# Framed Parabolic Sheaves on a Trinion

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, 1 Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>and  Jacques Hurtubise  Department of Mathematics, McGill University, Burnside
Hall, 805 Sherbrooke St. W., Montreal, Que. H3A 2K6, Canada
<EMAIL_ADDRESS>
###### Abstract.

We consider for structure groups ${\rm SU}(n)\,\subset\,{\rm
SL}(n,\mathbb{C})$ a densely defined toric structure on the moduli of framed
parabolic sheaves on a three-punctured sphere, which degenerates to an actual
toric structure. In combination with previous degeneration results, these
extend to similar moduli for arbitrary Riemann surfaces.

###### Key words and phrases:

Stable vector bundles, algebraic curves, framed parabolic structures, trinion

###### 2000 Mathematics Subject Classification:

14P99, 53C07, 32Q15

###### Contents

  1. 1 Introduction
  2. 2 The moduli of framed parabolic sheaves on a trinion
  3. 3 Okounkov bodies and toric degeneration
  4. 4 Strata, and birational equivalence
  5. 5 Hamiltonians and imploded cross-sections
  6. 6 Degenerations of bundle moduli

##  1\. Introduction

In 1992, Jeffrey and Weitsman, [JW], showed how the space of ${\rm SU}(2)$
representations of the fundamental group of a Riemann surface possessed an
integrable Hamiltonian system with continuous Hamiltonians, smooth away from a
small locus, and giving a torus action on an open dense set; in a sense it is
an “almost toric” structure on the space. This structure was associated to the
Goldman flows on representations associated to a trinion (three-punctured
spheres) decomposition of the surface (it is also called “pant
decomposition”). The construction allowed them to give an interpretation of
the Verlinde formulae, [Ve], in terms of Bohr-Sommerfeld quantization, by
counting lattice points in the moment polytope.

We recall that via the theorem of Narasimhan and Seshadri, [NS], the space of
representations of the fundamental group in ${\rm SU}(n)$ (more generally, in
a compact semi-simple group $G$) has another interpretation as a space of
stable holomorphic rank $n$ vector bundles equipped with a global volume form
(more generally, stable principal $G_{\mathbb{C}}$-bundles, where
$G_{\mathbb{C}}$ is the complexificaton of $G$). In the holomorphic
interpretation of the moduli space as stable principal ${\rm
SL}(2,\mathbb{C})$ bundles the Verlinde formulae give the dimensions of spaces
of sections of tensor powers of a natural determinant line bundle over the
moduli space, and if the variety were actually toric, the point count would
naturally correspond to this dimension, by the general theory of toric
varieties (e.g. as in Fulton [Fu]). It is thus as if the actual moduli space
were a deformation of another moduli space that is actually toric. We now have
the good fortune of being able to refer to a beautiful series of results on
the structure of these toric degenerations, valid in large generality, both in
the algebraic framework (see the papers of Anderson [An], Kaveh and Khovansky
[KK])) and in the symplectic framework (see the paper of Harada and Kaveh
[HK]).

Indeed, in our case, a suitable toric candidate for ${\rm SU}(2)$ was produced
in [HJS1], [HJS2]. In these papers, a moduli space of “framed parabolic
sheaves” was constructed over punctured Riemann surfaces, with the standard
parabolic structures of bundles at the puncture (i.e., flags) being enriched
by an additional framing variable (a trivialization of the top exterior powers
of the subquotients of the flag). This space also has a symplectic
interpretation as an “imploded cross-section” of the space of all framed
representations of the fundamental group of the punctured surface, these
representations being thought of as flat connections on a bundle, and the
framing being the additional data of a trivialization at each puncture. The
space of framed parabolic sheaves is a “master space” for parabolic bundles:
it gives the standard parabolic moduli, either symplectically by a torus
reduction, or holomorphically by the corresponding geometric invariant
theoretic (GIT) quotient by a complex torus; in both cases the tori act on the
framings.

Using the actions of tori at the punctures, we can perform a glueing
procedure: we first build a nodal curve from a (possibly disconnected)
punctured curve by joining some or all of the punctures in pairs. Above this
nodal curve, for each node, we can build “partially glued” sheaves from framed
parabolic sheaves by identifying the top exterior powers of the subquotients
of the flags at the two punctures by their trivializations. This actually
amounts to a diagonal symplectic quotient in the symplectic picture or a
diagonal geometric invariant theoretic quotient in the holomorphic picture by
the appropriate torus actions. (In fact, with the appropriate normalizations,
the quotients are by an “anti-diagonal” torus, because the order of the
eigenvalues at one puncture is the opposite of that at the other puncture.)

Of course, we need to see that this space is the actual result of a
degeneration. This was carried out in [BHu], where the glued space was
obtained from the standard moduli on smooth curves as a limit associated to a
nodal degeneration of the curve into a “balloon animal” made up of trinions.
That paper considers both a symplectic and a holomorphic version. It may be
mentioned that the degenerations obtained are different from those considered
by Bhosle in [Bh2].

In the case of ${\rm SL}(2,\mathbb{C})$ it is already shown, [HJ], that the
trinion building block is $\mathbb{P}^{3}(\mathbb{C})$, and so the glued space
over the nodal curve corresponding to the glued trinions is indeed toric,
being obtained by symplectic (or GIT) torus quotients from a product of the
trinion spaces ($\mathbb{P}^{3}(\mathbb{C})$) : the toric variety
corresponding to ${\rm SU}(2)$ representations for a Riemann surface is a
moduli space of “partially glued” ${\rm SL}(2,\mathbb{C})$ sheaves over a
degeneration of the Riemann surface into a nodal curve whose each irreducible
component is a trinion.

The construction of Jeffrey and Weitsman does not work in the ${\rm
SL}(n,\mathbb{C})$, $n\,>\,2$, case. The reason is that they use the Goldman
flows associated to the trinion decomposition in their construction, and there
are simply not enough of the flows. While in this more general case also we
can obtain the limiting space for ${\rm SL}(n,\mathbb{C})$ as a moduli space
associated to the nodal curve, and we find that it is a torus reduction of the
trinion spaces, the latter show no sign of being toric. The subject of this
paper is to produce a further toric degeneration of the trinion spaces, giving
us then also a toric degeneration of the moduli spaces.

It is our pleasure to dedicate this paper to our friend and collaborator Oscar
García-Prada, a constant source of ideas and projects, on the occasion of his
sixtieth birthday.

##  2\. The moduli of framed parabolic sheaves on a trinion

###  2.1. Holomorphic and symplectic moduli

From the complex geometric point of view, a trinion, which will be denoted by
$X$, is a Riemann sphere with three marked distinct points
$p_{1},\,p_{2},\,p_{3}$.

Our aim is to see how to degenerate the moduli space of ${\rm
SL}(n,\mathbb{C})$-framed parabolic sheaves on a trinion $X$. The theory of
semistable framed parabolic ${\rm SL}(n,\mathbb{C})$ sheaves, which is given
in [HJS2], is briefly recalled below.

We first consider the case when the underlying sheaf is actually locally free.
On the trinion $X$, the data are:

  * •

A rank $n$ holomorphic vector bundle $E$ on $X$ of degree $0$, with a fixed
isomorphism

$\Phi\,:\,\bigwedge\nolimits^{n}E\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\mathcal{O}_{X}$
(2.1)

(the ${\rm SL}(n,\mathbb{C})$ structure).

  * •

Elements

$\beta_{j}^{i}\,\in\,\bigwedge^{i}E^{*}_{p_{j}},\ \ j\,=\,1,\,2,\,3,\ \
i\,=\,1,\,\cdots,\,n.$

They may vanish, but must be decomposable (so the annihilator of any nonzero
$\beta_{j}^{i}$ is a subspace $E_{j}^{n-i}\,\subset\,E_{p_{j}}$ of dimension
$n-i$), and are compatible in the sense that if
$\beta_{j}^{i},\,\beta_{j}^{i^{\prime}}$ are both nonzero, with
$i\,<\,i^{\prime}$, then there is an element

$\gamma^{i,i^{\prime}}\,\in\,\bigwedge\nolimits^{i^{\prime}-i}E^{*}_{p_{j}}$

such that
$\beta_{j}^{i^{\prime}}\,=\,\gamma^{i,i^{\prime}}\bigwedge\beta_{j}^{i}$.
Furthermore, $\beta_{j}^{n}\,=\,\Phi(p_{j})^{-1}(1)$, for $j\,=\,1,\,2,\,3$,
where $\Phi$ is the fixed isomorphism in (2.1).

The relation to parabolic bundles is simply that each non-zero $\beta^{i}_{j}$
gives the above $n-i$ dimensional subspace $E^{n-i}_{j}$ of the fiber
$E_{p_{j}}$, defining a flag at each $p_{j}$. The fact that some of the
$\beta^{i}_{j}$ can be zero allows for non-complete flags. Note that the above
“quotients” $\gamma^{i,i^{\prime}}_{j}$ define volume forms on the quotients
$E_{j}^{n-i}/E_{j}^{n-i^{\prime}}$, hence the term “framed”.

These framed bundles, even if when semi-stable, can degenerate to acquire
torsion. Fortunately not much torsion is necessary, and in fact all of the
properties of the torsion described below follow from the semistability
condition and being a closed orbit in the moduli space; see Lemma 3.4 of
[HJS2]. (In fact, in terms of eventual parabolic weights, the torsion only
appears when the gap $\alpha_{1}-\alpha_{n}$ is $1$.) The set of structures
extends to allow:

  * •

A rank $n$ sheaf $E$ on $X$, locally free away from the marked points, and of
degree $0$. The torsion sheaf is supported on the reduced marked points
$p_{1},\,p_{2},\,p_{3}$. Therefore, the torsion at $p_{j}$ is of the form
$\mathcal{O}_{p_{j}}^{\oplus s_{j}}$.

  * •

An ${\rm SL}(n,\mathbb{C})$ structure on $E$, meaning an isomorphism $\det
E\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,{\mathcal{O}}_{X}$ (see
[Ko, Ch. V, § 6] for determinant bundle of coherent sheaves). This is
equivalent to giving an isomorphism

$\Phi\,:\,\bigwedge\nolimits^{n}(E/{\rm
torsion})\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\mathcal{O}(-s_{1}p_{1}-s_{2}p_{2}-s_{3}p_{3}),$

where $s_{j}$ are the above integers.

  * •

Elements $\beta_{j}^{i}\,\in\,\bigwedge^{i}E^{*}_{p_{j}}$, $j\,=\,1,\,2,\,3$
and $i\,=\,1,\,\cdots,\,n-1,\,n$. They may vanish, but must be decomposable
(so that the annihilator of any nonzero $\beta_{j}^{i}$ is a subspace
$E_{j}^{n-i}\,\subset\,E_{p_{j}}$ of dimension $n-i$) and if
$\beta_{j}^{i},\,\beta_{j}^{i^{\prime}}$ are both nonzero, with
$i\,<\,i^{\prime}$, there is

$\gamma^{i,i^{\prime}}_{j}\,\in\,\bigwedge\nolimits^{i^{\prime}-i}E^{*}_{p_{j}}$
(2.2)

such that
$\beta_{j}^{i^{\prime}}\,=\,\gamma^{i,i^{\prime}}_{j}\wedge\beta_{j}^{i}$. If
the rank $s_{j}$, of the torsion sheaf at $p_{j}$, is positive, then
$\beta_{j}^{n}\,=\,0$, $\beta_{j}^{i}\,=\,0$ for $i\,<\,s_{j}$ and
$\beta_{j}^{s_{j}}$ is a nonzero product of elements
$e_{j}^{1},\,e_{j}^{2},\,\cdots,\,e_{j}^{s_{j}}$ generating the dual of
torsion.

In [HJS2] a moduli space $\mathcal{M}_{SL}\,=\,\mathcal{M}_{SL}(X)$ of these
framed sheaves is constructed. It also has a symplectic version, defined using
imploded cross-sections, expressed in terms of representations and framings of
the eigenspaces of the monodromies at the punctures [HJS1]. For our case this
amounts to the following:

  * •

A representation of the fundamental group of the trinion into ${\rm SU}(n)$.

  * •

Trivializations of the top exterior powers of the generalized eigenspaces
associated to each eigenvalue of the monodromy around each puncture. If we
lift the eigenvalues of the monodromy around the puncture $p_{i}$ to the
fundamental alcove in the Lie algebra, i.e., take logarithms, these are given
by

$\alpha^{1}(i)\,\geq\,\alpha^{2}(i)\,\geq\,\alpha^{3}(i)\,\geq\,\cdots\,\geq\,\alpha^{n}(i)\,\geq\,\alpha^{1}(i)-1,\
\ \sum_{j=1}^{n}\alpha^{j}(i)\,=\,0.$

Up to a normalization factor, the norms of the trivializations in the
holomorphic picture are given by the differences
$\alpha^{j}(i)-\alpha^{j+1}(i)$. These $\alpha^{j+}(i)-\alpha^{j+1}(i)$ serve
as the moment maps for the natural torus actions on the trivializations. Under
the standard equivalence with parabolic bundles, the $\alpha^{j}(i)$ also give
the weights associated to the geometric invariant theoretic construction of
the moduli space of parabolic bundles.

On the holomorphic side, the addition of torsion means that we can obtain, for
example, moduli spaces of bundles of different degrees from the same space of
framed sheaves; the difference of degrees is “made up” by torsion. This occurs
when the equality $\alpha^{1}(i)-\alpha^{n}(i)\,=\,1$ holds for eigenvalues
(recall that the $\sum_{j=1}^{n}\alpha^{j}(i)\,=\,0$). The point here is that
while we can have locally free bundles with flags for
$\alpha^{1}(i)-\alpha^{n}(i)\,=\,1$, the orbits of these are not closed in the
GIT construction, and we can degenerate so that the bundle acquires nontrivial
torsion; these will represent the closed orbits in the GIT construction.

The moduli space of these framed sheaves is represented by a standard quotient
construction: twist the sheaf $E$ by ${\mathcal{O}}_{X}(k)$ for $k$
sufficiently large, and let $Z$ be the space of sections
$H^{0}(X,\,E(k))\,=\,H^{0}(X,\,E\otimes{\mathcal{O}}_{X}(k))$. We have the
homomorphism

$\bigwedge\nolimits^{n}Z\,=\,\bigwedge\nolimits^{n}H^{0}(X,\,E(k))\,\longrightarrow\,H^{0}(X,\,\det
E(k))\,=\,H^{0}(X,\,(\det E)\otimes\mathcal{O}_{X}(nk)).$

Following Bhosle [Bh1], and Hurtubise-Jeffrey-Sjamaar [HJS2], using the
trivialization of $\det E$ the above homomorphism gives a homomorphism

$\Phi\,\,\in\,\,{\rm
Hom}\left(\bigwedge\nolimits^{n}Z,\,H^{0}(X,\,\mathcal{O}_{X}(nk))\right).$
(2.3)

This $\Phi$ represents the sheaf $E(k)$, and so it represents $E$. Indeed, we
can define a subbundle $Ann(\Phi)$ of the trivial vector bundle
$\mathcal{Z}\,=\,Z\times X\,\longrightarrow\,X$ by

$Ann(\Phi)\,=\,\left\\{(s,\,x)\,\in\,\mathcal{Z}\,\,\big{|}\,\,\Phi(s\wedge\alpha)(x)\,=\,0\
\text{ for all
}\,\alpha\,\in\,\bigwedge\nolimits^{n-1}\mathcal{Z}\right\\}\,\subset\,\mathcal{Z}.$
(2.4)

This gives an exact sequence

$0\,\longrightarrow\,Ann(\Phi)\,\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}\,\mathcal{Z}\,=\,{\mathcal{O}}_{X}^{\oplus
N}\,\longrightarrow\,E(k)/{\rm torsion}\,\longrightarrow\,0,$ (2.5)

where $N\,=\,\dim H^{0}(X,\,E(k))$. We note that the global sections of
$Ann(\Phi)$ correspond to the sections of $E(k)$ with values in its torsion
subsheaf. Let

$T\,:=\,\eta(H^{0}(X,\,Ann(\Phi)))\,\subset\,\mathcal{Z}\,=\,{\mathcal{O}}_{X}^{\oplus
N}$

be the trivial subsheaf of $\mathcal{Z}$ generated by $H^{0}(X,\,Ann(\Phi))$,
where $\eta$ is the map in (2.5).

For $j\,=\,1,\,2,\,3$, consider the evaluation map

$Z\,=\,H^{0}(X,\,E(k))\,\longrightarrow\,E(k)_{p_{j}}.$

Take the dual of the $i$-th exterior product of it

$\varphi_{j}^{i}\,:\,\bigwedge\nolimits^{i}E(k)^{*}_{p_{j}}\,\longrightarrow\,\bigwedge\nolimits^{i}H^{0}(X,\,E(k))^{*}\,.$

Let

$\psi_{j}^{i}\,:=\,\varphi_{j}^{i}(\beta^{i}_{j})\,\in\,\bigwedge\nolimits^{i}H^{0}(X,\,E(k))^{*}$
(2.6)

be the image, under $\varphi_{j}^{i}$, of the exterior algebra element
$\beta^{i}_{j}$ in the definition of framed flags.

The planes in the flags are given by the evaluation at $p_{j}$ of
$Ann(\psi_{j}^{i})$, where $\psi^{i}_{j}$ are constructed in (2.6). The moduli
space is the GIT quotient under ${\rm GL}(N,\mathbb{C})$ of tuples
$\Phi,\psi_{j}^{i}$. Note that these elements are monomials in the exterior
algebra (point-wise, in the case of $\Phi$), that is they satisfy Plücker
relations, and that there are compatibility conditions between the
$\psi_{j}^{i}$ at each $p_{j}$. In particular, the fact that we are getting
flags at $p_{j}$ is due to the compatibility of $\Phi,\,\psi_{j}^{i}$, so that

$Ann(\Phi(p_{j}))\,\subset\,T+Ann(\psi_{j}^{i}),$

and $Ann(\psi_{j}^{i})\,\subset\,Ann(\psi_{j}^{i^{\prime}})$ whenever
$i\,<\,i^{\prime}$.

We have two lemmas from [HJS2]:

######  Lemma 2.1 ([HJS2, p. 359, Lemma 3.3]).

Let $\left(\Phi,\,(\psi^{i}_{j})_{i=1,\cdots,m,j=1,2,3}\right)$ be a semi-
stable point. Then the following two hold:

  1. (1)

$\Phi\,\neq\,0$, and $Ann(\Phi)$ is a subbundle of $\mathcal{Z}$ rank $N-n$.

  2. (2)

The intersection
$H^{0}(\Sigma,\,Ann(\Phi))\bigcap(\bigcap_{i}Ann(\psi_{j}^{i}))$ is zero for
all $j\,=\,1,\,2,\,3$.

The following lemma is on the torsion part of a semi-stable sheaf.

######  Lemma 2.2 ([HJS2, pp. 359–360, Lemma 3.4]).

  1. (1)

The torsion subsheaf of $E$ is concentrated over the points
$p_{1},\,p_{2},\,p_{3}$.

  2. (2)

For any $j\,=\,1,\,2,\,3$, let $x$ be a local holomorphic coordinate function
on $X$ around $p_{j}$, with $x(p_{j})\,=\,0$. The torsion subsheaf at $p_{j}$
is a sum of modules ${\mathcal{O}}/x^{l_{\mu}}{\mathcal{O}}$. Then all the
$l_{\mu}$ are one.

  3. (3)

The torsion submodule $T_{j}$ of $E$ at $p_{j}$ is then a sum of terms
${\mathcal{O}}/x{\mathcal{O}}$, which we write as vector spaces
$\mathbb{C}^{s_{j}}$. One then has that $s_{j}\,<\,n$. If
$e_{1},\,\cdots,\,e_{s_{j}}$ is a basis of $T_{j}$, then the contraction of
$e_{1}\wedge\cdots\wedge e_{s_{j}}$ with $\psi_{j}^{i}$ is non-zero for at
least one $i\,\geq\,s_{j}$.

  4. (4)

Let $\widehat{e}_{1},\,\widehat{e}_{2},\,\cdots,\,\widehat{e}_{s_{j}}$ be
elements of $(\mathbb{C}^{N})^{*}\,=\,H^{0}(\Sigma,\,E(k))^{*}$ corresponding
to the duals of $e_{1},\,\cdots,\,e_{s_{j}}$ under evaluation. If the orbit of
$\left(\Phi,(\psi_{j}^{i})_{i=1,\cdots,n;j=1,2,3}\right)$ is closed, then for
each $p_{j}$ we can choose the bases $e_{1},\,\cdots,\,e_{s_{j}}$ and
$\widehat{e}_{1},\,\widehat{e}_{2},\,\cdots,\,\widehat{e}_{s_{j}}$, and
complete the basis of $\mathbb{C}^{N}$ by elements
$\widehat{e}_{s_{j}+1},\,\cdots,\,\widehat{e}_{N}$, so that there exist
constants $c_{j}$, possibly zero, with
$\psi_{j}^{i}\,=\,c_{j}\widehat{e}_{1}\wedge\widehat{e}_{2}\wedge\cdots\wedge\widehat{e}_{i}$,
for $i\,=\,s_{j},\,\cdots,\,n$ (“The torsion vectors come first”),
$\psi_{j}^{i}\,=\,0$ for $i\,<\,s_{j}$, and $\psi_{j}^{s_{j}}\,\neq\,0$.

We note that in consequence the rank $s_{j}$ of the torsion simply records for
closed orbits the vanishing of $\psi_{j}^{n}$, $\psi_{j}^{i}$,
$i\,=1,\,\cdots,\,s_{j}-1$.

The result of [HJS2] tells us that there is a moduli space of these framed
sheaves on any punctured Riemann surface $\Sigma$ — with both a holomorphic
and a symplectic realization. In the holomorphic version, there is a natural
complex torus action on the framings for each marked point. Symplectically,
there is a real torus action on unitary framings, well defined as we are
imploding. Its moment map is simply the set of eigenvalues of the holonomy at
the marked point, and takes values in the fundamental alcove of ${\rm SU}(n)$.

###  2.2. Glueing

The moduli space admits a torus action at each marked point. The various
moduli spaces of parabolic bundles can be obtained as torus quotients of
$\mathcal{M}_{SL}$, or rather of strata inside $\mathcal{M}_{SL}$ given by the
vanishing of various $\beta_{j}^{i}$: in essence, we quotient out the scale
given by $\gamma^{i,i^{\prime}}_{j}$ (see (2.2)). Indeed, the stability theory
of parabolic bundles for ${\rm SL}(n,\mathbb{C})$ involves a simplex of
weights

$\Delta\,=\,\left\\{(\alpha^{1}\,\geq\,\alpha^{2}\,\geq\,\cdots\,\geq\,\alpha^{n}\,\geq\,\alpha^{1}-1)\,\,\big{|}\,\,\sum_{i=1}^{n}\alpha^{i}\,=\,0\right\\}$

for each marked point. This simplex of weights decomposes into faces

$\Delta^{I},\,\,\,I\,=\,\\{i_{1}\,<\,i_{2}\,<\,\cdots\,<\,i_{k}\\}$

consisting of the weights for which $\alpha^{i}\,>\,\alpha^{i+1}$ if and only
if $i\,\in\,I$. For the trinion, there are three simplices $\Delta_{j}$ (one
per marked point), and the faces $\Delta_{j}^{I}$ will correspond to strata in
the framed moduli space for which only the
$\beta_{j}^{i_{l}},\,\,i_{l}\,\in\,I$, are non-zero, i.e., for which there is
a partial flag at $p_{j}$ indexed by $I$. With that the stability for the
framed moduli is given by the induced parabolic bundle being stable for one
choice of weights in the corresponding $\Delta_{j}^{I}$.

There is an alternative operation: given a framed sheaf on a surface $\Sigma$
with punctures at $p_{1},\,p_{2}$ with identical vanishing patterns for the
$\beta_{1}^{j},\,\beta_{2}^{j}$ (more precisely, mapped to one another by the
reversal of order in $\\{1,\,\cdots,\,n\\}$), we can use the trivializations
to identify the respective subquotients of the flags at $p_{1},\,p_{2}$. This
amounts to taking a anti-diagonal torus quotient on the framings at
$p_{1},\,p_{2}$; symplectically, it amounts to setting
$\alpha_{1}(j)+\alpha_{2}(j)\,=\,0$, and then quotienting by the real torus.

The same operation is possible if the two points $p_{1}$, $p_{2}$ are on
different surfaces $\Sigma_{i}$. If we start with a surface $\Sigma_{0}$ built
from a set of trinions by glueing at the punctures, we can obtain a moduli
space associated to $\Sigma_{0}$ as a torus quotient of a ($3g-3$)-fold
product of the moduli $\mathcal{M}_{SL}$ associated to a trinion by iterating
the above glueing construction. Thus a toric structure on a deformation of the
space associated to $\Sigma_{0}$ depends on finding one for
$\mathcal{M}_{SL}$.

###  2.3. A generic set

We now turn to understanding an “almost toric” structure on the moduli
$\mathcal{M}_{SL}$. As this is a bit easier to describe if the top forms
$\beta_{i}^{n}$ are allowed not to coincide with the ${\rm SL}(n,\mathbb{C})$
structure at the marked points, we define

$\mathcal{M}_{GL}\,\,=\,\,\\{(E,\,\beta_{i}^{j})\,\big{|}\,\,(E,\,\beta_{i}^{1},\,\cdots,\,\beta_{i}^{n-1},\,c_{i}\beta_{i}^{n})\,\in\,\mathcal{M}_{SL}\,\
\text{ for }\,\ c_{i}\,\in\,\mathbb{C}^{*}\\}.$

Note that the sheaf $E$ retains its ${\rm SL}(n,\mathbb{C})$ structure.

There is a Zariski open subset $U\,\subset\,\mathcal{M}_{GL}$ of the moduli
space, consisting of points for which

  * •

the sheaf $E$ is a vector bundle, and is trivial,

  * •

there are full flags at the three points $p_{1},\,p_{2},\,p_{3}$, and

  * •

they are all mutually transverse in the global trivializations by sections of
$E$.

We can then normalize. It is easiest to think in terms of the equally trivial
$E^{*}$, so that our framed flags $\beta^{j}_{i}$ are now $j$-vectors. We
choose global trivializations

$e_{1},\,\cdots,\,e_{n}$

of $E^{*}$ so that the flag at $p_{1}$ is the standard one
($(E^{*})_{1}^{i}\,=\,\langle e_{1},\,\cdots,\,e_{i}\rangle$), the flag at
$p_{2}$ is the “antistandard” one ($(E^{*})_{2}^{i}\,=\,\langle
e_{n-i+1},\,\cdots,\,e_{n}\rangle$). This form of the flags reduces the
choices of basis to the diagonal torus; incorporating the framing at $p_{1}$
so that it is the standard one then fixes the scale of
$e_{1},\,\cdots,\,e_{n}$, and then the framing at $p_{2}$ will be given by
normalizing the basis $e_{i}$ by non-zero constants $a_{1},\,\cdots,\,a_{n}$:

$f_{i}\,=\,a_{i}e_{i}$

so that $\beta_{2}^{i}\,=\,(a_{1}\cdots
a_{n-i})^{-1}e_{1}^{*}\wedge\cdots\wedge e_{n-i}^{*}$. The third framed flag
at $p_{3}$ can then be given a normalized basis

$\displaystyle g_{1}\,=\,$ $\displaystyle
m_{1,1}e_{1}+m_{1,2}e_{2}+\ldots+m_{1,n}e_{n}$ $\displaystyle g_{2}\,=\,$
$\displaystyle m_{2,1}e_{1}+m_{2,2}e_{2}+\ldots+m_{2,n-1}e_{n-1}$
$\displaystyle g_{3}\,=\,$ $\displaystyle
m_{3,1}e_{1}+m_{3,2}e_{2}+\ldots+m_{3,n-2}e_{n-2}$ $\displaystyle\cdots$
$\displaystyle\cdots\,\cdots$ $\displaystyle g_{n-1}\,=\,$ $\displaystyle
m_{n-1,1}e_{1}+m_{n-1,2}e_{2}$ $\displaystyle g_{n}\,=\,$ $\displaystyle
m_{n,1}e_{1}$

with $m_{i,j}\,\in\,{\mathbb{C}}^{*}\,=\,\mathbb{C}\setminus\\{0\\}$ so that
$E_{3}^{i}\,=\,\langle g_{1},\,\cdots,\,g_{i}\rangle$. Write this as

$g\,=\,Me,$

for a matrix $M$. The total dimension of the space of $a,\,M$ is then

$d\,:=\,n+(n+1)n/2\,=\,\frac{(n+2)(n+1)}{2}-1.$

The actions of the diagonal torus $(\mathbb{C}^{*})^{n}$ of ${\rm
GL}(n,\mathbb{C})$ acting on the framings associated to the points
$p_{1},\,p_{2},\,p_{3}$ is given by its natural action on the bases
$e_{i},\,f_{i},\,g_{i}$ respectively, and so on the coefficients
$a_{i},\,m_{i,j}$. This extends to a free action of

$(\mathbb{C}^{*})^{d}\,\,=\,\,(\mathbb{C}^{*})^{\frac{(n+2)(n+1)}{2}-1}$

on $U$, by adding $\mathbb{C}^{*}$ actions, acting individually on each
$m_{i,j},\,j\,\leq\,n-i$, or on the $a_{i}$.

###  2.4. A birational map to projective space

In the previous formulation, it is unclear how to extend the action globally,
as it depends on a normalization. We can give a different encoding of our data
(equivalent to the preceding one over $U$) which will give a birational map to
projective space on which $(\mathbb{C}^{*})^{d}$ will act, albeit with
different weights. Let

$\widetilde{\mathcal{M}_{GL}}\,\,=\,\,\\{(E,\,\beta_{i}^{j},\,V)\,\,\big{|}\,\,(E,\,\beta_{i}^{j})\,\in\,\mathcal{M}_{GL},\
V\,\subset\,H^{0}(X,\,E),\ \dim V\,=\,n\\}.$

Then $\widetilde{\mathcal{M}}_{GL}$ maps bijectively to $\mathcal{M}_{GL}$ on
the generic locus where $E$ is trivial, and even on the next-to-generic locus
where the splitting type of $E$ is minimally non-trivial, i.e., the degrees of
the line bundle components are $(1,\,0,\,\cdots,\,0,\,-1)$.

We have the evaluation maps $V\,\longrightarrow\,E_{p_{j}}$, and so on
exterior powers, which we compose with the maps $\beta^{i}_{j}$, to give us

$\widetilde{\beta}^{i}_{j}\,=\,\widetilde{\beta}^{i}_{j}(E,\,\beta^{i}_{j},\,V)\,\in\,\bigwedge\nolimits^{i}V^{*}.$

We then consider for each $j_{1},\,j_{2},\,j_{3}\,\in\,\\{0,\,\cdots,\,n\\}$
summing to $n$, the elements

$\widetilde{\beta}^{j_{1},j_{2},j_{3}}\,=\,\widetilde{\beta}_{1}^{j_{1}}\wedge\beta_{2}^{j_{2}}\wedge\beta_{3}^{j_{3}}\,\in\,\bigwedge\nolimits^{n}V^{*}.$

A change of basis of $V$ changes these by the same scalar for all indices
$j_{1},\,j_{2},\,j_{3}$; we thus have a well defined element
$\phi(E,\,\beta_{i}^{j},\,V)$ of $\mathbb{P}^{d}$, as long as one of the
$\widetilde{\beta}^{j_{1},j_{2},j_{3}}$ is non-zero. This is not always the
case: for example we can have a case when the bundle $E$ is non-trivial (so
that
$\widetilde{\beta}^{n,0,0}\,=\,\widetilde{\beta}^{0,n,0}\,=\,\widetilde{\beta}^{0,0,n}\,=\,0$)
and none of the other indices $j_{1},\,j_{2},\,j_{3}$ for which the
corresponding $\widetilde{\beta}_{i}^{j_{i}}$ are non-zero actually sum to
$n$. We thus must take the closure of a graph:

$\widehat{\mathcal{M}_{GL}}\,\,=\,\,\overline{\\{((E,\,\beta_{i}^{j},\,V),\,(\widetilde{\beta}^{j_{1},j_{2},j_{3}})\,\in\,\mathcal{M}_{GL}\times\mathbb{P}^{d}\,\big{|}\,\,\phi(E,\,\beta_{i}^{j},\,V)\,=\,(\widetilde{\beta}^{j_{1},j_{2},j_{3}})\\}}.$

This gives us a sequence of maps, which are birational:

$\mathcal{M}_{GL}\,\longleftarrow\,\widetilde{\mathcal{M}}_{GL}\,\longleftarrow\,\widehat{\mathcal{M}_{GL}}\,\longrightarrow\,\mathbb{P}^{d}.$

Under these correspondences, the complement of the coordinate hyperplanes in
$\mathbb{P}^{d}$ maps bijectively to our open subset $U$. There is a natural
action of $(\mathbb{C}^{*})^{d}$ (the maximal torus of ${\rm
PGL}(d+1,{\mathbb{C}})$) on $\mathbb{P}^{d}$; it is the complexification of an
$(S^{1})^{d}$ action on $\mathbb{P}^{d}$ which is Hamiltonian, with moment map
taking values in the standard simplex. Via our birational maps, the action
transfers over to the space $U$, in a way equivalent to what we have already
defined. We would hope to extend this to all of $\mathcal{M}_{GL}$; however
this cannot be done without deforming $\mathcal{M}_{GL}$.

##  3\. Okounkov bodies and toric degeneration

To see how this can be done, we recall the theory of Okounkov bodies [KK, EK]
and the theorems of Andersen, [An], and Harada-Kaveh [HK]. A beautiful
introduction to the theory can be found in the Notices survey article of
Escobar and Kaveh [EK]. The Okounkov body of a variety $X$ depends on two
pieces of data. The first is the choice of a space $S$ of sections of a very
ample line bundle bundle $L$, giving an embedding into projective space; as
such the elements of $S$ generate the homogeneous coordinate ring of $X$.
Typically, $S$ is simply $H^{0}(X,\,L)$. The second is some coordinate system
near a point $p$, given by an ordered set of functions
$f_{1},\,\cdots,\,f_{d}$ near the point. This can be thought of as
corresponding to $d$ divisors with normal crossing at the point; as we will be
thinking of them in terms of a lexicographic order, they can be thought of as
a flag (indeed, it suffices to consider the successive intersections
$D_{1},\,D_{1}\cap D_{2},\,\cdots$). Given any other function $g$ near $p$, we
can expand $g$ into a power series
$\sum_{\alpha=(\alpha_{1},\cdots,\alpha_{d})}c_{\alpha}f_{i}^{\alpha_{1}}\cdots
f_{d}^{\alpha_{d}}$. We can then define a $\mathbb{Z}^{d}$-valued valuation
$\nu(g)$ of $g$ by

$\nu(g)\,=\,\min\\{\alpha\,\,|\,\,c_{\alpha}\,\neq\,0\\}$

here the minimum is with respect to a lexicographical order. (There are more
general valuations which could be used, however this is what we need.)

Now consider the homogeneous coordinate ring

$\mathbb{C}[X]\,=\,\bigoplus_{m\in\mathbb{Z}_{\geq 0}}\mathbb{C}[X]_{m}$

generated by products of elements of $S$ (we fix a trivialization of $L$ near
$p$); $\mathbb{C}[X]_{m}$ corresponds to the $m$-fold product. We have a semi-
group $F_{S}\,\subset\,\mathbb{Z}_{\geq 0}\times\mathbb{Z}^{d}$

$F_{S}\,=\,\bigcup_{m\geq
0}\\{(m,\,\nu(g))\,\,|\,\,g\,\in\,\mathbb{C}[X]_{m}\\}.$

Let $C$ be the closure of the convex hull of $F_{S}\cup\\{0\\}$; define the
Okounkov body $\Delta_{S}$ as the intersection of $C$ with the plane
$m\,=\,1$. A standard example would be given by $\mathbb{P}^{n}(\mathbb{C})$,
with the $S$ being the sections of
$\mathcal{O}_{\mathbb{P}^{n}(\mathbb{C})}(1)$, and the standard coordinate
planes defining the valuation; we obtain the standard simplex as the Okounkov
body. This is of course the moment polytope for the standard torus action of
$\mathbb{P}^{n}$, and the same result holds in general for toric varieties.

There is a natural question: if the Okounkov body of a variety $X$ is the
polytope of a toric variety, can the variety be deformed to a toric one? This
imposes constraints on the body, since polytopes have a finite number of
sides, and the polytopes here are rational. When the body is of this form, the
answer, by a theorem of Anderson, is yes. For a variety $X$, let $X(S)$ denote
the variety defined as the closure of the image of $X$ in $\mathbb{P}(S)$.
Anderson produces a flat degeneration $\mathbb{X}$ of $X(S)$, a family
$\mathbb{X}\,\longrightarrow\,\mathbb{C}$ with fibers $X_{t}\,=\,X(S)$ for
$t\,\neq\,0$, and $X_{0}$ toric, with constant Okounkov body.

######  Theorem 3.1 ([An, pp. 1184–1185, Theorem 1]).

Let $\nu$ be the valuation associated to a flag of subvarieties of $X$, and
let $S\,\subset\,H^{0}(X,\,L)$ be a linear system such that $F_{S}$ is
finitely generated, and the Okounkov body $\Delta_{S}$ is a rational polytope.

  * •

The variety $X(S)$ admits a flat degeneration $\mathcal{X}$ to the (not
necessarily normal) toric variety $X_{0}$. The normalization of $X_{0}$ is the
(normal) toric variety corresponding to the polytope $\Delta_{S}$.

  * •

If a torus $T$ acts on X , such that $S$ is a $T$-invariant linear system and
the flag consists of $T$-invariant subvarieties, then the degeneration is
$T$-equivariant.

  * •

Suppose $S^{\prime}\,\subset\,S$ is a subsystem inducing a birational morphism

$\phi\,:\,X(S)\,\longrightarrow\,X(S^{\prime}),$

and whose semi-group is also finitely generated. The corresponding
degenerations of $X(S)$ and $X(S^{\prime})$ are compatible: there is a
commuting diagram of flat families

$\begin{matrix}\mathcal{X}&&{\buildrel\Phi\over{\longrightarrow}}&&\mathcal{X}^{\prime}\\\
&\searrow&&\swarrow\\\ &&\mathbb{C}&&\end{matrix}$

The proof proceeds by a degeneration of the ring structure, transforming a
ring with a filtration into a graded one. Once we have the toric degeneration
we can ask what the corresponding Hamiltonian picture would be here. The
answer is given by Harada and Kaveh, under the hypotheses that the family
$\mathcal{X}\,\longrightarrow\,\mathbb{C}$ has an expression as a restriction
of a uniform projection
$\mathbb{P}^{N}\times\mathbb{C}\,\longrightarrow\,\mathbb{C}$, that the toric
action on $X_{0}$ is the restriction of an action on $\mathbb{P}^{N}$, that
the Kähler forms on $\mathcal{X}$ are restrictions of the form on
$\mathbb{P}$, and invariant under the torus action, and finally that
$\mathcal{X}$ is smooth away from the origin:

######  Theorem 3.2 ([HK, Theorems 1 and 2]).

Let $X$ be a smooth $n$-dimensional projective variety, and let $\omega$ be a
Kähler structure on $X$. Suppose that there exists a toric degeneration
$\pi\,:\,\mathcal{X}\,\longrightarrow\,\mathbb{C}$ of $X$, satisfying the
above conditions. Then:

  * •

There exists a surjective continuous map $\phi\,:\,X\,\longrightarrow\,X_{0}$
which is a symplectomorphism restricted to a dense open subset
$U\,\subset\,X$.

  * •

There exists a completely integrable system $\mu\,=\,H_{1},\,\cdots,\,H_{n}$
on $(X,\,\omega)$, in the sense of the Hamiltonians being continuous on $X$,
smooth and defining a standard integrable system on $U$, such that its moment
image $\Delta$ coincides with the moment image of $(X_{0},\,\omega_{0})$,
which is a polytope.

  * •

On the open dense subset $U$ of $X$, the integrable system
$\mu\,=\,H_{1},\,\cdots,\,H_{n}$ generates a Hamiltonian torus action, and the
inverse image $\mu^{-1}(\Delta^{0})$ of the interior of $\Delta$ under the
moment map $\mu$ of the integrable system lies in the open subset $U$.

  * •

Under the hypothesis of the semi-group $F_{S}$ being finitely generated,
$\Delta$ is the Okounkov body of $X$ under the linear system $S$ giving the
projective embedding.

All the hypotheses of Theorem 3.2 seem quite likely to hold for the case which
concerns us, apart from smoothness.

##  4\. Strata, and birational equivalence

In order to apply Anderson’s theorem to the case which interests us, we would
like to be able to see how the property of having a rational finite Okounkov
body gets transported through the birational equivalences given above. There
is no question of the body being preserved: equivariantly birational toric
varieties do not have the same polytope. To see what is at stake here,
consider the simple case of $\mathbb{P}^{2}$, with its standard torus action,
standard affine coordinates and so on. We consider the identity projective
embedding, given by section of $\mathcal{O}_{\mathbb{P}^{2}}(1)$. In a
standard trivialization, its sections are $1,\,x,\,y$; those of
$\mathcal{O}_{\mathbb{P}^{2}}(m)$ are
$1,\,x,\,y,\,x^{2},\,xy,\,y^{2},\,\cdots,\,xy^{m-1},\,y^{m}$. The Okounkov
body, corresponding to the divisors $x\,=\,0$ and $y\,=\,0$ is a simplex. Now
blow up the origin $p$,

$\pi\,:\,\widetilde{\mathbb{P}}^{2}\,\longrightarrow\,\mathbb{P}^{2},$

with exceptional divisor $E$ sitting over the origin. We have coordinates
$(\widetilde{x},\,\widetilde{y})$ on the blowup, with
$(x,\,y)\,=\,(\widetilde{x},\,\widetilde{x}\widetilde{y})$, and we take the
divisors cut out by $(\widetilde{x},\,\widetilde{y})$ to define our Okounkov
body. An embedding of $\widetilde{\mathbb{P}}^{2}$ is given by sections of
$\pi^{*}({\mathcal{O}}_{\mathbb{P}^{2}}(2))(-E)$, spanned by
$\widetilde{x},\,\widetilde{x}\widetilde{y},\,\widetilde{x}^{2},\,\widetilde{x}^{2}\widetilde{y},\,\widetilde{x}^{2}\widetilde{y}^{2}$,
or, on $\mathbb{P}^{2}$ the space of sections by $x,\,y,\,x^{2},\,xy,\,y^{2}$,
i.e., the space of sections of
${\mathcal{O}}_{\mathbb{P}^{2}}(2)\otimes\mathcal{I}_{p}$, where $\mathcal{I}$
is the ideal sheaf of the origin. These sections and their powers gives an
Okounkov body which is a truncated simplex, obtained from the standard simplex
by lopping off a corner. The result is indeed the moment polytope for
$\widetilde{\mathbb{P}}^{2}$.

It would be tempting to assert that the property of having a finitely
generated rational polytope as Okounkov body survives any birational
equivalence; we do not see any easy way of carrying this through. In our case,
fortunately, we are assisted by the existence of a finite stratification of
our spaces, with equivalences on the strata being produced by group actions
which give a certain homogeneity to the space and reduce the number of added
constraints to the Okounkov polytope to a finite number. Indeed, our space of
bundles is a space of sheaves $E$, with additional structures given by the
elements $\beta_{i}^{j}$. The types of possible sheaves $E$ are

$\mathcal{O}(m_{1})^{\ell_{1}}\oplus\mathcal{O}(m_{2})^{\ell_{2}}\oplus\cdots\oplus\mathcal{O}(m_{s})^{\ell_{s}}\oplus((\mathbb{C}_{p_{1}})^{\oplus
k_{1}})\oplus((\mathbb{C}_{p_{2}})^{\oplus
k_{2}})\oplus((\mathbb{C}_{p_{3}})^{\oplus k_{3}})$

with the possibilities for $m_{i},k_{j}$ being bounded by a degree constraint
(summing to zero) and by stability. Stratifying the space $W$ used to build
the moduli space by these types, the moduli space construction is basically
the same in a neighborhood of any point in a stratum, one point mapping to any
other by the action of the group we quotient by. Going further, and choosing
an n-dimensional space $V$ of sections, there is a further stratification by
the intersection of $V$ with the space of sections of different types
$\mathcal{O}(m_{j})^{\ell_{j}},\,(\mathbb{C}_{p_{i}})^{\oplus k_{i}}$ and by
the dimensions of the pullbacks of $E_{p_{i}}$ to $V$, and by the intersection
patterns of all these spaces. Again, any neighborhood in the moduli
construction of one pair $(E,\,V)$ in a stratum is equivalent to any other;
what distinguishes them is the values of the coordinates
$\widetilde{\beta}^{j_{1},j_{2},j_{3}}\,=\,\widetilde{\beta}_{1}^{j_{1}}\wedge\widetilde{\beta}_{2}^{j_{2}}\wedge\widetilde{\beta}_{3}^{j_{3}}$.
Now consider for each point $p$ of a stratum the set $I_{p}$ of indices
$j_{1},\,j_{2},\,j_{3}$ for which $\widetilde{\beta}^{j_{1},j_{2},j_{3}}$
vanishes. It is easy to see that if $I_{p},\,I_{q}$ are the same for $p,\,q$
in the same stratum, they are equivalent by actions of ${\rm
GL}(n,\mathbb{C})$ on $E_{p_{i}}$, $i\,=\,1,\,2,\,3$ similar to those used in
Section 2.3; basically we normalize two of the three flags, and the
coordinates of the third give the different values.

We now consider the variety $\widehat{\mathcal{M}_{GL}}$ mapping to
$\mathbb{P}^{d}$, with an isomorphism from the set $U$, to the complement of
the coordinate hyperplanes in $\mathbb{P}^{d}$. The variety
$\widehat{\mathcal{M}_{GL}}$ can thus be thought of as sitting inside a blow-
up of various coordinate k-planes in $U$; the homogeneity argument of the
previous paragraph tells us that the blowing up needed is indeed constant on
the set of a stratum with constant $I_{p}$.

The variety $\widehat{\mathcal{M}_{GL}}$ is thus embedded by the linear system
$W$ of sections of a line bundle $L$, given as the pull-back of a line bundle
$\mathcal{O}(m)$ on $\mathbb{P}^{d}$, twisted by a divisor $-E$; it
corresponds on $\mathbb{P}^{d}$ to
$H^{0}(\mathbb{P}^{d},\,\mathcal{O}(m)\otimes\mathcal{I})$ for some ideal
corresponding to a family of coordinate $k$-planes, possibly with
multiplicity, but with a finite number of generators corresponding to
$\widetilde{\beta}^{j_{1},j_{2},j_{3}}$ vanishing to an appropriate order on
our strata. We choose as origin the point $p_{0}$ corresponding to the trivial
bundle with all $\beta_{i}^{j}$, $j\,=\,1,\,\cdots,\,n-1$, zero (i.e., no
flags) and $\beta_{i}^{n}$ given by the ${\rm SL}(m,\mathbb{C})$-structure,
blowing up further if necessary to ensure that the point and its images under
rescaling the $\beta_{i}^{n}$ are smooth; we pick a system of divisors $D_{i}$
in the preimage of the coordinate planes on $\mathbb{P}^{d}$, and again there
are a finite number of rational constraints on the order of the
$\widetilde{\beta}^{j_{1},j_{2},j_{3}}$. In short, we have a rational
subpolytope of the simplex, and we can apply Anderson’s theorem to
$\widehat{\mathcal{M}_{GL}}$.

We now want to project from $\widehat{\mathcal{M}_{GL}}$ to
${\mathcal{M}_{GL}}$, again desingularising the image in $\mathcal{M}_{GL}$ of
the point $p_{0}$ and its re-scalings; let the projection be called $\rho$. It
is of course an isomorphism on $\mathcal{M}_{GL}$ away from a set of
codimension at least two, and in particular in a neighborhood of the base
point, consequently mapping our coordinate planes $D_{i}$ down to
$\mathcal{M}_{GL}$. Now take the direct image under $\rho$ of $L$. As the
sections of $L$ are all products of length $m$ of the
$\widetilde{\beta}^{j_{1},j_{2},j_{3}}$, the direct image takes values in the
$m$-th power $L_{0}^{m}$ of the natural determinant line bundle $L_{0}$ over
the moduli space. Pushing down and pulling back the sections in $W$, we obtain
a subspace $W^{\prime}\,=\,\rho^{*}\rho_{*}W\,\subset\,W$ of the space $W$,
and we can think of $W^{\prime}$ as a subspace of
$H^{0}(\mathcal{M}_{GL},\,L_{0})$. Again, by the homogeneity of our
stratification, its Okounkov body is cut out in that of
$\widehat{\mathcal{M}_{GL}}$ by a finite number of constraints, and so is a
rational polytope.

In short, again by Anderson’s theorem:

######  Theorem 4.1.

The space ${\mathcal{M}_{GL}}$ admits a degeneration to a toric
$\mathcal{M}_{GL}^{0}$. The torus actions on the framings on
$\mathcal{M}_{GL}$ extend to torus actions on $\mathcal{M}_{GL}^{0}$.

##  5\. Hamiltonians and imploded cross-sections

###  5.1. Doubles and the construction of the moduli space

As we noted in the introduction, our space of framed sheaves admits a
symplectic interpretation, as a space of representations of the fundamental
group of the three-punctured sphere into ${\rm SU}(n)$, decorated with some
extra structure at the marked points. We would like in this section to explore
our “almost toric structure” from this symplectic point of view.

We first recall some constructions, made in [AMM] and [HJS1, Section 6], of
the double $DK\,=\,K\times K$ of a compact semi-simple group $K$, and of its
imploded cross-section. Setting $DK\,=\,\\{(u,\,v)\,\in\,K\times K\\}$, we can
think of $DK$ as the space of representations of the fundamental group of a
cylinder, i.e., flat $K$-connections, but on a bundle equipped with
trivializations at two points $q_{1},\,q_{2}$, one at each end of the
cylinder. The element $u$ represents parallel transport from $q_{1}$ to
$q_{2}$, and the element $v$ represents parallel transport around the end,
starting at $q_{2}$. The space of representations comes equipped with a two-
form, defined by:

$\omega\,\,=\,\,-\frac{1}{2}\bigl{(}Ad(v)u^{*}\theta_{L},\,u^{*}\theta_{L}\bigr{)}-\frac{1}{2}\bigl{(}u^{*}\theta_{L},\,v^{*}(\theta_{L}+\theta_{R})\bigr{)}.$

Here $\theta_{L}$ is the left Maurer-Cartan form, and $\theta_{R}$ is the
right form. The form $\omega$ is not closed, and $d\omega$ is given by the
pull-back under a suitable group-valued moment map of the canonical three-form
on the compact group. In our case, the group action is one of $K\times K$ on
$DK$:

$(k_{1},\,k_{2})(u,\,v)\,=\,(k_{1}uk_{2}^{-1},\,k_{2}vk_{2}^{-1}).$

In terms of flat connections on the cylinder, this is the action corresponding
to changes of the trivializations at $p_{1},\,p_{2}$. The moment map here is
given by $(Ad(u)v^{-1},\,v)\,\in\,K\times K$, and $DK$ is a quasi-Hamiltonian
$K\times K$ space. This three-form disappears under restriction to a torus or
under quasi-Hamiltonian reduction, and as this is what we will be doing, the
two form will end up being closed.

The next step will be to restrict $v$ to lie in the fundamental alcove
$\Delta$ of the torus $T$, the polytope representing conjugacy classes in $T$.
For ${\rm SU}(n)$, we take a logarithm and divide by $2\pi\sqrt{-1}$, so that

$v\,=\,\exp(2\pi\sqrt{-1}(\alpha_{1},\,\alpha_{2},\,\cdots,\,\alpha_{n})).$

The alcove $\Delta$ is precisely the set of ordered eigenvalues

$\Delta\,=\,\left\\{(\alpha_{1}\,\geq\,\alpha_{2}\,\geq\,\cdots\,\geq\,\alpha_{n}\,\geq\,\alpha_{1}-1)\,\,\big{|}\,\,\sum_{i=1}^{n}\alpha_{i}\,=\,0\right\\}$

for each marked point. This simplex of weights decomposes into faces

$\Delta^{I},\ \ \,I\,=\,\\{i_{1}\,<\,i_{2}\,<\,\cdots\,<\,i_{k}\\}$

consisting of the weights for which $\alpha_{i}\,>\,\alpha_{i+1}$ if and only
if $i\,\in\,I$. There is a similar decomposition for arbitrary $K$.

We now can restrict to $K\times\Delta\,\subset\,DK$. For $v$ in the interior
of $\Delta$, the form is non-degenerate; but on the boundary strata
$\Delta_{I}$ of $\Delta$ there are null directions, which are orbits of the
commutator group $[K_{I},\,K_{I}]$ of the stabilizer $K_{I}$ of the elements
of $\Delta_{I}$; see [HJS1]. We quotient these out, to obtain the imploded
cross-section:

$DK_{\mathrm{impl}}\,\,=\,\,\sqcup_{I}K_{I}/[K_{I},K_{I}]\times\Delta_{I}.$
(5.1)

This is a stratified quasi-Hamiltonian $K\times T$–space. The action at
$p_{2}$ is now restricted to the torus $T$.

We can use the $DK_{\mathrm{impl}}$ in (5.1) as building blocks for the
construction of the moduli spaces $M$ of framed representations on the
trinion. Indeed, take three copies of $DK_{\mathrm{impl}}$:

$\left\\{(u_{1},\,v_{1},\,u_{2},\,v_{2},\,u_{3},\,v_{3})\,\in\,DK_{\mathrm{impl}}^{3}\right\\}.$
(5.2)

Together, their elements represent holonomies $v_{i}\,\in\,\Delta$ at the
punctures $p_{i}$, $i\,=\,1,\,2,\,3$, along with parallel transport $u_{i}$ to
a common point $p_{0}$. The constraint that they define together a
representation of the fundamental group

$u_{1}v_{1}u_{1}^{-1}u_{2}v_{2}u_{2}^{-1}u_{3}v_{3}u_{3}^{-1}\,=\,1$

is equivalent to setting to $1$ the moment map for the diagonal action of $K$,
with a quotienting by $K$ eliminating the choice of an arbitrary
trivialization at $p_{0}$. Thus, the end result is:

######  Theorem 5.1 ([HJS1]).

The space $M$

$M\,=\,\left\\{(u_{1},\,v_{1},\,u_{2},\,v_{2},\,u_{3},\,v_{3})\,\in\,DK_{\mathrm{impl}}^{3}\,\,\big{|}\,\,u_{1}v_{1}u_{1}^{-1}u_{2}v_{2}u_{2}^{-1}u_{3}v_{3}u_{3}^{-1}\,=\,1\right\\}/K$

is a Hamiltonian $T^{3}$ space with moment map

$(u_{1},\,v_{1},\,u_{2},\,v_{2},\,u_{3},\,v_{3})\,\longmapsto\,(v_{1},\,v_{2},\,v_{3})\,=\,V.$

Its Hamiltonian reduction at $V=(v_{1},\,v_{2},\,v_{3})$ is the moduli space
$M_{V}$ of parabolic bundles with holonomy $v_{i}$ at $p_{i}$. The reduction
fixes the $v_{i}\,\in\,\Delta$, and quotients out the orbit of
$K_{i}/[K_{I},\,K_{I}]$, eliminating the framings of the subquotients at the
punctures.

###  5.2. Hamiltonian flows and vector fields

We would like the Hamiltonian flows on the generic locus of $M$, corresponding
to the toric structure on the degeneration. This locus will be given by first
asking that the $v_{i}$ lie in the interior $\Delta_{0}$ of $\Delta$ (over
which $DK_{\mathrm{impl}}$ is simply $K\times\Delta_{0}$), and also asking
that the frames represented by $u_{i}$ satisfy certain transversality
conditions. As this is simpler to see in the complex domain, we first
complexify $M$ and $DK_{\mathrm{impl}}$ to spaces
$M_{\mathbb{C}},(DK_{\mathrm{impl}})_{C}$ and consider everything over the
complex domain, so that the $u_{i}$ belong to the complexified group and the
$v_{i}$ to the complement of the root planes in $T_{\mathbb{C}}$.

We specialize to ${\rm SU}(n)$ and its complexification ${\rm
SL}(n,\mathbb{C})$. The interior $\Delta_{0}$ of the fundamental alcove
corresponds to distinct eigenvalues and in the complex picture (for $M$) to
full flags. With the usual conventions, these flags correspond in the torus
trivialization to $\langle e_{1}\rangle\,\subset\,\langle
e_{1},\,e_{2}\rangle\,\subset\,\cdots$. We will keep this ordering for
$p_{1}$, and take the inverse order $\langle e_{n}\rangle\,\subset\,\langle
e_{n},\,e_{n-1}\rangle\,\subset\,\cdots$ for $p_{2},\,p_{3}$. Matrices denoted
by $b$ will be triangular, either upper ($b^{+}$) or lower ($b^{-}$); those
denote by $n$ will be strictly triangular (i.e., with 1’s on the diagonal) and
either upper ($n^{+}$) or lower ($n^{-}$); those denoted by $d$ will be
diagonal. Corresponding Lie algebra elements will be denoted by
$\beta^{\pm},\mu^{\pm},\delta$, with $\beta^{\pm}=\mu^{\pm}+\delta$;
variations of the $v$ will denoted by $\zeta$.

Over a generic locus, then, we can factor the $u_{i}$ as
$b_{i}^{+}n_{i}^{-}=n_{i}^{+}d_{i}n_{i}^{-}$, or the other way round. We can
also suppose that the flag transported from $p_{1}$ and the flag transported
from $p_{2}$ intersect transversally at $p_{0}$, and so we can normalize our
frame at $p_{0}$ by choosing the intersections as a basis so that $u_{1}$ is
an upper triangular matrix $b_{1}^{+}=n_{1}^{+}$ (i.e., preserving the flag at
$p_{1}$), $u_{2}$ is a strictly lower triangular matrix $n_{2}^{-}$,
(preserving the flag and the trivialization from $p_{2}$), and, with some
further genericity, $u_{3}$ factors uniquely as $b_{3}^{+}n_{3}^{-}$, a
product of an upper triangular matrix and a strictly lower triangular matrix,
which we also write as $n_{3}^{+}d_{3}n_{3}^{-}$.

We then have the relation

$b_{1}^{+}v_{1}(b_{1}^{+})^{-1}n_{2}^{-}v_{2}(n_{2}^{-})^{-1}\,=\,b_{3}^{+}n_{3}^{-}v_{3}^{-1}(n_{3}^{-})^{-1}(b_{3}^{+})^{-1}.$
(5.3)

In this normalisation, $n_{3}^{+}$ represents the relative positions of the
flags, and the diagonal components $d_{1},d_{3}$ the framings. As we are on
the generic, stable, locus, we can vary these (framed) flags (i.e.,
$b_{3}^{+}$) freely while keeping the $v_{i}$ fixed, and there will be a
corresponding variation of the $u_{i}$ on the unitary locus; this, however,
will be accompanied by a corresponding variation of
$n_{3}^{-},b_{1}^{+},n_{2}^{-}$ in order to preserve 5.3.

We now evaluate the form $\omega$ on two-parameter infinitesimal variations
$u,\,v$ over $(K\times\Delta_{0})_{\mathbb{C}}$. We set

$\displaystyle
b^{+}(1+\epsilon\beta^{+}+\delta\widehat{\beta}^{+})n^{-}(1+\epsilon\mu^{-}+\delta\widehat{\mu}^{-})$
$\displaystyle=\,b^{+}n^{-}(1+\epsilon((n^{-})^{-1}\beta^{+}n^{-}+\mu^{-})+\delta((n^{-})^{-1}\widehat{\beta}^{+}n^{-}+\widehat{\mu}^{-}))$

and

$v(1+\epsilon\zeta+\delta\widehat{\zeta}),$

and so the form applied to the variations becomes

$\displaystyle\omega((\beta^{+},\,\mu^{-},\,\zeta),\,(\widehat{\beta}^{+},\,\widehat{\mu}^{-},\,\widehat{\zeta}))\,=$
$\displaystyle\frac{1}{2}tr((v(n^{-})^{-1}\beta^{+}n^{-}v^{-1})$
$\displaystyle((n^{-})^{-1}\widehat{\beta}^{+}n^{-})-(v(n^{-})^{-1}\widehat{\beta}^{+}n^{-}v^{-1})((n^{-})^{-1}\beta^{+}n^{-}))$
$\displaystyle+\frac{1}{2}tr((v(n^{-})^{-1}\beta^{+}n^{-}v^{-1})$
$\displaystyle\widehat{\mu}-(v(n^{-})^{-1}\widehat{\beta}^{+}n^{-}v^{-1})\mu^{-})$
$\displaystyle+\frac{1}{2\pi
i}tr((n^{-})^{-1}\beta^{+}n^{-}+\mu^{-})\widehat{\zeta}$
$\displaystyle-((n^{-})^{-1}\widehat{\beta}^{+}n^{-}+\widehat{\mu}^{-})\zeta).$

For $(\beta^{+},\,\mu^{-},\,\zeta)\,=\,(0,\,\mu^{-},\,0)$ this becomes:

$\displaystyle\omega((0,\mu^{-},0),(\widehat{\beta}^{+},\widehat{\mu}^{-},\widehat{\zeta}))\,=\,$
$\displaystyle\frac{1}{2}tr(-(v(n^{-})^{-1}\widehat{\beta}^{+}n^{-}v^{-1})\mu^{-})$
$\displaystyle=\,$
$\displaystyle\frac{1}{2}tr(-\widehat{\beta}^{+}(n^{-}v^{-1}\mu^{-}v(n^{-})^{-1}))$
$\displaystyle=\,$
$\displaystyle\frac{1}{2}tr(-\widehat{\mu}^{+}(n^{-}v^{-1}\mu^{-}v(n^{-})^{-1}))$
(5.4)

Note that this implies that the various vectors tangent to the space
$n^{+}B^{-}$, that is $\mu^{-},\,\delta$ pair amongst themselves to zero. The
formula (5.4) also shows that the basis tangent vector
$(\mu(i,\,j)^{+})_{k,l}\,=\,\delta_{k,i}\delta_{l,j}$, $i\,<\,j$, pairs non-
trivially with $(\mu(j,\,i)^{-})_{k,l}\,=\,\delta_{k,j}\delta_{l,i}$ with
coefficient $(b^{-})_{j,j}(b^{-})_{i,i}^{-1}v_{i}v_{j}^{-1}$. Furthermore,
again for $i\,<\,j$, $\mu(i,\,j)^{+}$ only pairs non-trivially with
$\mu(i^{\prime},\,j^{\prime})^{-}$ if
$j^{\prime}\,\leq\,i\,<\,j\,\leq\,i^{\prime}$. This ensures that, for
$i\,<\,j$ the symplectic form gives a nondegenerate pairing between the
$\mu(i,\,j)^{+}$ and $\mu(j^{\prime},\,i^{\prime})^{-}$, which we write as

$\omega(\mu(i,\,j)^{+},\mu(i^{\prime},j^{\prime})^{-})\,=\,A_{i,j;i^{\prime}j^{\prime}}$

so that on linear combinations $P\,=\,\sum_{i<j}p_{i,j}\mu(i,\,j)^{+}$,
$Q\,=\,\sum_{i>j}q_{i,j}\mu(i,j\,)^{-}$, we have, schematically

$\omega(P,\,Q)\,=\,p^{T}Aq.$

###  5.3. Hamiltonians

The formula above also suggest that the components of the group $N^{+}$ of
strictly upper triangular matrices should be the Hamiltonians generating the
vector fields tangent to $b^{+}N^{-}$. Now note that the group $N^{+}$ admits
a logarithm (inverting the exponential map) with values in the Lie algebra
$\mathfrak{n}^{+}$ of strictly upper triangular matrices. Let
$\xi\,\in\,\mathfrak{n}^{+}$ with $\exp(\xi)\,=\,n^{+}$, so that
$\xi\,=\,\log(n^{+})$. We have the formula

$\displaystyle\frac{d}{dt}\exp(\xi+t\zeta)|_{t=0}\,=$
$\displaystyle\,\exp(\xi)\cdot\Big{(}\frac{1-\exp(-ad(\xi))}{ad(\xi)}\Big{)}\zeta$
(5.5) $\displaystyle=$
$\displaystyle\,\exp(\xi)(1-\frac{ad(\xi)}{2}+\frac{ad(\xi)^{2}}{6}-\ldots)\zeta.$

Since the adjoint action is nilpotent, this gives a bijective map between the
tangent vectors to $\mathfrak{n}^{+}$ and the tangent vectors to $N^{+}$.
Consider the components of $\xi$, for $i\,<\,j$:

$H_{i,j}(n^{+},b^{-},v)\,=\,\log(n^{+})_{i,j}\,=\,\xi_{i,j}.$

By the formula (5.5) above, the differential of $H_{i,j}$ evaluated on the
left invariant vector fields $P\,=\,\sum_{i<j}p_{i,j}\beta(i,j)^{+}$ is a
combination of coefficients:

$dH_{i,j}\,=\,p_{i,j}+\sum_{i^{\prime}<i,j^{\prime}>j}C_{i^{\prime},i;j^{\prime},j}(b^{+})p_{i^{\prime},j^{\prime}}$

or, schematically,

$dH_{i,j}\,=\,(p^{T}C)_{i,j}.$

Combining, since we want for our Hamiltonian vector fields
$X_{H_{i,j}}\,=\,\sum_{i^{\prime}j^{\prime}}M_{i^{\prime},j^{\prime};i,j}\mu(j^{\prime},i^{\prime})^{-}$
that, for all $P$

$dH_{i,j}(P)\,=\,-\omega(P,X_{H_{i,j})}$

we have

$(p^{T}C)_{i,j}\,=\,-(p^{T}AM)_{i,j}$

giving $M\,=\,A^{-1}C$ and

$X_{H_{i,j}}\,=\,-\sum_{i^{\prime}j^{\prime}}(A^{-1}C)_{i^{\prime},j^{\prime};i,j}\mu(j^{\prime},i^{\prime})^{-}$

for $i\,<\,j$.

Finally, we have two ”symplectically dual” quantities. First, the Hamiltonians
associated to the torus coefficients $v$: set as Hamiltonians

$H^{V}_{l}\,=\,\frac{1}{2\pi i}\log(v)_{l},\ \ \,l\,=\,1,\,\cdots,\,n-1.$

In the complex domain these would be multi-valued. They give the torus actions
associated to $M$ by Theorem 5.1, acting on the $u_{i}$ by right
multiplication. Their vector fields are given by left invariant vector fields
$\delta$ corresponding to the diagonal action on the right. By (5.4), they
Poisson commute with the $H_{i,j}$. Dually, we can also take as Hamiltonians
the entries of the diagonal matrix $d$ in a decomposition $n^{+}dn^{-}$; again
they Poisson commute with the $H_{i,j}$. Their vector fields will be vectors
$\zeta$ tangent to the diagonal matrices $v$.

Summarizing:

######  Theorem 5.2.

The Hamiltonians $H_{i,j},\,1\,\leq\,i\,<\,j\,\leq\,n$ and $H^{V}_{l}$,
$l\,=\,1,\,\cdots,\,n-1$ generate a completely integrable Hamiltonian system
(i.e., a holomorphic Lagrangian foliation) in the generic set of
$(DK_{\mathrm{impl}})_{\mathbb{C}}$, with leaves $n^{+}B^{-}\times\\{v\\}$.
Similarly, the Hamiltonians $H_{i,j},\,1\,\leq\,i\,<\,j\,\leq\,n$ and the
components $d_{i}$, $l\,=\,1,\,\cdots,\,n$ of $d$ generate a completely
integrable Hamiltonian system (i.e., a holomorphic Lagrangian foliation) in
the generic set of $(DK_{\mathrm{impl}})_{\mathbb{C}}$, with leaves
$n^{+}N^{-}\times\ (\Delta_{0})_{\mathbb{C}}$

The Hamiltonians $H_{i,j}$ are of course defined only on a generic locus; we
give other Hamiltonians, more globally defined, which give the same ring, and
which descend to $M$.

###  5.4. Global and reducible Hamiltonians

We now have Lagrangian foliations, with associated Hamiltonians, on each of
the three $DK_{\mathrm{impl}}$ factors. We would like to combine these into
one set. Thinking of $DK_{\mathrm{impl}}$ or its complexification as
parametrizing framed representations, or their complexification, we have at
each puncture $p_{i}$ a torus-valued endomorphism $v_{i}$, and a flag of
subspaces

$E_{i,j}\,=\,\langle e_{1},\,e_{2},\,\cdots,\,e_{j}\rangle,\,\ \text{ for }\
i\,=\,1$

and

$E_{i,j}\,=\,\langle e_{n},\,e_{n-1},\,\cdots,\,e_{n-j+1}\rangle,\,\ \text{
for }\ i\,=\,2,\,3.$

The map $v_{i}$ generates an automorphism $V_{i,j}\,=\,\bigwedge^{j}(v_{i})$
(a scalar) of $\bigwedge^{j}E_{i,j}$, essentially as the product of the first
$j$ eigenvalues. Now transport the basis $e_{j}$ of $E_{i,j}$ to $p_{0}$ using
the $u_{i}$, obtaining vectors $u_{i}(e_{j})$ in the fiber $E_{p_{0}}$ of the
bundle $E_{p_{0}}$, and so, taking their product, a $j$-vector
$u_{i}(E_{i,j})$ in $\bigwedge^{j}(E_{p_{0}})$. The fiber $E_{p_{0}}$ has a
top $n$-form $dvol$ given by the ${\rm SL}(n,{\mathbb{C}})$ structure; for
$j_{1}+j_{2}+j_{3}\,=\,n$, consider the matrix

$G_{j_{1},j_{2},j_{3}}\,=\,(u_{1}(e_{1},\,\cdots,\,e_{j}),\,u_{2}(e_{n},\,\cdots,\,e_{n-j_{2}+1}),\,u_{3}(e_{n},\,\cdots,\,e_{n-j_{3}+1})$

and take the volume:

$dvol(u_{1}(E_{1,j_{1}})\wedge u_{2}(E_{2,j_{2}})\wedge
u_{3}(E_{3,j_{3}}))\,=\,\det G_{j_{1},j_{2},j_{3}}.$

Note that decomposing $u_{1}$ as $b_{1}^{-}n_{1}^{+}$ we can replace $u_{1}$
by $b_{1}^{-}$; likewise, setting $u_{j}=b_{j}^{+}n_{j}^{-},j=2,3$, we can
replace $u_{j}$ by $b_{j}^{+}$ for $j=2,3$ This is invariant under the left
action of ${\rm SL}(n,\mathbb{C})$ on the $u_{i}$. It is not invariant under
the action of the tori associated to the punctures. (On our generic set we
could, for example, use the right torus action on $u_{i}$ to normalize its
principal minors $\det(u_{i})_{s,t}$, $s,\,t\,\in\,\\{1,\,\cdots,\,j\\}$,
$i\,=\,1,\,3$ and $\det(u_{2})_{s,t}$, $s,\,t\,\in\,\\{n-j+1,\,\cdots,\,n\\}$,
$i\,=\,1,\,3$ to one; but then you lose the ${\rm
SL}(n,\mathbb{C})$-invariance) We set for $j_{1}+j_{2}+j_{3}\,=\,n$,
$j_{i}\,\in\,\\{0,\,\cdots,\,n-1\\}$, ,

$\widetilde{H}^{G}_{j_{1},j_{2},j_{3}}\,=\,\det G_{j_{1},j_{2},j_{3}}.$ (5.6)

In our normalization of
$u_{1}\,=\,b_{1}^{+}=\,n_{1}^{+}d_{1},\,u_{2}\,=\,n_{2}^{-},$ and
$u_{3}\,=\,b_{3}^{+}n_{3}^{-}=\,n_{3}^{+}d_{3}n_{3}^{-}=$, the Hamiltonians
become determinants of submatrices of $b_{3}^{+}$, times a product of entries
of $d_{1}$:

$H^{G}_{j_{1},j_{2},j_{3}}\,=\,d_{1,1}...d_{1,j_{1}}\det((b_{3}^{+})_{i,j},\,i\,=\,j_{1}+1,\,\cdots,\,j_{1}+j_{2},\,j\,=\,n-j_{2}+1,\,\cdots,\,n).$

######  Proposition 5.3.

In the above normalization on the generic set there is a biholomorphism
relating the $H^{G}_{j_{1},j_{2},j_{3}}$ and (the Hamiltonians $H_{i,j}$ and
the coefficients of $d_{1},d_{3}$) so that they represent a reparametrization
of our Hamiltonians.

###### Proof.

Note that in both cases there are $(n+4)(n-1)/2$ functions; now let us fix the
$d_{i}$; this leaves $n(n-1)/2$ functions on both sides.. We first remark that
under the exponential map, the $H_{i,j}$ map biholomorphically to the
coefficients of $n_{3}^{+}$. The map giving $H^{G}_{j_{1},j_{2},j_{3}}$ from
$(n_{3}^{+})_{i,j}$ is given above; conversely, first remark that for
$j_{3}\,=\,1$, the Hamiltonians $H^{G}_{j_{1},j_{2},j_{3}}$ are simply (up to
the diagonal elements of $b_{1}^{+}$, and a sign) the coefficients
$(n_{3}^{+})_{i,n}$. Now take $j_{3}\,=\,2$, $j_{1}\,=\,n-3$. The value of
$(n_{3}^{+})_{n-1,n-1}$ being one, we can compute $(n_{3}^{+})_{n-2,n-1}$ from
the determinant $H^{G}_{n-3,1,2}$; keeping $j_{3}\,=\,2$, and decreasing
$j_{1}$, we can then compute successively the coefficients
$(n_{3}^{+})_{i,n-1}$ from $H^{G}_{i-1,1,2}$. Now begin again with
$j_{3}\,=\,3$, we can again work our way up, computing successively
$(n_{3}^{+})_{n-3,n-2},\,(n_{3}^{+})_{n-4,n-2},\,\cdots$ and so on. We then
proceed in the same fashion for $j_{3}\,=\,4,\,\cdots,\,n-1$. ∎

The $H^{G}_{j_{1},j_{2},j_{3}}$, by the previous section, Poisson commute.
They are invariant under the left action of ${\rm SL}(n,\mathbb{C})$, and so
descend to commuting Hamiltonians on the quotiented space $M$, by the results
of [AMM], section 5. Note that the $H^{G}_{j_{1},j_{2},j_{3}}$ give us well
defined functions on ${\rm SL}(n,\mathbb{C})\times T_{\mathbb{C}}$; however,
when restricted to ${\rm SU}(n)\times\Delta$, they will not be invariant under
$[K_{I},\,K_{I}]$, except on the generic locus $\Delta_{0}$ where
$[K_{I},\,K_{I}]$ is trivial.

###  5.5. Reality

We thus have an isotropic holomorphic foliation along (an open dense subset)
of the complexification $M_{\mathbb{C}}$ of our space $M$, cut out by the
functions $H^{G}_{j_{1},j_{2},j_{3}}$. This does not yet give us a real
Lagrangian foliation on $M$, as indeed, even on $M$, this foliation is very
far from being real. We must adjust.

The subset $M\,\subset\,M_{\mathbb{C}}$ is the fixed point locus of an
involution $I$, induced from the involution on ${\rm SL}(n,\mathbb{C})$ given
by

$I(g)\,=\,(g^{*})^{-1}.$

Our symplectic form $\omega$ is invariant under $I$, in the sense that
$\overline{I^{*}(\omega)}\,=\,\omega$. We will want as Hamiltonians functions
invariant under the involution $I$. For the torus Hamiltonians, this is
straightforward: set

$\alpha_{k}^{l}\,=\,Re(H^{V}_{k,l}).$

These are the coefficients $\alpha$ in the fundamental alcove, but with the
order reversed for $k\,=\,2,\,3$. For the other Hamiltonians, we take

$H^{GG}_{j_{1},j_{2},j_{3}}\,=\,A(\alpha_{1}^{j_{1}+1}-\alpha_{1}^{j_{1}})(\alpha_{2}^{n-j_{2}}-\alpha_{2}^{n-j_{2}+1})(\alpha_{3}^{n-j_{3}}-\alpha_{3}^{n-j_{3}+1})(I^{*}H^{G}_{j_{1},j_{2},j_{3}})^{*}H^{G}_{j_{1},j_{2},j_{3}};$
(5.7)

here $A$ is the product
$(\alpha_{1}^{1}-\alpha_{1}^{n}-1)(\alpha_{2}^{1}-\alpha_{2}^{n}-1)(\alpha_{3}^{1}-\alpha_{3}^{n}-1)$.
The $\alpha$–factors and $A$–factors are there to ensure that the Hamiltonian
is invariant under the action of $[K_{I},\,K_{I}]$, on the strata where this
is non-trivial. When we are on the unitary locus, the $H^{GG}$ are up to a
real factor the determinants of a Hermitian matrix of the form

$\begin{pmatrix}a_{1,1}&a_{1,2}&a_{1,3}\\\ a_{1,2}^{*}&a_{2,,2}&a_{2,3}\\\
a_{1,3}^{*}&a_{2,3}^{*}&a_{3,3}\end{pmatrix},$

with the $a_{i,i}$ diagonal and real. As the matrix is Hermitian, the
determinant is real; it is invariant both under the ${\rm SU}(n)$ action and
the unitary torus actions. It is the product of $H^{G}_{j_{1},j_{2},j_{3}}$
and its complex conjugate, and so represents the norm squared of
$H^{G}_{j_{1},j_{2},j_{3}}$, again up to some factors.

There remains to see that these Hamiltonians commute. Denote
$\\{j_{1},\,j_{2},\,j_{3}\\}$ by $\sigma$, and $\\{k_{1}\,,k_{2},\,k_{3}\\}$
by $\rho$. We already have $X_{H^{G}_{\sigma}}(H^{G}_{\rho})\,=\,0$; also
$H^{G}_{\rho}$ is constant along the $H^{G}_{\sigma}$ flow. If a function is
constant, its norm squared is too, and so
$X_{H^{G}_{\sigma}}(H^{GG}_{\rho})\,=\,0$. Likewise,
$\overline{X_{H^{G}_{\sigma}}}(H^{GG}_{\rho})\,=\,0$, and so

$X_{H^{GG}_{\sigma}}(H^{GG}_{\rho})\,=\,0.$

The functions Poisson commute amongst themselves, and evidently commute with
the torus Hamiltonians.

##  6\. Degenerations of bundle moduli

Let us start with the moduli space $\mathcal{M}_{SL}(Y_{1})$ of ${\rm
SL}(n,\mathbb{C})$ bundles on a smooth genus $g$ curve $Y_{1}$, or
equivalently, the space of representations of the fundamental group of $Y_{1}$
into ${\rm SU}(n)$. The results of [BHu] give us a degeneration of the moduli
spaces ${\mathcal{M}_{SL}}(Y_{t})$ associated to a degeneration of $Y_{1}$
through smooth curves $Y_{t}$ into a “totally nodal” curve $Y_{0}$. This is a
curve obtained by glueing $(2g-2)$ three-punctured spheres $X_{i}$ pairwise at
their punctures, with $3g-3$ glueings in all. The limiting moduli space
${\mathcal{M}_{SL}}(Y_{0})$ is a quotient of the product of the moduli spaces
$\mathcal{M}_{SL}(X_{i})$ of our framed parabolic sheaves over trinions, where
the quotient is by the anti-diagonal torus actions on the trivializations
associated to the elements of $\mathcal{M}_{SL}(X_{i})$: that is, if the
puncture $p_{i}$ of $X_{i}$ is being identified, to $p_{j}$ of $X_{j}$, we
identify the trivializations associated to elements of
$\mathcal{M}_{SL}(X_{i})$ at $p_{i}$ with the trivializations associated to
elements of $\mathcal{M}_{SL}(X_{j})$ at $p_{j}$; note that the glueing is
only partial, of subquotients of the bundle. We then quotient by the anti-
diagonal torus action preserving these identifications. This quotienting can
either be done symplectically, or holomorphically. Note that symplectically,
there is a reversal of the $\alpha_{i}$ on the two punctures being glued:
$\alpha^{i}_{1}=-\alpha_{2}^{n-i+1}$, hence the anti-diagonal quotient instead
of the diagonal one.

Thus we have a degeneration of ${\mathcal{M}_{SL}}(Y_{t})$ into a limit
${\mathcal{M}_{SL}}(Y_{0})$, which is a quotient, symplectic or holomorphic,
of a product of $\mathcal{M}_{SL}(X_{i})$. Note that this is not a
degeneration with a constant fiber away from $0$; the
${\mathcal{M}_{SL}}(Y_{t})$ are different, and indeed there is a Torelli
theorem telling us that ${\mathcal{M}_{SL}}(Y_{t})$ determines $Y_{t}$ [BGL],
[GL].

The ${\mathcal{M}_{SL}}(X_{i})$ themselves are not toric, except for ${\rm
SL}(2,\mathbb{C})$, when they are identified with
$\mathbb{P}^{3}(\mathbb{C})$. However, since we now have a toric degeneration
of the $\mathcal{M}_{SL}(X_{i})$ which is equivariant for the torus actions at
the punctures, we now have one for ${\mathcal{M}_{SL}}(Y_{0})$, and so a
degeneration for ${\mathcal{M}_{SL}}(Y_{t})$.

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