{n}}$
on ${\mathbb{C}}^{2n}$ through the infinitesimal action of
$\mathbb{T}_{\mathbb{C}}^{n}$. Such vector fields preserve $\pi_{B}$ in (3.3)
because $\pi_{B}$ is a linear combination of wedge products of these fields.
The fields are locally, but not globally, Hamiltonian because their primitives
involve $\log(z_{k})$ or $\log(w_{k})$ which are not even single valued on
${\mathbb{T}}_{\mathbb{C}}^{2n}$ let alone extend to $z_{k}=0$ or $w_{k}=0$.
In the algebraic category, a polynomial Hamiltonian then does not exist.
Rather surprisingly, every other polynomial vector field in the kernel has a
polynomial Hamiltonian.

######  Theorem 3.2.

If $[B_{pq}]$ in (3.3) is symmetric and invertible, then

$\displaystyle
H^{1}\big{(}{\mathbb{C}}^{2n},\pi_{B}\big{)}={\mathbb{C}}\langle
z_{1}\partial_{z_{1}},\dots,z_{n}\partial_{z_{n}},w_{1}\partial_{w_{1}},\dots,w_{n}\partial_{w_{n}}\rangle$

for the algebraic cohomology.

###### Proof.

We start by computing the action of $\sigma$ on the natural
${\mathbb{C}}$-basis elements of $R\otimes\bigwedge V$, i.e., those elements
of the form $z^{\alpha}w^{\beta}\partial_{z_{j}}$ or
$z^{\alpha}w^{\beta}\partial_{w_{k}}$. A straightforward computation with the
Schouten bracket shows that

$\displaystyle\sigma\big{(}z^{\alpha}w^{\beta}\partial_{z_{k}}\big{)}=z^{\alpha}w^{\beta}\partial_{z_{k}}\wedge\left(\sum_{p=1}^{n}(B_{p}\cdot\beta)z_{p}\partial_{z_{p}}-\sum_{q=1}^{n}(B_{q}\cdot(\alpha-
e_{k}))w_{q}\partial_{w_{q}}\right)$ (3.4)

and likewise

$\displaystyle\sigma\big{(}z^{\alpha}w^{\beta}\partial_{w_{k}}\big{)}=z^{\alpha}w^{\beta}\partial_{w_{k}}\wedge\left(\sum_{p=1}^{n}(B_{p}\cdot(\beta-
e_{k}))z_{p}\partial_{z_{p}}-\sum_{q=1}^{n}(B_{q}\cdot\alpha)w_{q}\partial_{w_{q}}\right)$
(3.5)

for each $\alpha,\beta\in\mathbb{N}^{n}$ and $k=1,2,\dots,n$, where $B_{p}$
denotes the $p^{th}$ row or column of $B$.

Let us next observe that we can decompose a general linear combination of
basis elements in a useful manner. To describe the decomposition, it will be
convenient to briefly suppress the splitting of variables into $z$ and $w$ and
instead using a single variable $\zeta$ for both: we write $X=\sum
c_{\mu}^{k}\zeta^{\mu}\partial_{\zeta_{k}}$, where $\mu=(\alpha,\beta)$,
$\zeta_{k}=z_{k}$ if $k\leq n$ or $w_{k-n}$ if $k>n$, and
$\zeta^{\mu}=z^{\alpha}w^{\beta}$. We partition the terms of $X$ by defining

$\displaystyle X_{\lambda}=\sum_{\begin{subarray}{c}\mu,k\\\
\mu=\lambda+e_{k}\end{subarray}}c_{\mu}^{k}\zeta^{\mu}\partial_{\zeta_{k}},$

as $\lambda$ ranges over integral vectors of length $2n$. Although it appears
in a different form, this decomposition technique is used in the general work
of [5]. Note that $X_{\lambda}\neq 0$ for only finitely many $\lambda$.
Furthermore, an inspection of (3.4) and (3.5) reveals that if $\lambda$ and
$\lambda^{\prime}$ are distinct, then $\sigma(X_{\lambda})$ and
$\sigma(X_{\lambda^{\prime}})$ have no like terms, that is, no terms with the
same wedge and homogeneous degrees. This means that if $\sigma(X)=0$, then
$\sigma(X_{\lambda})=0$ for each $\lambda$.

By considering one such $X_{\lambda}$ at a time, we may assume without loss of
generality that $X=X_{\lambda}$. Since for each $\partial_{\zeta_{k}}$ and
fixed $\lambda$, the exponent vector $\mu$ is uniquely determined, we can
rewrite $X$ as

$\displaystyle
X=\sum_{k=1}^{2n}c^{k}_{\lambda+e_{k}}\zeta^{\lambda+e_{k}}\partial_{\zeta^{k}}.$

In general, the coordinates of $\lambda+e_{k}$ must be nonnegative, so the
coordinates of $\lambda$ must be greater than or equal to $-1$ with at most
one coordinate equal to $-1$. When $\lambda=0$ in ${\mathbb{Z}}^{2n}$, then
$X_{\lambda}$ is a linear combination of
$\zeta_{1}\partial_{\zeta_{1}},\dots,\zeta_{2n}\partial_{\zeta_{2n}}$. If
$\lambda_{\ell}=-1$, then $X$ is the single term
$X=c\zeta^{\lambda+e_{\ell}}\partial_{\zeta_{\ell}}$ for some
$c\in{\mathbb{C}}$.

Suppose $\lambda_{\ell}=-1$ with $\ell\leq n$ and translate back in terms of
$z$ and $w$, so that $X=cz^{\alpha}w^{\beta}\partial_{z_{\ell}}$ and
$\alpha_{\ell}=0$. Applying (3.4), we can compute $\sigma(X)$ and see that the
coefficient of $\partial_{z_{\ell}}\wedge\partial_{w_{q}}$ in $\sigma(X)$ is
$(B_{q}\cdot(\alpha-e_{\ell}))w_{q}z^{\alpha}w^{\beta}$ for each $q$. Since
$B$ is invertible and $\alpha-e_{\ell}\neq 0$, then $\sigma(X)\neq 0$.
Similarly, by applying (3.5) we can deduce the same conclusion if $\ell>n$.
Thus if $X=X_{\lambda}$ is in the kernel of $\sigma$, we may assume that all
coordinates of $\lambda$ are nonnegative and so

$\displaystyle
X=X_{\lambda}=\sum_{j=1}^{n}a^{j}z^{\alpha+e_{j}}w^{\beta}\partial_{z_{j}}+\sum_{j=1}^{n}b^{j}_{\alpha,\beta}z^{\alpha}w^{\beta+e_{j}}\partial_{w_{j}}$
(3.6)

for $\lambda=(\alpha,\beta)$ and some coefficients $a^{j}_{\alpha,\beta}$ and
$b^{j}_{\alpha,\beta}$. Applying (3.4) and (3.5) to (3.6), we find that

$\displaystyle\sigma(X)=X\wedge\left(\sum_{p=1}^{n}(B_{p}\cdot\beta)z_{p}\partial_{z_{p}}-\sum_{q=1}^{n}(B_{q}\cdot\alpha)w_{q}\partial_{w_{q}}\right).$
(3.7)

Another straightforward computation with the Schouten bracket shows that

$\displaystyle\sigma\big{(}z^{\alpha}w^{\beta}\big{)}=z^{\alpha}w^{\beta}\left(\sum_{p=1}^{n}(B_{p}\cdot\beta)z_{p}\partial_{z_{p}}-\sum_{q=1}^{n}(B_{q}\cdot\alpha)w_{q}\partial_{w_{q}}\right).$
(3.8)

Thus, if $X=X_{\lambda}\not=0$ satisfies $\sigma(X)=0$, then
$X\wedge\sigma(z^{\alpha}w^{\beta})=0$ by (3.7) and (3.8). Hence, if
$\sigma(z^{\alpha}w^{\beta})\not=0$ then $X=c\sigma(z^{\alpha}w^{\beta})$ for
some $c\not=0$. If $(\alpha,\beta)=(0,0)$, then
$\sigma(z^{\alpha}w^{\beta})=0$ by inspection of (3.8). On the other hand, if
$(\alpha,\beta)\not=(0,0)$, then $\sigma(z^{\alpha}w^{\beta})\not=0$ because
$[B_{pq}]$ is invertible. Thus,
$X=c\sigma(z^{\alpha}w^{\beta})=\sigma(cz^{\alpha}w^{\beta})$ when
$(\alpha,\beta)\not=(0,0)$. This completes the proof. ∎

###  3.2 Algebraic cohomology for $\boldsymbol{n=1}$

When $n=1$, the matrix $[B_{pq}]$ is $1\times 1$ consisting of a single non-
zero complex number. Hence $\pi_{B}=bzw\partial_{z}\wedge\partial_{w}$. The
cohomology of $({\mathbb{C}}^{2},bzw\partial_{z}\wedge\partial_{w})$ has been
computed by a number of methods (cf. [2, 6]). We supply another algebraic
method. As mentioned before, an overall non-zero scalar multiple of a Poisson
tensor does not affect its cohomology, so we can more simply consider
$\pi=zw\partial_{z}\wedge\partial_{w}$. By Proposition 3.1 and Theorem 3.2, we
know that

$\displaystyle
H^{0}\big{(}{\mathbb{C}}^{2},bzw\partial_{z}\wedge\partial_{w}\big{)}={\mathbb{C}}\langle
1\rangle\qquad\text{and}\qquad
H^{1}\big{(}{\mathbb{C}}^{2},bzw\partial_{z}\wedge\partial_{w}\big{)}={\mathbb{C}}\langle
z\partial_{z},w\partial_{w}\rangle$

for each $b\not=0$. It remains then to determine
$H^{2}({\mathbb{C}}^{2},bzw\partial_{z}\wedge\partial_{w})$ which is the space
of all bi-vector fields on ${\mathbb{C}}^{2}$ modulo the subspace of the image
of $\sigma$ in wedge degree $2$. At this point, let us observe that the
cochain complex $R\otimes\bigwedge V$ can be graded not only by the wedge
degree from the second factor $\bigwedge V$ but also using the total
homogeneous degree on the polynomial ring $R$. The Poisson differential
$\sigma$, which raises the wedge degree by $1$ also raises the homogeneous
degree by $1$ because its action on a multi-vector field involves 1st order
differentiation of coefficients post-multiplied by a homogeneous quadratic
expression. The cohomology $H({\mathbb{C}}^{2},\pi)$ then has a direct sum
decomposition

$\displaystyle
H\big{(}{\mathbb{C}}^{2},\pi\big{)}=\bigoplus_{d\in\mathbb{N}}\bigoplus_{p=0}^{2}H^{p}_{[d]}\big{(}{\mathbb{C}}^{2},\pi\big{)},$

where

$\displaystyle
H^{p}_{[d]}\big{(}{\mathbb{C}}^{2},\pi\big{)}=\frac{\ker\sigma\colon
R_{[d]}\otimes\bigwedge^{p}V\to
R_{[d+1]}\otimes\bigwedge^{p+1}V}{\mathrm{im}\,\sigma\colon
R_{[d-1]}\otimes\bigwedge^{p-1}V\to R_{[d]}\otimes\bigwedge^{p}V}.$

In particular, the bi-vector field $\partial_{z}\wedge\partial_{w}$ which
generates $R_{[0]}\otimes\bigwedge^{2}V$ is certainly not in the image of
$\sigma$ and so
$H^{2}_{[0]}({\mathbb{C}}^{2},\pi)={\mathbb{C}}\langle\partial_{z}\wedge\partial_{w}\rangle$.
The next result completes the computation of the complex algebraic cohomology.
See Fig. 1. Intriguingly, this recovers the same result found by Nakanishi in
[7],

$\displaystyle
H\big{(}{\mathbb{C}}^{2},zw\partial_{z}\wedge\partial_{w}\big{)}={\mathbb{C}}\langle
1,z\partial_{z},w\partial_{w},\partial_{z}\wedge\partial_{w},zw\partial_{z}\wedge\partial_{w}\rangle$

except that he considered real coefficients, with $z$ and $w$ as real
variables, and used analytic techniques to classify the solutions to the
partial differential equations encoded in $\sigma(Y)=0$.

$\begin{array}[]{c|ccc}\dim H_{[d]}^{p}&0&1&2\\\ \hline\cr 0&1&0&1\\\
1&0&2&0\\\ 2&0&0&1\\\ 3&0&0&0\\\ 4&0&0&0\\\ 5&0&0&0\\\ 6&0&0&0\\\ \end{array}$

Figure 1: Dimensions of
$H^{p}_{[d]}({\mathbb{C}}^{2},zw\partial_{z}\wedge\partial_{w})$ up to $d=6$.

######  Proposition 3.3.

For $b\not=0$,

$\displaystyle
H^{2}\big{(}{\mathbb{C}}^{2},bzw\partial_{z}\wedge\partial_{w}\big{)}\simeq{\mathbb{C}}\langle\partial_{z}\wedge\partial_{w},zw\partial_{z}\wedge\partial_{w}\rangle$

for the complex algebraic cohomology.

###### Proof.

As we observed before,
$H({\mathbb{C}}^{2},bzw\partial_{z}\wedge\partial_{w})=H({\mathbb{C}}^{2},zw\partial_{z}\wedge\partial_{w})$.
We will compute
$H^{2}_{[d]}({\mathbb{C}}^{2},bzw\partial_{z}\wedge\partial_{w})$ for each
$d\geq 1$. Let $d\geq 1$ be given. Since
$\ker\sigma\cap(R_{[d]}\otimes\bigwedge^{2}V)=R_{[d]}\otimes\bigwedge^{2}V$,
it suffices to show that every basis element of the form
$z^{p}w^{q}\,\partial_{z}\wedge\partial_{w}$, where $p+q=d$ with $d\not=2$,
has a primitive with respect to $\sigma$. Consider a general element $X\in
R_{[d-1]}\otimes\bigwedge^{1}V$ of the form

$\displaystyle
X=\sum_{\alpha,\beta}a_{\alpha\beta}z^{\alpha}w^{\beta}\partial_{z}+\sum_{\gamma,\delta}b_{\gamma,\delta}z^{\gamma}w^{\delta}\partial_{w},$

where the sums are over pairs of non-negative integers which sum to $d-1$. A
straightforward calculation with the Schouten bracket shows that

$\displaystyle\sigma(X)=-\Big{(}{}-a_{0,d-1}w^{d}+b_{0,d-1}(d-2)zw^{d-1}+\sum(a_{p,q-1}(p-1)+b_{p-1,q}(q-1))z^{p}w^{q}$
$\displaystyle\hphantom{\sigma(X)=}{}+a_{d-1,0}(d-2)z^{d-1}w-b_{d-1,0}z^{d}\Big{)}\partial_{z}\wedge\partial_{w},$

where the sum is over $p,q\geq 2$ with $p+q=d$. From this formula, it follows
that the basis elements

$\displaystyle w^{d}\partial_{z}\wedge\partial_{w},\qquad
z^{d}\partial_{z}\wedge\partial_{w}$

have unique primitives with respect to $\sigma$ and the basis elements

$\displaystyle zw^{d-1}\partial_{z}\wedge\partial_{w},\qquad
z^{d-1}w\partial_{z}\wedge\partial_{w}$

have unique primitives with respect to $\sigma$ provided $d\not=2$. The basis
elements $z^{p}w^{q}\partial_{z}\wedge\partial_{w}$ with $p,q\geq 2$ have one
parameter families of primitives. The only basis vectors of
$R\otimes\bigwedge^{2}V$ which are not accounted for are
$\partial_{z}\wedge\partial_{w}$ and $zw\partial_{z}\wedge\partial_{w}$. As we
already observed, the first cannot be in the image of $\sigma$. Examining the
formula above for $\sigma(X)$ when $d=2$, we see that
$zw\partial_{z}\wedge\partial_{w}$ does not appear either. Therefore,
$H_{[d]}^{2}=\mathbf{0}$ when $d\geq 1$, unless $d=2$ in which case
$H_{[2]}^{2}={\mathbb{C}}\langle zw\partial_{z}\wedge\partial_{w}\rangle$. ∎

###  3.3 Numerical results for $\boldsymbol{H^{*}}$ when $\boldsymbol{n=2}$

When $n=2$, we are considering the Poisson cohomology
$H({\mathbb{C}}^{4},\pi_{B})$ in the complex algebraic category where

$\displaystyle\pi_{B}=B_{11}z_{1}w_{1}\partial_{z_{1}}\wedge\partial_{w_{1}}+B_{12}z_{1}w_{2}\partial_{z_{1}}\wedge\partial_{w_{2}}+B_{21}z_{2}w_{1}\partial_{z_{2}}\wedge\partial_{w_{1}}+B_{22}z_{2}w_{2}\partial_{z_{2}}\wedge\partial_{w_{2}}$

for some matrix

$\displaystyle B=\begin{pmatrix}B_{11}&B_{12}\\\ B_{21}&B_{22}\end{pmatrix}.$

With the increase in the number of variables and the number of terms in
$\pi_{B}$, comes an increase in complexity of the cohomological computations,
even for the algebraic problem. Still, $\sigma$ in this context raises wedge
degree by $1$ and total homogeneous degree by $1$, so the cochain complex
$R\otimes\bigwedge V$ decomposes as a direct sum of cochain complexes
$(\bigoplus_{p=0}^{4}R_{[d+p]}\otimes\bigwedge^{p}V,\sigma)$ over $d$ from
$-2n$ to $\infty$, where we set $R_{[\ell]}\otimes\bigwedge^{p}V=\mathbf{0}$
if $\ell<0$. While $R\otimes\bigwedge^{p}V$ is infinite dimensional, each
$R_{[d]}\otimes\bigwedge^{p}V$ is finite dimensional with

$\displaystyle\dim_{\mathbb{C}}R_{[d]}\otimes\textstyle{\bigwedge^{p}V}={d+2n-1\choose
2n-1}{2n\choose p},\qquad\text{where}\quad n=2.$

Thus, $\sigma\colon R_{[d]}\otimes\bigwedge^{p}V\to
R_{[d+1]}\otimes\bigwedge^{p+1}V$ has a matrix representation, and linear
algebra can be used to determine each $H^{p}_{[d]}({\mathbb{C}}^{4},\pi_{B})$.
For brevity, let’s write $\sigma^{p}_{[d]}$ for $\sigma\colon
R_{[d]}\otimes\bigwedge^{p}V\to R_{[d+1]}\otimes\bigwedge^{p+1}V$.

$B=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\hskip
85.35826ptB=\begin{pmatrix}2&1\\\ 1&2\end{pmatrix}$
$\begin{array}[]{c|ccccc}\dim H_{[d]}^{p}&0&1&2&3&4\\\ \hline\cr
0&1&0&2&0&1\\\ 1&0&4&0&4&0\\\ 2&0&0&6&0&2\\\ 3&0&0&0&4&0\\\ 4&0&0&0&0&1\\\
5&0&0&0&0&0\\\ 6&0&0&0&0&0\\\ 7&0&0&0&0&0\\\ 8&0&0&0&0&0\end{array}\ \ \ \
\begin{array}[]{c|ccccc}\dim H_{[d]}^{p}&0&1&2&3&4\\\ \hline\cr 0&1&0&0&0&1\\\
1&0&4&0&0&0\\\ 2&0&0&6&0&0\\\ 3&0&0&0&4&0\\\ 4&0&0&2&0&1\\\ 5&0&0&0&4&0\\\
6&0&0&0&0&2\\\ 7&0&0&0&0&0\\\ 8&0&0&0&0&0\end{array}$

Figure 2: Cohomological dimensions up to $d=8$ for the two cases shown.

We were able to use Mathematica 10 to perform these computations for small $d$
up to $d=8$ with fairly fast computing times on a 2.7 GHz processor with 8 GB
of RAM. The dimension of $R_{[d]}\otimes\bigwedge^{p}V$ grows fairly quickly
with $d$ and consequently the matrices representing each $\sigma^{p}_{[d]}$
get large fairly quickly. For example, $\sigma^{2}_{[8]}$ is represented by an
$880\times 990$ matrix.

The computations involved a mix of symbolic computation and linear algebra,
but the methods are standard. If we write $z^{p}w^{q}$ for the monomial
$z_{1}^{p_{1}}z_{2}^{p_{2}}w_{1}^{q_{1}}w_{2}^{q_{2}}$ determined by
$(p,q)\in\mathbb{N}^{2}\times\mathbb{N}^{2}$ and likewise write
$\partial_{z}^{\gamma}\partial_{w}^{\delta}$ for
$\partial_{z_{1}}^{\gamma_{1}}\wedge\partial_{z_{2}}^{\gamma_{2}}\wedge\partial_{w_{1}}^{\delta_{1}}\wedge\partial_{w_{2}}^{\delta_{2}}$
in $\bigwedge V$ determined by
$(\gamma,\delta)\in\\{0,1\\}^{2}\times\\{0,1\\}^{2}$, then we obtain a
${\mathbb{C}}$-basis
$\\{z^{p}w^{q}\partial_{z}^{\gamma}\partial_{w}^{\delta}\\}$ for
$R_{[d]}\otimes\bigwedge^{p}$ indexed by all $(p,q)$ with
$p_{1}+p_{2}+q_{1}+q_{2}=d$ and $(\gamma,\delta)$ with
$\gamma_{1}+\gamma_{2}+\delta_{1}+\delta_{2}=p$. Determining the matrix
representing $\sigma^{p}_{[d]}$ in terms of these bases involved symbolic
computation, implementing calculations with the Schouten bracket. Once the
matrix representations were found, computing $H^{p}_{[d]}$ reduced to the
following linear algebra.

  1. 1.

Using Gauss–Jordan elimination, we can determine a basis for the column space
of the matrix representing $\sigma^{p-1}_{[d-1]}$ and determine the
corresponding symbolic basis for $\mathrm{im}\,\sigma^{p-1}_{[d-1]}$.

  2. 2.

Next we find a basis for the null space of the matrix representing
$\sigma^{p}_{[d]}$ and determine the corresponding symbolic basis for
$\ker\sigma_{[d]}^{p}$.

  3. 3.

If we form a matrix whose rows are the basis vectors of the column space for
$\sigma_{[d-1]}^{p-1}$ (written as rows) followed by the basis vectors for the
null space of the matrix for $\sigma^{p}_{[d]}$ (written as rows) and then
perform Gauss–Jordan elimination on this matrix we obtain a matrix in reduced
row echelon form.

  4. 4.

If the basis for the column space of $\sigma_{[d-1]}^{p-1}$ had $k$ elements,
then the upper left $k\times k$ block of this reduced matrix will be the
identity. The remaining non-zero rows, rewritten as columns, form a basis for
a subspace of the null space of the matrix for $\sigma^{p}_{[d]}$ which is
complementary to column space of the matrix for $\sigma^{p-1}_{[d-1]}$. The
corresponding symbolic vectors are representatives for a basis of
$H^{p}_{[d]}$.

$B=\begin{pmatrix}3&-1\\\ -1&2\end{pmatrix}\hskip
56.9055ptB=\begin{pmatrix}6&-2\\\ -2&2\end{pmatrix}\hskip
56.9055ptB=\begin{pmatrix}11&-3\\\ -3&2\end{pmatrix}$
$\begin{array}[]{c|ccccc}\dim H_{[d]}^{p}&0&1&2&3&4\\\ \hline\cr
0&1&0&0&0&1\\\ 1&0&4&0&0&0\\\ 2&0&0&6&0&0\\\ 3&0&0&0&4&0\\\ 4&0&0&0&0&1\\\
5&0&0&0&0&0\\\ 6&0&0&0&0&0\\\ 7&0&0&0&0&0\\\ 8&0&0&0&0&0\end{array}\ \ \ \
\begin{array}[]{c|ccccc}\dim H_{[d]}^{p}&0&1&2&3&4\\\ \hline\cr 0&1&0&0&2&1\\\
1&0&4&0&2&2\\\ 2&0&0&6&2&2\\\ 3&0&0&0&6&2\\\ 4&0&0&0&2&3\\\ 5&0&0&0&2&2\\\
6&0&0&0&2&2\\\ 7&0&0&0&2&2\\\ 8&0&0&0&2&2\end{array}\ \ \ \
\begin{array}[]{c|ccccc}\dim H_{[d]}^{p}&0&1&2&3&4\\\ \hline\cr 0&1&0&0&0&1\\\
1&0&4&0&0&0\\\ 2&0&0&6&0&0\\\ 3&0&0&0&4&0\\\ 4&0&0&0&0&1\\\ 5&0&0&0&0&0\\\
6&0&0&0&0&0\\\ 7&0&0&0&0&0\\\ 8&0&0&0&0&0\end{array}$

Figure 3: Cohomological dimensions up to $d=8$ for the three cases shown.

We considered quadratic forms $B$ whose matrices $[B_{ij}]$ had one of five
forms

$\displaystyle\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\qquad\begin{pmatrix}2&1\\\
1&2\end{pmatrix},\qquad\begin{pmatrix}2+m^{2}&-m\\\
-m&2\end{pmatrix}\qquad\text{for}\quad m=1,2,3.$

These give local forms for the toric Poisson structures constructed in [1] on
the smooth toric varieties ${\mathbb{C}}P^{1}\times{\mathbb{C}}P^{1}$,
${\mathbb{C}}P^{2}$, and the Hirzebruch surfaces $X_{m}$ for $m=1,2,3$,
respectively. Figs. 2 and 3 show the cohomological dimensions computed for
each of these cases for $0\leq p\leq 4$ and $0\leq d\leq 8$. In each of these
cases, the elements
$z^{\gamma}w^{\delta}\partial_{z}^{\gamma}\partial_{w}^{\delta}$ in
$R\otimes\bigwedge V$ represent classes in the basis for $H$. Furthermore, the
diagonal $\bigoplus_{k=0}^{4}H_{[k]}^{k}=\bigwedge{\mathbb{C}}\langle
z_{1}\partial_{z_{1}},z_{2}\partial_{z_{2}},w_{1}\partial_{w_{1}},w_{2}\partial_{w_{2}}\rangle$
is generated by the basis vector fields generating the
${\mathbb{T}}_{\mathbb{C}}^{4}$ action on ${\mathbb{C}}^{4}$. Let us refer to
the basis vectors of $R\otimes\bigwedge V$ of this type as those of Type I.
The additional classes in each basis were, remarkably, also represented by
basis vectors of $\mathcal{V}$ which lie in $\ker\sigma$ but are of one of two
other types. The generators of Type II were
$z^{\alpha}w^{\beta}\partial_{z^{\gamma}w^{\delta}}$ in $\ker\sigma$ with the
property that $\alpha\cdot\gamma=0$ and $\beta\cdot\delta=0$. Those of the
remaining Type III were non-zero wedge products of generators of Type I and
Type II. Thus, it appears that the cohomology is generated as a module over
the diagonal $\bigoplus_{k=0}^{4}H_{[k]}^{k}=\bigwedge{\mathbb{C}}\langle
z_{1}\partial_{z_{1}},z_{2}\partial_{z_{2}},w_{1}\partial_{w_{1}},w_{2}\partial_{w_{2}}\rangle$
by $1$ and the basis vectors of $R\otimes\bigwedge V$ which lie in the kernel
of $\sigma$ and are of Type II. In a forthcoming paper, we use representation
theory to show that such a presentation of the cohomology as a module over
$\bigwedge{\mathfrak{t}}_{\mathbb{C}}$ is possible for any $n$ and $B$.

######  Example 3.4.

For $B=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$, the cohomology up to
homogeneous degree $8$ generated as a module over the diagonal by $1$,
$\partial_{z_{1}}\wedge\partial_{w_{1}}$,
$\partial_{z_{2}}\wedge\partial_{w_{2}}$,
$\partial_{z_{1}}\wedge\partial_{z_{2}}\wedge\partial_{w_{1}}\wedge\partial_{w_{2}}$.

######  Example 3.5.

For $B=\begin{pmatrix}2&1\\\ 1&2\end{pmatrix}$, the cohomology up to
homogeneous degree $8$ is generated as a module over the diagonal by $1$,
$z_{1}^{2}w_{2}^{2}\partial_{z_{2}}\wedge\partial_{w_{1}}$,
$z_{2}^{2}w_{1}^{2}\partial_{z_{1}}\wedge\partial_{w_{2}}$, and
$\partial_{z_{1}}\wedge\partial_{z_{2}}\wedge\partial_{w_{1}}\wedge\partial_{w_{2}}$.

######  Example 3.6.

For $B=\begin{pmatrix}3&-1\\\ -1&2\end{pmatrix}$, the cohomology up to
homogeneous degree $8$ is generated as a module over the diagonal by $1$ and
$\partial_{z_{1}}\wedge\partial_{z_{2}}\wedge\partial_{w_{1}}\wedge\partial_{w_{2}}$.

######  Example 3.7.

For $B=\begin{pmatrix}6&-2\\\ -2&2\end{pmatrix}$, the cohomology up to
homogeneous degree $8$ is generated as a module over the diagonal by $1$ and
$\partial_{z_{1}}\wedge\partial_{z_{2}}\wedge\partial_{w_{1}}\wedge\partial_{w_{2}}$
and
$z_{2}^{d}\partial_{z_{1}}\wedge\partial_{w_{1}}\wedge\partial_{w_{2}},w_{2}^{d}\partial_{z_{1}}\wedge\partial_{z_{2}}\wedge\partial_{w_{1}}$
for each $0\leq d\leq 8$. In fact, we can show that these latter expressions
are generators of the full cohomology for every degree $d$. Thus, the
algebraic cohomology of this Poisson structure on ${\mathbb{C}}^{4}$ is
infinite dimensional despite the fact that $B$ is integral and symmetric
positive definite.

######  Example 3.8.

For $B=\begin{pmatrix}11&-3\\\ -3&2\end{pmatrix}$, the cohomology up to
homogeneous degree $8$ is generated as a module over the diagonal by $1$ and
$\partial_{z_{1}}\wedge\partial_{z_{2}}\wedge\partial_{w_{1}}\wedge\partial_{w_{2}}$.

### Acknowledgements

Portions of this work were completed independently by the two authors during
independent sabbatical leaves from California State Polytechnic University
Pomona and, separately, while supported by the Provost’s Teacher-Scholar
Program. We appreciate this support and the suggestions from the referees
which improved the paper.

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