# Polyhedral symplectic maps and hyperKähler geometry

Yann Rollin  Yann Rollin, Laboratoire Jean Leray, Université de Nantes
<EMAIL_ADDRESS>
###### Abstract.

We provide an interpretation of the space of symplectic diffeomorphism of a
$4$-dimensional real torus $T^{4}$, as the vanishing locus of a hyperKähler
moment map. This construction leads to a notion of modified hyperKähler moment
map flow, whose regular fixed points coincide with the symplectic
diffeomorphisms of $T^{4}$.

Given a Euclidean triangulation of $T^{4}$, we introduce a notion of
polyhedral maps and polyhedral symplectic maps of the torus. We show that all
the construction carried out in the smooth setting have analogues in the
polyhedral setting. In this context, the hyperKähler modified moment map flow
is an ordinary differential equation, which is always convergent. Under a non-
degeneracy assumption of the limit, this provides an effective construction of
polyhedral symplectic maps of the torus.

This paper lays out the ground work for some experimental symplectic geometry.
In an upcoming work with François Jauberteau [3], we write a computer program
that runs a numerical version of the flow and generates non trivial examples
of symplectic polyhedral maps of $T^{4}$.

###### Key words and phrases:

Polyhedral geometry, piecewise linear geometry, symplectic maps, hyperKähler,
moment map

###### 2010 Mathematics Subject Classification:

Primary 53D05, 53D20; Secondary 53D30, 57K17

##  1\. Introduction

###  1.1. Background

The group of _symplectic diffeomorphisms_ is a central object for _Hamiltonian
mechanics_ and _classical differential geometry_. The famous _Arnold
conjecture_ is about the number of _fixed points_ of a symplectic
diffeomorphism and can be understood as a natural extension of the _Poincaré-
Birkhoff fixed point theorem_. The Arnold conjecture has attracted a lot of
interest and stirred deep results in the field of symplectic topology,
intimately related to the existence of _pseudoholomorphic curves_ , _Floer
homology_ , and the idea of _soft versus hard_ symplectic geometry developed
by Gromov. These questions naturally related to _rigidity_ or _flexibility_
properties of the group of _symplectic diffeomorphisms_.

In particular, Eliasberg and Gromov proved a surprising result: although it is
possible to approximate any diffeomorphism by _volume preserving
diffeomorphisms_ , in the $C^{0}$-sense, it is not possible to do so with
symplectic diffeomorphisms. More precisely, the Eliashberg-Gromov theorem
states that every smooth diffeomorphism, which is a $C^{0}$-limit of a
sequence of symplectic diffeomorphism, must be symplectic as well. This
rigidity result is a strong incentive to relax the regularity properties of
_symplectic maps_. This gave rise to the field of _$C^{0}$ -symplectic
geometry_, which is, in some sense, an exploration of the closure of the group
of symplectic diffeomorphisms.

From a very down to earth perspective, this paper is concerned with _piecewise
linear_ or _polyhedral_ symplectic geometry, understood as some kind of
$C^{0}$-approximation of the _smooth_ symplectic geometry. It should be
pointed out that the definition of a _piecewise linear symplectic manifold_ is
neither clear nor widely accepted at this stage. However, we give here a
tentative definition: let $M$ be a _piecewise linear manifold_. Then $M$ can
be realized as a _simplicial complex_ ${\mathscr{T}}$ given as a set of
_Euclidean simplices_ glued together using _affine maps_. A _Whitney form_
consists of a collection of differential forms on each simplex of
${\mathscr{T}}$, whose pull-backs agree along common faces. A _piecewise
linear symplectic form_ is a collection of _constant_ symplectic forms on each
simplex of ${\mathscr{T}}$, which satisfies the Whitney property. Very little
is known in this context. For instance, we can ask if such piecewise linear
symplectic form is _locally trivial_. In other words, can we define a local
piecewise linear homeomorphism with values in ${\mathbb{R}}^{2n}$ in a
neighborhood of every point, that sends the symplectic form onto the Euclidean
symplectic form of ${\mathbb{R}}^{2n}$? This question is an obvious analogue
of the well known _Darboux local coordinates theorem_ , in the smooth setting.
This basic result of smooth symplectic geometry turns out to be an _open
question_ in the piecewise linear setting.

From a broader scope, this paper is part of an ongoing research project,
concerned with _symplectic polyhedra_ piecewise linear symplectic maps, and
their relations with smooth symplectic geometry (cf. [5, 8, 4, 3]).
Constructions of piecewise linear symplectic maps are scarce. The only work in
this direction seems to be a local polyhedral approximation result of smooth
symplectic maps due to Gratza [2]. In this paper, we provide an effective
construction of polyhedral symplectic maps of a $4$-dimensional symplectic
torus, obtained as limits of a _modified hyperKähler moment map flow_ , on the
moduli space of differential forms on the torus. The flow is an _ordinary
differential equation_ and is always _convergent_. In particular
approximations of the limits of the flow can be investigated with a computer.
Such a program is provided in the upcoming joint work with François Jauberteau
[3].

###  1.2. Statement of results

The $4$-dimensional real torus $M=V/\Gamma$ is understood as a quotient of an
affine space $V$ by a lattice $\Gamma$ of the underlying vector space
$\overrightarrow{V}$. Furthemore, the vector space $\overrightarrow{V}$ is
equiped with an identification with the space of quaternions ${\mathbb{H}}$.
The quaternionic multiplication by $i$ _on the right_ provides a complex
structure on $V$ and $M$. The Euclidean structure of ${\mathbb{H}}$ induces a
Kähler structure on $V$ and $M$ with Kähler metrics $g_{V}$, $g_{M}$ and
Kähler forms denoted $\omega_{V}$ and $\omega_{M}$. From this point, we focus
on the space of symplectic diffeomorphisms of the symplectic torus

$(M,\omega_{M}).$

In addition $M$ carries a hyperKähler structure $(M,g_{M},I,J,K)$, with almost
complex structures $I$, $J$ and $K$ induced by the quaternionic multiplication
_on the left_ by $i$, $j$ and $k$ on $\overrightarrow{V}\simeq{\mathbb{H}}$.
The corresponding Kähler forms are denoted $\hat{\omega}_{I}$,
$\hat{\omega}_{J}$ and $\hat{\omega}_{K}$.

The following observation is the central idea of our construction. It is
stated in the smooth setting firs, for the sake of simplicity. However, we
shall see that there is an completely similar analogue of this statement in
the _polyhedral setting_.

######  Theorem A.

The space ${\mathscr{F}}$ of $\overrightarrow{V}$-valued differential
$1$-forms ${\mathscr{F}}=\Omega^{1}(M,\overrightarrow{V})$ on $M$, carries a
natural induced hyperKähler structure

$({\mathscr{F}},{\mathcal{G}},{\mathcal{I}},{\mathcal{J}},{\mathcal{K}}),$

with Kähler forms $\Omega_{I}$, $\Omega_{J}$ and $\Omega_{K}$. The hyperKähler
structure of ${\mathscr{F}}$, is invariant under the action of the infinite
dimensional torus ${\mathbb{T}}=C^{\infty}(M,S^{1})$ on ${\mathscr{F}}$, by
complex multiplication on the right, where $S^{1}$ is the unit circle in
${\mathbb{C}}$.

Furthermore, the ${\mathbb{T}}$-action is Hamiltonian with respect to
$\Omega_{I},\Omega_{J}$ and $\Omega_{K}$ and the three corresponding moment
maps

$\mu_{I},\mu_{J},\mu_{K}:{\mathscr{F}}\to\mathfrak{t}\simeq
C^{\infty}(M,{\mathbb{R}}),$

are given by the formulas

$\mu_{\bullet}(F)=-\frac{(F^{*}\omega_{V})\wedge\hat{\omega}_{\bullet}}{\sigma_{M}},$

where $\sigma_{M}=\frac{\omega_{M}^{2}}{2}$ is the volume form of $M$.

A smooth map $f:M\to M$ defines a _tangent map_ $f_{*}:TM\to TM$ on the
tangent bundle. Using the triviality of the tangent bundle of a quotient
$M=V/\Gamma$, the tangent map $f_{*}$ can be understood as a
$\overrightarrow{V}$-valued differential form, denoted

${\mathscr{D}}f\in\Omega^{1}(M,\overrightarrow{V})={\mathscr{F}}.$

Thus ${\mathscr{D}}$ defines a linear map between _the moduli spaces_

${\mathscr{D}}:{\mathscr{M}}\to{\mathscr{F}},$

where

${\mathscr{M}}=\\{f:M\to M,f\in C^{\infty}\\}$

is the moduli space of smooth maps of $M$. It is convenient to gather the
various moment maps into a single map

${\bm{\mu}}=(\mu_{I},\mu_{J},\mu_{K}),$

called the _hyperKähler moment map_. Then we have the following result:

######  Theorem B.

Let $f:M\to M$ be a smooth map of the torus, homotopic to a symplectic
diffeomorphism of $(M,\omega_{M})$. Let $F={\mathscr{D}}f\in{\mathscr{F}}$ be
the associated differential. The following properties are equivalent:

  1. (1)

The map $f$ is a symplectic diffeomorphism of the symplectic torus
$(M,\omega_{M})$.

  2. (2)

${\bm{\mu}}(F)=0$.

In view of Theorem B, it is tempting to consider the _downward gradient flow_
of the functional

$\displaystyle\phi:{\mathscr{F}}$ $\displaystyle\to{\mathbb{R}}$
$\displaystyle F$ $\displaystyle\mapsto\frac{1}{2}\|{\bm{\mu}}(F)\|^{2}$

in order to interpret symplectic maps as limits of a flow. This classical
approach, in the context of _hyperKähler moment map geometry_ , fails to work
here. Indeed, the gradient flow of $\phi$ does not preserve the image of
${\mathscr{D}}$, so that the flow does not induce a flow on ${\mathscr{M}}$.
To get around this issue, we introduce the subspace of closed forms, whose
cohomology class is a multiple of some $\alpha\in
H^{1}(M,\overrightarrow{V})$, denoted

${\mathscr{F}}^{c}_{\alpha}=\\{F\in{\mathscr{F}},dF=0\mbox{ and
}[F]\in{\mathbb{R}}\alpha\\},$

and consider the downward gradient flow of the functional $\phi$, restricted
to ${\mathscr{F}}^{c}_{\alpha}$. This flow is called the _modified moment map
flow_ (cf. §2.14 for details).

######  Theorem C.

A differential form $F\in{\mathscr{F}}^{c}_{\alpha}$ is a fixed point of the
modified moment map flow if, and only if, ${\bm{\mu}}(F)=0$.

Assuming that $\alpha$ is a symplectic cohomology class, a fixed point $F$ of
the flow satisfies the following dichotomy:

  1. (1)

Either $F=0$, and we say that $F$ is a degenerate fixed point,

  2. (2)

or $\alpha=\tau[F]$ for some $\tau\in{\mathbb{R}}^{*}$. Then there exists
$h\in{\mathscr{M}}$, such that ${\mathscr{D}}h=\tau F$ and
$[{\mathscr{D}}h]=\alpha$, with the property that $h$ is a symplectic
diffeomorphism of $(M,\omega_{M})$. In this case, we say that $F$ is a generic
fixed point.

The modified moment map flow admits some nice _existence and decay_
properties:

######  Theorem D.

The modified moment map flow along ${\mathscr{F}}^{c}_{\alpha}$ exists and is
unique for short time in the following sense: given a smooth form
$F\in{\mathscr{F}}^{c}_{\alpha}$ and $k\geq 1$, there exists $\varepsilon>0$
and a unique family $F_{t}$ of $\overrightarrow{V}$-valued $1$-forms, with
regularity of type $C^{k}$ on $M$, defined for
$t\in(-\varepsilon,\varepsilon)$, which is a solution of the modified moment
map flow, with initial condition $F_{0}=F$.

Furthermore the norm $\|F\|_{\mathcal{G}}$ can only decrease along flow line.

Then we provide a polyhedral definition of all the above constructions at
Section 3. In a nutshell, given a Euclidean triangulation of $M$, we introduce
${\mathscr{M}}$ as the space of polyhedral maps of $M$. We can make sens of
${\mathscr{F}}$ and ${\mathscr{F}}^{c}_{\alpha}$ and all the previous
hyperKähler moment map constructions. In this finite dimensional setting, the
modified moment maps flow is an ordinary differential equation so that we can
prove much stronger results:

######  Theorem E.

The statements of Theorems A, B and C hold for the polyhedral setting, once
spaces of smooth objects have been replaced by their polyhedral analogues. In
addition, the modified moment map flow exists for arbitrarily large time and
is convergent. In the case were the cohomology class $\alpha$ symplectic, the
renormalized generic limits of the flow are the differentials of polyhedral
symplectic maps of $(M,\omega_{M})$.

Building on Theorem E, we provide a computer sofware in [3], which constructs
approximations of the limitis of the modified moment map flow in the
polyhedral setting. We can deduce and visualize many effective examples of
polyhedral symplectic maps of the $4$-torus.

##  2\. Smooth symplectic geometry

###  2.1. Symplectic maps

We first consider the general case of a _connected symplectic manifold_
$(M,\omega_{M})$. The moduli space of _smooth maps_ of $M$ is denoted

${\mathscr{M}}=\\{f:M\to M,f\in C^{\infty}\\}.$

The moduli space ${\mathscr{M}}$ contains the _group of diffeomorphisms_ of
$M$, denoted $\mathrm{Diff}(M)$. A smooth map $f\in{\mathscr{M}}$ is called a
_symplectic map_ , if it respects the symplectic form, in the sense that

$f^{*}\omega_{M}=\omega_{M}.$

The non-degeneracy of $\omega_{M}$ implies that the tangent map is a fiberwise
isomorphism of the tangent bundle. Thus every symplectic map is a _local
diffeomorphism_ of $M$. If $M$ is closed, it follows that
$f\in\mathrm{Diff}(M)$ as well, for topological reasons:

######  Proposition 2.1.1.

Assume that $(M,\omega_{M})$ is a closed symplectic manifold and let $f:M\to
M$, be a symplectic map of $M$. Then $f$ is a diffeomorphism of $M$.

###### Proof.

Since $f$ is symplectic, it is a local diffeomorphism. Furthermore
$f^{*}\omega_{M}=\omega_{M}$ implies $f^{*}\omega_{M}^{n}=\omega_{M}^{n}$,
where $2n$ is the real dimension of $M$. Hence
$f^{*}[\omega_{M}]^{n}=[\omega_{M}]^{n}$ for the induced cohomology class.
Since $M$ is a closed connected symplectic manifold

$H^{2n}(M,{\mathbb{R}})={\mathbb{R}}\cdot[\omega_{M}]^{n},$

which shows that $f$ has degree $1$. Since $f$ is a local diffeomorphism of
degree $1$ of a compact manifold, we have $f\in\mathrm{Diff}(M)$. ∎

In the rest of this paper, the symplectic manifold $(M,\omega_{M})$ is always
a torus, in particular it is closed and connected. Hence, the space of
symplectic maps of $M$ is also a subgroup of the group of diffeomorphisms of
$M$ denoted,

$\mathrm{Symp}(M,\omega_{M})\subset\mathrm{Diff}(M)\subset{\mathscr{M}}.$

###  2.2. Hermitian tori

From now on, we restrict our attention to a quotient torus

$M=V/\Gamma,$

where $V$ is an affine space, with underlying vector space
$\overrightarrow{V}$ endowed with a _Hermitian structure_ , and
$\Gamma\subset\overrightarrow{V}$ is a _lattice_. As usual, the quotient torus
$M$ admits a canonical action of $\overrightarrow{V}$ by translations.

The complex vector space $\overrightarrow{V}$ carries a _Hermitian metric_
$h_{V}$, which is a sesquilinear form, anti-${\mathbb{C}}$-linear in the first
variable. Then we can write for every $v_{1},v_{2}\in\overrightarrow{V}$

$h_{V}(v_{1},v_{2})=g_{V}(v_{1},v_{2})+i\omega_{V}(v_{1},v_{2}),$

where $g_{V}$ and $\omega_{V}$ are respectively the real and imaginary parts
of $h_{V}$. By definition, $g_{V}$ is a _Euclidean metric_ on $V$ and
$\omega_{V}$ is a symplectic form. Furthermore, $V$ is endowed with an _almost
complex structure_ deduced from the multiplication by $i$ on
$\overrightarrow{V}$. Thus, $i$ is sometimes understood a linear endomorphism
$i:\overrightarrow{V}\to\overrightarrow{V}$, given by the map $v\mapsto iv$.

The above structures on $V$ descend to the quotient $M=V/\Gamma$. Hence $M$
admits a complex manifold, where $i:TM\to TM$ is the almost complex structure
deduced from the complex structure of $V$. In addition, the Hermitian
structure $h_{V}$, the Riemannian metric $g_{V}$ and the symplectic form
$\omega_{V}$ descend respectively on $M$ as $h_{M}$, $g_{M}$ and $\omega_{M}$.
This provides a flat _Kähler structure_ $(M,g_{M},i)$ on the torus $M$, with
Kähler metric $g_{M}$ and Kähler form $\omega_{M}$. We also obtain a Hermitian
structure $h_{M}=g_{M}+i\omega_{M}$ on the tangent bundle of $M$, understood
as a complex vector bundle.

###  2.3. Differentials and differentiation

The tangent bundle

$\pi:TV\to V$

is canonically isomorphic to the trivial bundle

$\pi_{1}:V\times\overrightarrow{V}\to V,$

where $\pi_{1}$ is the canonical first projection. The vector space
$\overrightarrow{V}$ acts by translations on $V$ and the induced action on the
tangent bundle $TV$ is trivial on the fibers. The tangent bundle $\pi:TM\to M$
is obtained as a quotient of $TV\to V$ by the $\Gamma$ action. Therefore
$TM\to M$ is also canonically isomorphic to the trivial bundle
$\pi_{1}:M\times\overrightarrow{V}\to M$. We obtain a smooth
$\overrightarrow{V}$-invariant bundle map

$\rho:TM\to\overrightarrow{V}$ (2.1)

given by $\pi_{2}:M\times\overrightarrow{V}\to\overrightarrow{V}$, where
$\pi_{2}$ is the canonical projection on the second factor. In addition, the
map $\rho$ respects the Kähler structures, in the sense that it is complex
linear, and:

$\rho^{*}\omega_{V}=\omega_{M},\quad\rho^{*}g_{V}=g_{M},\quad\rho^{*}h_{V}=h_{M},$

where the pullbacks are understood in the sense

$(\rho^{*}\omega_{V})(v_{1},v_{2})=\omega_{V}(\rho(v_{1}),\rho(v_{2})),\mbox{
etc\ldots}$

Any map $f\in{\mathscr{M}}$ admits a tangent map

$f_{*}:TM\to TM,$

lifting $f:M\to M$. Using $\rho$, the tangent map $f_{*}$ can be regarded as a
differential $1$-form with values in $\overrightarrow{V}$

${\mathscr{D}}f:TM\to\overrightarrow{V},$ (2.2)

defined by

${\mathscr{D}}f=\rho\circ f_{*}.$ (2.3)

In other words

${\mathscr{D}}f\in{\mathscr{F}}=\Omega^{1}(M,\overrightarrow{V}),$

and the operator ${\mathscr{D}}$ can be regarded as a complex linear map

${\mathscr{D}}:{\mathscr{M}}\to{\mathscr{F}}.$

######  Remark 2.3.1.

Alternatively, ${\mathscr{D}}f$ can be constructed as follows: we consider a
lift $\tilde{f}:V\to V$ of $f:M\to M$ on the universal cover $p:V\to M$, that
is a smooth map such that $p\circ\tilde{f}=f\circ p$. The differential
$d\tilde{f}$ is a $\overrightarrow{V}$-valued $1$-form on $V$, in other words
an element of $\Omega^{1}(V,\overrightarrow{V})$. By construction $\tilde{f}$
is $\Gamma$-equivariant, so that $d\tilde{f}$ descends to the quotient as an
element of $\Omega^{1}(M,\overrightarrow{V})$, which agrees with
${\mathscr{D}}f$.

The space of maps ${\mathscr{M}}$ admits a canonical action of
$\overrightarrow{V}$, by translations: given $f\in{\mathscr{M}}$ and
$v\in\overrightarrow{V}$, we define $f+v\in{\mathscr{M}}$ by

$(f+v)(x)=f(x)+v$

for every $x\in M$. By definition, we have the following proposition:

######  Proposition 2.3.2.

The ${\mathscr{D}}$-operator is translation invariant, in the sense that for
all $v\in\overrightarrow{V}$ and $f\in{\mathscr{M}}$

${\mathscr{D}}f={\mathscr{D}}(f+v).$

Conversely, the ${\mathscr{D}}$-operator is almost injective:

######  Proposition 2.3.3.

Suppose that ${\mathscr{D}}f_{1}={\mathscr{D}}f_{2}$ for some
$f_{1},f_{2}\in{\mathscr{M}}$. Then, there exists $v\in\overrightarrow{V}$
such that $f_{2}=f_{1}+v$. In other words, we have an isomorphism

${\mathscr{D}}({\mathscr{M}})\simeq{\mathscr{M}}/\overrightarrow{V},$

where $\overrightarrow{V}$ acts on ${\mathscr{M}}$ by translation.

###### Proof.

The maps $f_{1}$ and $f_{2}$ admit lifts $\tilde{f}_{1}$ and
$\tilde{f}_{2}:V\to V$ to the universal cover of $M$, according to Remark
2.3.1. The assumption ${\mathscr{D}}f_{1}={\mathscr{D}}f_{2}$ implies that
$d\tilde{f}_{1}=d\tilde{f}_{2}$, where $d$ is the usual differential. Hence
$\tilde{f}_{1}$ and $\tilde{f}_{2}$ agree up to a translation
$v\in\overrightarrow{V}$. Therefore $f_{1}=f_{2}+v$. ∎

###  2.4. Cohomology

In this section, we recall some basic facts on the cohomology of a torus and
homotopy classes of maps of the torus.

The usual $d$-operator for real valued differential forms is denoted
$d_{k}:\Omega^{k}(M)\to\Omega^{k+1}(M)$, for $k\geq 0$. This operator admits a
canonical extension to $\overrightarrow{V}$-valued differential forms
$\Omega^{k}(M,\overrightarrow{V})$ denoted

$d_{k}:\Omega^{k}(M,\overrightarrow{V})\to\Omega^{k+1}(M,\overrightarrow{V})$

and defined by the property that for any real valued differential $k$-form
$\gamma$ and vector $v\in\overrightarrow{V}$, we have

$d_{k}(\gamma\otimes v)=(d_{k}\gamma)\otimes v.$

Furthermore we have a complex

$0\stackrel{{\scriptstyle
d_{-1}}}{{\longrightarrow}}\Omega^{0}(M,\overrightarrow{V})\stackrel{{\scriptstyle
d_{0}}}{{\longrightarrow}}\Omega^{1}(M,\overrightarrow{V})\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}\cdots\quad\cdots\stackrel{{\scriptstyle
d_{k-1}}}{{\longrightarrow}}\Omega^{k}(M,\overrightarrow{V})\stackrel{{\scriptstyle
d_{k}}}{{\longrightarrow}}\cdots$

with associated _De Rham cohomology groups_ denoted

$H^{k}(M,\overrightarrow{V})=\frac{\ker d_{k}}{\mathrm{Im}\;d_{k-1}}\mbox{ for
all }k\geq 0.$

and canonical isomorphisms

$H^{k}(M,{\mathbb{R}})\otimes\overrightarrow{V}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{k}(M,\overrightarrow{V}).$

######  Proposition 2.4.1.

The image of the map ${\mathscr{D}}:{\mathscr{M}}\to{\mathscr{F}}$ consists of
closed forms. In other words

$d_{1}\circ{\mathscr{D}}=0,$

which is to say

$\mathrm{Im}\;{\mathscr{D}}\subset\ker d_{1}.$

###### Proof.

Every smooth map $f:M\to M$ can be lifted to the universal cover $p:V\to M$ as
a map $\tilde{f}:V\to V$ by Remark 2.3.1. The $\overrightarrow{V}$-valued
differential $1$-form ${\mathscr{D}}f$ satisfies the property
$p^{*}{\mathscr{D}}f=d\tilde{f}$, by definition. Since $d^{2}=0$, we deduce
that $d\circ p^{*}{\mathscr{D}}(f)=d^{2}\tilde{f}=0$. Since $p$ is a
submersion, we conclude that $d{\mathscr{D}}f=0$. ∎

By Proposition 2.4.1, every ${\mathscr{D}}f$ is a closed
$\overrightarrow{V}$-valued form. Hence, ${\mathscr{D}}f$ defines a cohomology
class denoted $[{\mathscr{D}}f]\in H^{1}(M,\overrightarrow{V})$ and we have a
canonical map

${\alpha:{\mathscr{M}}}$${H^{1}(M,\overrightarrow{V})}$${f}$${{[{\mathscr{D}}f]}.}$
(2.4)

Quotient tori come equipped with a canonical isomorphism

$\psi:\overrightarrow{V}^{*}\to H^{1}(M,{\mathbb{R}}).$

Indeed, any element of $\gamma\in\overrightarrow{V}^{*}$ can be understood as
a parallel differential $1$-form on $V$. But such a form $\gamma$ descends to
the quotient $M$ as a $1$-form denoted $\gamma$ as well. Since this form is
parallel, it is closed and defines a cohomology class $[\gamma]$ in
$H^{1}(M,{\mathbb{R}})$, and $\psi(\gamma)=[\gamma]$, by definition. The fact
that $\psi$ is an isomorphism is well known.

The dual lattice $\Gamma^{*}\subset\overrightarrow{V}^{*}$ is given by

$\Gamma^{*}=\\{\phi\in\overrightarrow{V}^{*},\forall\gamma\in\Gamma,\phi(\gamma)\in{\mathbb{Z}}\\}.$

Using the isomorphism $\psi$, the dual lattice
$\Gamma^{*}\subset\overrightarrow{V}^{*}$ is understood as a lattice of
$H^{1}(M,{\mathbb{R}})$. It is a well known fact, that $\Gamma^{*}\subset
H^{1}(M,{\mathbb{R}})$ is identified to the lattice
$H^{1}(M,{\mathbb{Z}})\subset H^{1}(M,{\mathbb{R}})$.

Every map $f\in{\mathscr{M}}$, defines a morphism

$f^{*}:H^{1}(M,{\mathbb{Z}})\to H^{1}(M,{\mathbb{Z}}).$

which depends only on the homotopy class of the map $f:M\to M$. Using the
isomorphism $\psi$, the map $f^{*}$ can be understood as a map
$f^{*}:\Gamma^{*}\to\Gamma^{*}$. In other words $f^{*}$ is an endomorphism of
the lattice $\Gamma^{*}$ denoted

$f^{*}\in\mathrm{End}(\Gamma^{*})=\mathrm{End}(H^{1}(M,{\mathbb{Z}})).$

Hence, the $f^{*}$ can be regarded as an element of
$\Gamma^{*}\otimes_{\mathbb{Z}}\Gamma^{**}\simeq
H^{1}(M,{\mathbb{Z}})\otimes_{\mathbb{Z}}\Gamma^{**}$. Using the canonical
isomorphism $\Gamma^{**}\simeq\Gamma$, this is an element $f^{*}\in
H^{1}(M,{\mathbb{Z}})\otimes_{\mathbb{Z}}\Gamma$. Hence

$f^{*}\in H^{1}(M,{\mathbb{Z}})\otimes_{\mathbb{Z}}\Gamma\subset
H^{1}(M,{\mathbb{R}})\otimes_{\mathbb{R}}\overrightarrow{V}=H^{1}(M,\overrightarrow{V}).$

Then we have the classical result of topology for a torus, well known in the
case of a circle :

######  Proposition 2.4.2.

Using the above notations, we have

$\alpha(f)=f^{*}.$

and $\alpha(f)$ belongs to the lattice $\mathrm{End}(\Gamma^{*})$ of
$H^{1}(M,\overrightarrow{V})$. In addition, two smooth maps $f_{1}$ and
$f_{2}:M\to M$ are homotopic if, and only if, $\alpha(f_{1})=\alpha(f_{2})$
and the map $\alpha:{\mathscr{M}}\to\mathrm{End}(\Gamma^{*})$ is surjective.
In particular, the map $\alpha$ provides a bijection between
$\mathrm{End}(\Gamma^{*})$ and the set of homotopy classes of smooth maps
$f:M\to M$.

######  Definition 2.4.3.

If $\alpha\in\mathrm{End}(H^{1}(M,{\mathbb{Z}}))$ is the cohomology class
$f^{*}$ of a continuous map $f^{*}$, such that
$f^{*}[\omega_{M}]=[\omega_{M}]$, we say that $\alpha$ is a symplectic class.

######  Remark 2.4.4.

The identity map $\mathrm{id}_{M}\in{\mathscr{M}}$ is a symplectic map. Let

$\alpha_{1}=\alpha(\mathrm{id}_{M})\in H^{1}(M,\overrightarrow{V})$ (2.5)

be its corresponding cohomology class. If $f\in{\mathscr{M}}$ satisfies
$\alpha(f)=\alpha_{1}$, then $f$ is homotopic to $\mathrm{id}_{M}$. In
particular

$f^{*}[\omega_{M}]=[\omega_{M}]$

and $\alpha_{1}$ is a symplectic class.

###  2.5. Primitive of an integral form

Assume that $\alpha$ is a cohomology class in
$H^{1}(M,{\mathbb{Z}})\otimes_{\mathbb{Z}}\Gamma$. Let
$F\in\mathrm{Im}\;{\mathscr{D}}\subset{\mathscr{F}}$ be a closed differential
form such that

$\alpha=[F]$

and $\tilde{F}=p^{*}F$ be the lift of $F$ to the universal cover $p:V\to M$.
Let $z_{0}\in V$ be any fixed point, then let $f:V\to V$ be the function
defined by

$\tilde{f}(z)=\int_{z_{0}}^{z}\tilde{F},$

where the integral is taken along any smooth path from $z_{0}$ to $z$. Since
$F$ is closed, the integral is independent of the choice of path between
$z_{0}$ and $z$. Moreover, for every $\gamma\in\Gamma$,
$\tilde{f}(z+\gamma)-\tilde{f}(z)\in\Gamma$, since $F$ belongs to
$H^{1}(M,{\mathbb{Z}})\otimes\Gamma$. In particular, $\tilde{f}$ descends as a
smooth map $f:M\to M$ such that ${\mathscr{D}}f=F$. By definition
$\alpha(f)=\alpha$, which gives the following result:

######  Proposition 2.5.1.

Given $\alpha\in H^{1}(M,{\mathbb{Z}})\otimes\Gamma$ represented by a closed
differential form $F\in{\mathscr{F}}$, there exists map $f\in{\mathscr{M}}$
such that ${\mathscr{D}}f=F$. In addition $f$ is unique up to the action of
translations and $\alpha(f)=\alpha$.

###  2.6. Structures on vector bundles

Recall that the torus $M=V/\Gamma$ is equipped with a Kähler structure
$(M,g_{M},i)$ with Kähler form $\omega_{M}$ deduced from the Hermitian
structure of $\overrightarrow{V}$.

Let $J_{V}$ be another almost complex structure on $\overrightarrow{V}$,
compatible with the metric $g_{V}$, in the sense that
$g_{V}(J_{V}v_{1},J_{V}v_{2})=g_{V}(v_{1},v_{2})$ for every
$v_{1},v_{2}\in\overrightarrow{V}$. Thus, we have an alternate symplectic form
on $V$ defined by

$\hat{\omega}_{V}=g_{V}(J_{V}\cdot,\cdot).$

We stress here the fact that the $g_{V}$-compatible symplectic structure
$\hat{\omega}_{V}$ and $\omega_{V}$ are not related in an obvious way. For
instance, they may induce opposite orientations.

The almost complex structure $J_{V}$ and Kähler structure $\hat{\omega}_{V}$
descend to the quotient $M$. Thus, they provide an alternate Kähler structure
$(M,g_{M},J_{M})$ on $M$, with underlying Kähler form denoted
$\hat{\omega}_{M}$. In conclusion, we have two competing Kähler structures
$(M,g_{M},i)$ and $(M,g_{M},J_{M})$ on $M$, with Kähler forms $\omega_{M}$ and
$\hat{\omega}_{M}$.

The cotangent bundle $T^{*}M\to M$ is identified to the tangent bundle $TM\to
M$ thanks to the duality provided by the Riemannian metric $g_{M}$. Thus
$T^{*}M$ and all the bundles of $k$-forms $\Lambda^{k}M\to M$ are endowed with
an induced Euclidean inner product denoted $g$ and deduced from $g_{M}$. It
follows that the complexified bundles

$\Lambda^{k}M\otimes_{{\mathbb{R}}}\overrightarrow{V}\to M$

have several induced structures:

  * •

The bundles $\Lambda^{k}M\otimes_{\mathbb{R}}\overrightarrow{V}\to M$ carry
fiberwise Euclidean inner products denoted $g$ as well and induced by $g$ and
$g_{V}$.

  * •

The bundles $\Lambda^{k}M\otimes_{\mathbb{R}}\overrightarrow{V}\to M$ carry a
canonical structure of complex vector bundle, since $\overrightarrow{V}$ is a
complex vector space. Indeed, complex valued forms $F$ admit a canonical
action of ${\mathbb{C}}$, where multiplication by $i$ is trivially given by
pointwise multiplication

$(i\cdot F)(v)=iF(v).$

  * •

The bundles $\Lambda^{1}M\otimes\overrightarrow{V}\to M$ carry a fiberwise
almost complex structure $J$ induced by the right action of $J_{M}$ and
defined by

$J\cdot F=-F\circ J_{M}.$

Furthermore, $J_{M}$ is complex linear in the sense that it commutes with the
multiplicative action of ${\mathbb{C}}$ of complex bundles:

$i\cdot J\cdot F=J\cdot i\cdot F.$

  * •

The metrics are compatible with complex structures in the sense that

$g(iF_{1},iF_{2})=g(F_{1},F_{2})\quad\mbox{ and }\quad g(J\cdot F_{1},J\cdot
F_{2})=g(F_{1},F_{2}).$

  * •

The bundles $\Lambda^{1}M\otimes_{\mathbb{R}}\overrightarrow{V}\to M$ carry
fiberwise symplectic forms denoted $\hat{\omega}$ deduced from $g$ and $J$,
and defined by

$\hat{\omega}(F_{1},F_{2})=g(J\cdot F_{1},F_{2}).$

###  2.7. An involution

We define a fiberwise involution $R$ of the vector bundle
$T^{*}M\otimes_{\mathbb{R}}\overrightarrow{V}\to M$ by

$RF=iJ\cdot F.$

The eigenspaces decomposition is written

$T^{*}M\otimes_{\mathbb{R}}\overrightarrow{V}=\Lambda^{1,+}\oplus\Lambda^{1,-}$

where the eigenspaces $\Lambda^{1,\pm}$ are associated to the eigenvalues $\pm
1$. Notice that an eigenvector associated to the eigenvalue $+1$ (resp. $-1$)
correspond to $J_{M}$-complex (resp. anti-complex) bundle morphisms from $TM$
to $\overrightarrow{V}$. In other words

$RF=\pm F\Leftrightarrow F\circ J_{M}=\pm iF.$

The involution $R$ induces a canonical involution denoted ${\mathscr{R}}$ on
the space of differential forms ${\mathscr{F}}$ and we have a similiar
splitting

${\mathscr{F}}={\mathscr{F}}^{+}\oplus{\mathscr{F}}^{-}$

into complex and anti-complex differential forms. Accordingly, a differential
form $F\in{\mathscr{F}}$ admits a decomposition

$F=F^{+}+F^{-},$

with components $F^{\pm}\in{\mathscr{F}}^{\pm}$.

###  2.8. Group action

The bundle $T^{*}M\otimes_{\mathbb{R}}\overrightarrow{V}\to M$ is a complex
vector bundle. In particular it admits a fiberwise multiplication by complex
numbers. Thus complex valued functions on $M$ act by multiplication on
sections of $T^{*}M\otimes_{\mathbb{R}}\overrightarrow{V}$ and we may define
an action of the _infinite dimensional torus_

${\mathbb{T}}^{\mathbb{C}}=C^{\infty}(M,{\mathbb{C}}^{*})$

on ${\mathscr{F}}$, by the formula

$\lambda\cdot F={\bar{\lambda}}^{-1}F^{+}+\lambda F^{-},$

for every $F\in{\mathscr{F}}$ and $\lambda\in{\mathbb{T}}^{\mathbb{C}}$, where
multiplication in the RHS are the canonical complex multiplication and
$\bar{\lambda}$ denotes the complex conjugate. Then the restriction of the
action of ${\mathbb{T}}^{\mathbb{C}}$ to the subgroup

${\mathbb{T}}=C^{\infty}(M,S^{1}),$

where $S^{1}$ is identified to complex numbers of module $1$, is simply given
by the standard multiplication $\lambda\cdot F=\lambda F$.

We can introduce the exponential map $\exp(\xi)=e^{i\xi}$ for every complex
valued function $\xi$ on $M$. Thus, the Lie algebra
$\mathfrak{t}^{\mathbb{C}}$ of ${\mathbb{T}}^{\mathbb{C}}$ is identified to
$C^{\infty}(M,{\mathbb{C}})$ whereas the Lie algebra $\mathfrak{t}$ of
${\mathbb{T}}$ is identified to $C^{\infty}(M,{\mathbb{R}})$ via the
exponential map. It follows that the infinitesimal action of
$\xi\in\mathfrak{t}^{\mathbb{C}}$ is given by the vector field $X_{\xi}$

$X_{\xi}(F)=i\bar{\xi}F^{+}+i\xi F^{-}.$

In particular $X_{\xi}(F)=i\xi F$ if $\xi\in\mathfrak{t}$, whereas
$X_{\xi}(F)=-i\xi{\mathscr{R}}F=\xi J\cdot F$ if $\xi\in i\mathfrak{t}$. This
implies that for every $\xi\in\mathfrak{t}$, we have

$J\cdot X_{\xi}=i\xi J\cdot F=X_{i\xi}.$ (2.6)

###  2.9. Induced structures on the moduli space

We now focus on the moduli space ${\mathscr{F}}$, which is the space of smooth
sections of the vector bundle of
$\Lambda^{1}M\otimes_{\mathbb{R}}\overrightarrow{V}\to M$. The space
${\mathscr{F}}$ is understood as an infinite dimensional complex vector space,
or a module on the space of complex valued functions, which carries a
canonical Kähler structure:

  * •

For $F_{1}$ and $F_{2}$ in ${\mathscr{F}}$, we define a Riemannian metric
${\mathcal{G}}$ on ${\mathscr{F}}$ by

${\mathcal{G}}(F_{1},F_{2})=\int_{M}g(F_{1},F_{2})\sigma_{M},$

where $\sigma_{M}$ is the volume form associated to $g_{M}$ and its canonical
orientation deduced from $i$. In other words
$\sigma_{M}=\frac{\omega_{M}^{n}}{n!}$.

  * •

We define an almost complex structures ${\mathcal{J}}$ on ${\mathscr{F}}$ by

$({\mathcal{J}}\cdot F)|_{z}=J\cdot F_{z}.$

  * •

We define a symplectic form $\Omega$ on ${\mathscr{F}}$ by

$\Omega(F_{1},F_{2})={\mathcal{G}}({\mathcal{J}}\cdot
F_{1},F_{2})=\int_{M}\hat{\omega}(F_{1},F_{2})\sigma_{M}.$

We summarize our construction in the next proposition.

######  Proposition 2.9.1.

Let $M=V/\Gamma$ be a quotient torus, where $\overrightarrow{V}$ is a complex
vector space endowed with a Hermitian structure. For each almost complex
structure $J_{V}$ of $\overrightarrow{V}$ compatible with the Euclidean metric
$g_{V}$, there exists a natural Kähler structure
$({\mathscr{F}},{\mathcal{G}},{\mathcal{J}})$ with Kähler form $\Omega$ on
${\mathscr{F}}=\Omega^{1}(M,\overrightarrow{V})$.

######  Proposition 2.9.2.

The ${\mathbb{T}}^{\mathbb{C}}$-action preserves the complex structure
${\mathcal{J}}$ of ${\mathscr{F}}$. Furthermore, the
${\mathbb{T}}^{\mathbb{C}}$-action is the complexified action of
${\mathbb{T}}$.

###### Proof.

This is an observation that the ${\mathbb{T}}^{\mathbb{C}}$ action commutes
with ${\mathcal{J}}$, which proves the first part of the proposition. By
construction $\mathfrak{t}^{\mathbb{C}}=\mathfrak{t}\oplus i\mathfrak{t}$. By
(2.6), the infinitesimal action satisfies ${\mathcal{J}}X_{\xi}=X_{i\xi}$ for
every $\xi\in\mathfrak{t}$, which proves the second statement of the lemma. ∎

######  Proposition 2.9.3.

The ${\mathbb{T}}$-action preserves the Kähler structure
$({\mathscr{F}},{\mathcal{G}},{\mathcal{J}})$. In particular, it preserves the
Kähler form $\Omega$.

###### Proof.

For $\lambda\in{\mathbb{T}}$, the action is just the usual complex
multiplication $\lambda\cdot F=\lambda F$, so that $g(\lambda F,\lambda
F)=|\lambda|^{2}g(F,F)=f(F,F)$. Therefore ${\mathcal{G}}(\lambda\cdot
F,\lambda\cdot F)={\mathcal{G}}(F,F)$ and the ${\mathbb{T}}$-action preserves
${\mathcal{G}}$. By Proposition 2.9.2, the action preserves the almost complex
structure ${\mathcal{J}}$, hence the Kähler form $\Omega$ is preserved as
well. ∎

###  2.10. Hamiltonian action

In this section, we show that the ${\mathbb{T}}$-action on the symplectic
torus $(M,\omega_{M})$ is Hamiltonian and determine its moment map.

For each choice of $g_{V}$-compatible almost complex structure $J_{V}$ on $V$,
${\mathscr{F}}$ is endowed with a structure of Kähler manifold
$({\mathcal{G}},{\mathcal{J}},\Omega)$, by Proposition 2.9.1. Furthermore, the
Kähler strcture in invariant under the ${\mathbb{T}}$-action by Proposition
2.9.3. Let

$\mu:{\mathscr{F}}\to\mathfrak{t}=C^{\infty}(M,{\mathbb{R}})$

be the map given by the formula

$\mu(F)=-\frac{1}{2}g({\mathscr{R}}F,F),$

where $C^{\infty}(M,{\mathbb{R}})$ is identified to the Lie algebra
$\mathfrak{t}$ of the group ${\mathbb{T}}$, via the exponential map
$\exp:\mathfrak{t}\to{\mathbb{T}}$ given by $\exp(\xi)=e^{i\xi}$.

######  Proposition 2.10.1.

The map $\mu$ is a moment map for the action of ${\mathbb{T}}$ on
${\mathscr{F}}$.

###### Proof.

We compute

$\displaystyle D\mu|_{F}\cdot\dot{F}$
$\displaystyle=-\frac{1}{2}\Big{(}g({\mathscr{R}}F,\dot{F})+g({\mathscr{R}}\dot{F},F)\Big{)}$
(2.7) $\displaystyle=-g({\mathscr{R}}F,\dot{F})$ (2.8)
$\displaystyle=-g(i{\mathcal{J}}F,\dot{F})$ (2.9)
$\displaystyle=-\hat{\omega}(iF,\dot{F}),$ (2.10)

hence

$D\mu|_{F}\cdot\dot{F}=-\hat{\omega}(iF,\dot{F}).$

Using the inner product
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle:\mathfrak{t}\times\mathfrak{t}\to{\mathbb{R}}$
defined by

$\langle\\!\langle\zeta,\xi\rangle\\!\rangle=\int_{M}\zeta\xi\;\sigma_{M}$

we have

$\displaystyle\langle\\!\langle D\mu|_{F}\cdot\dot{F},\zeta\rangle\\!\rangle$
$\displaystyle=-\int_{M}\hat{\omega}(iF,\dot{F})\zeta\;\sigma_{M}$
$\displaystyle=-\int_{M}\hat{\omega}(i\zeta F,\dot{F})\;\sigma_{M}$
$\displaystyle=-\Omega(i\zeta F,\dot{F})$
$\displaystyle=-\iota_{X_{\zeta}(F)}\Omega(\dot{F}),$

where $X_{\zeta}$ is the vector field on ${\mathscr{F}}$, given by

$X_{\zeta}(F)=\zeta iF.$

The above identity shows that $\mu$ is a moment map for the action of
${\mathbb{T}}$. ∎

We summarize our construction as follows:

######  Theorem 2.10.2.

Let $M=V/\Gamma$ be a quotient torus, where $\overrightarrow{V}$ is an affine
space with $\overrightarrow{V}$, a complex vector space endowed with a
Hermitian structure and an alternate almost complex structure $J_{V}$
compatible with the metric. Then the moduli space ${\mathscr{F}}$ of
$\overrightarrow{V}$-valued differential $1$-forms carries a canonical Kähler
structure $({\mathscr{F}},{\mathcal{J}},{\mathcal{G}},\Omega)$ with a
Hamiltonian action of the torus ${\mathbb{T}}=C^{\infty}(M,S^{1})$, such that
$\mu$ is the moment map. In other words, if $X_{\zeta}$ is the vector field on
${\mathscr{F}}$ corresponding to the infinitesimal action of
$\zeta\in\mathfrak{t}$, we have

$D\langle\\!\langle\mu,\zeta\rangle\\!\rangle=-\iota_{X_{\zeta}}\Omega.$

###  2.11. Moment map and symplectic density

The moment map $\mu$ is closely related to symplectic densities. For
$F\in{\mathscr{F}}$, we have by definition

$2\mu(F)=g(iF\circ J_{M},F).$

We choose a local oriented orthonormal frame
$(e_{1},f_{1},\cdots,e_{n},f_{n})$ of $TM$ adapted to the Kähler structure
$(g_{M},J_{M},\hat{\omega}_{M})$. Then by definition of the metric $g$

$\displaystyle 2\mu(F)$ $\displaystyle=\sum_{k=1}^{n}g_{V}(iF\circ
J_{M}(e_{k}),F(e_{k}))+g_{V}(iF\circ J_{M}(f_{k}),F(f_{k}))$
$\displaystyle=\sum_{k=1}^{n}ig_{V}(iF(f_{k}),F(e_{k}))-g_{V}(iF(e_{k}),F(f_{k}))$
$\displaystyle=-2\sum g_{V}(iF(e_{k}),F(f_{k}))$
$\displaystyle=-2\sum\omega_{V}(F(e_{k}),F(f_{k}))$

In particular, if The differential form $F$ can be regarded as a familly of
linear maps from the fibers of $TM$ to $\overrightarrow{V}$. Thus we may
define the pullback of the Euclidean symplectic form $\omega_{V}$ by $F$:

$F^{*}\omega_{V}(u,v)=\omega_{V}(F(u),F(v)),$

and the above computation shows that

$\mu(F)=-\sum_{k=1}^{n}F^{*}\omega_{V}(e_{k},f_{k}).$

If $n=1$, we deduce that $\mu(F)=\frac{F^{*}\omega_{V}}{\sigma_{M}}$. More
generally, this formula is a trace. Indeed, assuming $n=2$ for simplicity, we
have

$\hat{\omega}_{M}=e_{1}^{*}\wedge f_{1}^{*}+e_{2}^{*}\wedge f_{2}^{*}$

so that

$F^{*}\omega_{V}\wedge\hat{\omega}_{M}=-\mu(F)\sigma_{M}$

Hence we obtain the following result:

######  Proposition 2.11.1.

In the case where $M=V/\Gamma$ is a torus of real dimension $4$, we have

$\mu(F)=-\frac{(F^{*}\omega_{V})\wedge\hat{\omega}_{M}}{\sigma_{M}},$ (2.11)

for every $F\in{\mathscr{F}}$. In particular, if $F={\mathscr{D}}f$ for some
$f\in{\mathscr{M}}$, then

$\mu({\mathscr{D}}f)=-\frac{(f^{*}\omega_{M})\wedge\hat{\omega}_{M}}{\sigma_{M}}.$

###  2.12. The hyperKähler torus

The space of quaternions ${\mathbb{H}}$ is a $4$-dimensional real vector
space: every $q\in{\mathbb{H}}$ can be written under the form

$q=a+bi+cj+dk,$

where $a,b,c,d$ are real numbers and the quaternions $i$, $j$ and $k$ satisfy
the quaternionic relations

$i^{2}=j^{2}=k^{2}=ijk=-1.$

There is a canonical inclusion ${\mathbb{C}}\subset{\mathbb{H}}$, since every
complex number $\lambda=a+bi$ is a quaternion. For $\lambda\in{\mathbb{C}}$,
we define a complex action over ${\mathbb{H}}$, by

$\lambda\cdot q=q\lambda,$

where the product in the RHS is the quaternionic multiplication. Using this
structure of complex vector space on ${\mathbb{H}}$, we have a complex linear
isomorphism

${\varpi:{\mathbb{C}}^{2}}$${{\mathbb{H}}}$${(z_{1},z_{2})}$${z_{1}+jz_{2}.}$
(2.12)

We define $3$ almost complex structures $I,J,K$ on ${\mathbb{H}}$, given by
the quaternionic multiplication by $i,j$ and $k$ _on the left_ :

$I\cdot q=iq,\quad J\cdot q=jq\quad\mbox{ and }\quad K\cdot q=kq,$

and we also denote by $I$, $J$ and $K$ the almost complex structures deduced
on ${\mathbb{C}}^{2}$ using the isomorphism $\varpi$.

######  Remark 2.12.1.

Notice that the almost complex structure $I$ on ${\mathbb{C}}^{2}$, defined
above, is not the standard multiplication by $i$ on the complex vector space
${\mathbb{C}}^{2}$. In fact $I\cdot(z_{1},z_{2})=(iz_{1},-iz_{2})$.

We consider from now on an affine space $V$ with underlying complex vector
space $\overrightarrow{V}$, of complex dimension $2$, endowed with a complex
isomorphism with ${\mathbb{H}}$. Then $\overrightarrow{V}$ can be identified
to ${\mathbb{C}}^{2}$ with its canonical Hermitian structure. We deduce three
additional almost complex structure $I$, $J$ and $K$ on $V$, deduced from the
one on ${\mathbb{H}}$. The three almost complex structures are compatible with
the metric $g_{V}$ and the corresponding symplectic forms $\hat{\omega}_{I}$,
$\hat{\omega}_{J}$ and $\hat{\omega}_{K}$ are given by

$\hat{\omega}_{I}=g_{V}(I\cdot,\cdot),\quad\hat{\omega}_{J}=g_{V}(J\cdot,\cdot)\quad\mbox{
and }\quad\hat{\omega}_{K}=g_{V}(K\cdot,\cdot).$

Recall that the metric $g_{V}$ and the orientation induced by $\omega_{V}$
provide a Hodge operator, which is an involution
$*:\Lambda^{2}\overrightarrow{V}^{*}\to\Lambda^{2}\overrightarrow{V}^{*}$
defined by

$\gamma_{1}\wedge*\gamma_{2}=g(\gamma_{1},\gamma_{2})\sigma_{V}.$

We have the well known eigenspaces decomposition

$\Lambda^{2}\overrightarrow{V}^{*}=\Lambda^{2,+}\overrightarrow{V}^{*}\oplus\Lambda^{2,-}\overrightarrow{V}^{*}$

into selfdual forms and anti-selfdual $2$-forms, corresponding to the $\pm 1$
eigenvalues of the Hodge operator. The isomorphism between
$\overrightarrow{V}$ and ${\mathbb{C}}^{2}$ together with the canoncial
coordinates $z_{1}=x_{1}+iy_{1}$ and $z_{2}=x_{2}+iy_{2}$, gives the fomulas

$\displaystyle\hat{\omega}_{I}$ $\displaystyle=dx_{1}\wedge
dy_{1}-dx_{2}\wedge dy_{2},$ $\displaystyle\hat{\omega}_{J}$
$\displaystyle=dx_{1}\wedge dx_{2}+dy_{1}\wedge dy_{2},$
$\displaystyle\hat{\omega}_{K}$ $\displaystyle=-dx_{1}\wedge
dy_{2}-dx_{2}\wedge dy_{1}.$

Notice that these forms are quite different from $\omega_{V}=dx_{1}\wedge
dy_{2}+dx_{2}\wedge dy_{2}$. Using the fact that $*dx_{1}\wedge
dy_{1}=dx_{2}\wedge dy_{2}$, $*dx_{1}\wedge dx_{2}=-dy_{1}\wedge dy_{2}$ and
$*dx_{1}\wedge dy_{2}=dy_{1}\wedge dx_{2}$, we observe that the forms
$\hat{\omega}_{\bullet}$ span the space of anti-selfdual $2$-forms
$\Lambda^{2,-}\overrightarrow{V}^{*}$, whereas $\omega_{V}$ is a selfdual
$2$-form:

$\Lambda^{2,-}\overrightarrow{V}^{*}={\mathbb{R}}\hat{\omega}_{I}\oplus{\mathbb{R}}\hat{\omega}_{J}\oplus{\mathbb{R}}\hat{\omega}_{K}$

and we have

$\hat{\omega}_{I}^{2}=\hat{\omega}_{J}^{2}=\hat{\omega}_{K}^{2}=-\omega_{V}^{2}=-2\sigma_{M}.$

Given a lattice $\Gamma$ of $\overrightarrow{V}$, we obtain a $4$-torus
$M=V/\Gamma$. All of the almost complex structures, metrics and Kähler forms
descend to the quotient $M$. Thus $M$ is endowed with a flat metric $g_{M}$,
the complex structure $i$ deduced from the fact that $\overrightarrow{V}$ is a
complex vector space and its Kähler form $\omega_{M}$. The three almost
complex structures $I,J$ and $K$ on $M$ are compatible with $g_{M}$. The
associated Kähler forms are denoted $\hat{\omega}_{I}$, $\hat{\omega}_{J}$ and
$\hat{\omega}_{K}$ as well.

###  2.13. HyperKähler moment map

According to Section 2.9, the almost complex structures $I,J$ and $K$ induce
three almost complex structures ${\mathcal{I}}$, ${\mathcal{J}}$ and
${\mathcal{K}}$ on the moduli space ${\mathscr{F}}$. Together with the metric
${\mathcal{G}}$, they provide a hyperKähler structure, with three
corresponding Kähler forms denoted $\Omega_{I},\Omega_{J}$ and $\Omega_{K}$ on
${\mathscr{F}}$. The action of ${\mathbb{T}}$ on ${\mathscr{F}}$ preserves the
hyperKähler structure and is Hamiltonian with respect to the three symplectic
forms. According to Section 2.10 and Formula (2.11), the corresponding moment
maps are explicitly given by

$\mu_{\bullet}(F)=-\frac{(F^{*}\omega_{V})\wedge\hat{\omega}_{\bullet}}{\sigma_{M}}$
(2.13)

for $\bullet=I,J,K$. This motivates the introduction of the hyperKähler moment
map

${\bm{\mu}}=(\mu_{I},\mu_{J},\mu_{K})$

######  Proposition 2.13.1.

For every $F\in{\mathscr{F}}$, the following properties are equivalent:

  1. (1)

$F^{*}\omega_{V}$ is selfdual;

  2. (2)

${\bm{\mu}}(F)=0$.

###### Proof.

The equivalence is an immediate consequence of Formula (2.13) and the fact
that the symplectic forms $\hat{\omega}_{\bullet}$ span the space of anti-
selfdual forms. ∎

In the case where $F={\mathscr{D}}f$, we have a stronger result:

######  Proposition 2.13.2.

Let $f\in{\mathscr{M}}$ be a smooth map. Then the following properties are
equivalent:

  1. (1)

${\bm{\mu}}({\mathscr{D}}f)=0$;

  2. (2)

$f^{*}\omega_{M}$ is a selfdual harmonic form.

In particular, if $f^{*}[\omega_{M}]=[\omega_{M}]$, then $f$ is a
symplectomorphism if, and only if ${\bm{\mu}}({\mathscr{D}}f)=0$.

###### Proof.

By Lemma 2.13.1, ${\bm{\mu}}({\mathscr{D}}f)=0$ is equivalent to the fact that
$F^{*}\omega_{V}$ is a selfdual for, where ${\mathscr{D}}f=F$. By definition
$F^{*}\omega_{V}=f^{*}\omega_{M}$, hence $f^{*}\omega_{M}$ is selfdual and
closed. In particular, it is harmonic and the first part of the proposition
follows.

If $f^{*}[\omega_{M}]=[\omega_{M}]$, then $f^{*}\omega_{M}-\omega_{M}$ is an
exact form. The Kähler form $\omega_{M}$ is selfdual. If $f^{*}\omega_{M}$ is
selfdual as well, then $f^{*}\omega_{M}-\omega_{M}$ is an exact selfdual form.
In particular, it is hamonic and exact, hence it must vanish. We conclude that
$f$ is a symplectic map. ∎

###  2.14. HyperKähler flow

Let $\alpha\in H^{1}(M,\overrightarrow{V})$ be a fixed cohomology class.
Typically we are interested in the cohomology class $\alpha_{1}$ of the
identity map of $M$. For this purpose, we introduce the space of closed forms

${\mathscr{F}}^{c}=\\{F\in{\mathscr{F}},dF=0\\}$

and the subspace of closed forms whose cohomology class is a multiple of
$\alpha$

${\mathscr{F}}^{c}_{\alpha}=\\{F\in{\mathscr{F}}^{c},\exists
t\in{\mathbb{R}},[F]=t\alpha\\}.$

Hodge theory provides an orthogonal projections

$\Pi:{\mathscr{F}}\to{\mathscr{F}}^{c}\quad\mbox{ and
}\quad\Pi_{\alpha}:{\mathscr{F}}\to{\mathscr{F}}^{c}_{\alpha}$

onto the space ${\mathscr{F}}^{c}$ of closed differential forms and its
subspace ${\mathscr{F}}^{c}_{\alpha}$. Indeed, any $1$-form
$F\in{\mathscr{F}}$ can be written as

$F=F_{h}+d\phi+d^{*}\gamma,$

where $F_{h}$ is harmonic, $\phi\in{\mathscr{M}}$ and $\gamma$ is a
$\overrightarrow{V}$-valued $2$-form on $M$. Then we put

$\Pi F=F_{h}+d\phi.$

Usint the orthogonal decomposition $F_{h}=F_{\alpha}+F^{\perp}$ into harmonic
forms, where $[F_{\alpha}]\in{\mathbb{R}}\alpha$, we can define the orthogonal
projection

$\Pi_{\alpha}:{\mathscr{F}}\to{\mathscr{F}}$

by $\Pi_{\alpha}(F)=F_{\alpha}+d\phi$.

We introduce the functional

$\phi:{\mathscr{F}}\to{\mathbb{R}}$

defined by

$\displaystyle\phi(F)$
$\displaystyle=\frac{1}{2}\|{\bm{\mu}}(F)\|^{2}=\frac{1}{2}\|\mu_{I}(F)\|+\frac{1}{2}\|\mu_{J}(F)\|+\frac{1}{2}\|\mu_{J}(F)\|,$

where $\|\cdot\|$ denotes the norm associated to the inner product
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle$ on $\mathfrak{t}$. The
differential of $\phi$ is readily computed:

$\displaystyle d\phi|_{F}\cdot\dot{F}=$ $\displaystyle\langle\\!\langle
d\mu_{I}|_{F}\cdot\dot{F},\mu_{I}(F)\rangle\\!\rangle$
$\displaystyle+\langle\\!\langle
d\mu_{J}|_{F}\cdot\dot{F},\mu_{K}(F)\rangle\\!\rangle$
$\displaystyle+\langle\\!\langle
d\mu_{K}|_{F}\cdot\dot{F},\mu_{K}(F)\rangle\\!\rangle$

By Formula (2.8), $d\mu_{I}|_{F}\cdot\dot{F}=-g({\mathscr{R}}_{I}F,\dot{F})$,
hence

$\mu_{I}(F)d\mu_{I}|_{F}\cdot\dot{F}=-g(\mu_{I}(F){\mathscr{R}}_{I}F,\dot{F})$

and more generally

$\mu_{\bullet}(F)d\mu_{\bullet}|_{F}\cdot\dot{F}=-g(\mu_{\bullet}(F){\mathscr{R}}_{\bullet}F,\dot{F})$

thus

$\langle\\!\langle
d\mu_{\bullet}|_{F}\cdot\dot{F},\mu_{\bullet}(F)\rangle\\!\rangle=\langle\\!\langle-\mu_{\bullet}(F){\mathscr{R}}_{\bullet}F,\dot{F}\rangle\\!\rangle=-\Omega_{\bullet}(\mu_{\bullet}(F)iF,\dot{F})=-\iota_{X_{\mu_{\bullet}(F)}(F)}\Omega_{\bullet}(\dot{F}).$

In particular,

$W_{\bullet}(F)=-\mu_{\bullet}(F){\mathscr{R}}_{\bullet}F$ (2.14)

is the gradient vector field of the functional $\|\mu_{\bullet}\|$ on
${\mathscr{F}}$. Eventually, the gradient vector field of $\phi$ on
${\mathscr{F}}$ is the sum

$\nabla\phi=W_{I}+W_{J}+W_{K}=-{\mathcal{I}}X_{\mu_{I}}-{\mathcal{J}}X_{\mu_{J}}-{\mathcal{K}}X_{\mu_{K}}.$
(2.15)

The downward gradient flow of $\phi$ along ${\mathscr{F}}$ given by the
evolution equation

$\frac{\partial F}{\partial t}=-\nabla\phi(F).$ (2.16)

is just the usual hyperKähler moment map flow on ${\mathscr{F}}$. Since this
flow may not preserve the closedness of $F$, we can consider the flow along
the space of closed differential forms ${\mathscr{F}}^{c}$, defined by

$\frac{\partial F}{\partial t}=-\Pi\nabla\phi(F).$ (2.17)

or along ${\mathscr{F}}^{c}_{\alpha}$, given by

$\frac{\partial F}{\partial t}=-\Pi_{\alpha}\nabla\phi(F).$ (2.18)

The latter is called the _modified moment map flow_. This is merely the
downward gradient flow of the function $\phi$ restricted to the subspace
${\mathscr{F}}^{c}_{\alpha}$.

###  2.15. Fixed points and decay of the flow

We investigate the basic, yet crucial, properties of the modified moment map
flow.

######  Proposition 2.15.1.

Fixed points the above flows are the zeroes of the hyperKähler moment map
${\bm{\mu}}$.

###### Proof.

If ${\bm{\mu}}(F)=0$, then $\nabla\phi(F)=0$ according to Formulas(2.15) and
(2.14). It follows that $F$ is a fixed point of the flows.

Conversely, let $F\in{\mathscr{F}}^{c}_{\alpha}$ be a fixed point of the
modified moment map flow (2.18). Then $\Pi_{\alpha}\nabla\phi(F)=0$, so that

$\langle\\!\langle\Pi_{\alpha}\nabla\phi(F),F\rangle\\!\rangle=\langle\\!\langle\nabla\phi(F),\Pi_{\alpha}F\rangle\\!\rangle=\langle\\!\langle\nabla\phi(F),F\rangle\\!\rangle,$

since $F\in{\mathscr{F}}^{c}_{\alpha}$. By Formula (2.15)

$\langle\\!\langle\Pi_{\alpha}\nabla(F),F\rangle\\!\rangle=\langle\\!\langle
W_{I}(F)+W_{J}(F)+W_{K}(F),F\rangle\\!\rangle.$

An easy computation shows that

$\langle\\!\langle
W_{\bullet}(F),F\rangle\\!\rangle=-\langle\\!\langle\mu_{\bullet}(F){\mathscr{R}}_{\bullet}F,F\rangle\\!\rangle=2\|\mu_{\bullet}(F)\|^{2},$

and we deduce that

$\langle\\!\langle\Pi_{\alpha}\nabla\phi(F),F\rangle\\!\rangle=2\sum\|\mu_{\bullet}(F)\|^{2}.$
(2.19)

Then $\mu_{\bullet}(F)=0$ and ${\bm{\mu}}(F)=0$. The proof of the lemma is
identical for the two other flows and left to the reader. ∎

######  Proposition 2.15.2.

Let $F_{t}$ be a solution of one of the above flows, for $t$ in some interval.
Then

$\frac{\partial}{\partial
t}\|F_{t}\|^{2}_{\mathcal{G}}=-4\|{\bm{\mu}}(F_{t})\|^{2}.$

In particular $\|F_{t}\|^{2}_{\mathcal{G}}$ cannot increase along the
interval.

###### Proof.

We deal with the case of the modified moment map flow, since the two others
are treated in the same way. We compute

$\frac{\partial\|F_{t}\|^{2}_{\mathcal{G}}}{\partial
t}=2\langle\\!\langle\frac{\partial F_{t}}{\partial
t},F_{t}\rangle\\!\rangle=-2\langle\\!\langle\Pi_{\alpha}\nabla\phi(F_{t}),F_{t}\rangle\\!\rangle.$

By Equation (2.19), we deduce

$\frac{\partial\|F_{t}\|^{2}_{\mathcal{G}}}{\partial
t}=-4\|{\bm{\mu}}(F)\|^{2}.$

∎

###  2.16. Short time existence

In this section, we explain how the Cauchy-Lipschitz theorem applies, once
suitable Hölder spaces have been introduced, and provides solutions of the
modified moment map flow.

We consider the $C^{k,\beta}$-norm $\|\cdot\|_{k,\beta}$ defined for functions
and tensors on $M$, where $\beta\in(0,1)$ is the Hölder regularity parameter
and $k\in{\mathbb{N}}$ is the number of derivatives controlled by the norm.
Spaces of smooth functions and differential forms can be completed into Banach
spaces, with respect to Hölder norms. For example, we denote by
${\mathscr{F}}(k,\beta)$ the completion of the smooth moduli space
${\mathscr{F}}$, with respect to the $\|\cdot\|_{k,\beta}$-Hölder norm. etc…

######  Theorem 2.16.1.

Given some Hölder space parameters $(k,\beta)$ and
$F\in{\mathscr{F}}^{c}_{\alpha}(k,\beta)$, there exists $\varepsilon_{0}>0$,
such that for every $\varepsilon\in(0,\varepsilon_{0})$, there exists a unique
differentiable map

$\displaystyle[-\varepsilon,\varepsilon]$
$\displaystyle\to{\mathscr{F}}^{c}_{\alpha}(k,\beta)$ $\displaystyle t$
$\displaystyle\mapsto F_{t}$

which is a solution of the modified moment map flow, with initial condition
$F_{0}=F$.

######  Lemma 2.16.2.

The projection $\Pi_{\alpha}$ defines a continuous map with respect to the
$(k,\beta)$-Hölder norm. In other words, there exists a constant $C>0$, such
that for every $F\in{\mathscr{F}}(k,\beta)$

$\|\Pi_{\alpha}F\|_{k,\beta}\geq C\|F\|_{k,\beta}.$

###### Proof of Lemma 2.16.2.

By classical Hodge theory, every $1$-form $F\in{\mathscr{F}}(k,\beta)$ admits
a ${\mathcal{G}}$-orthogonal decomposition

$F=F_{H}+\Delta G,$

where $G$ is a $1$-form orthogonal to harmonic forms and $F_{H}$ is the
harmonic part of $F$. A $C^{k,\beta}$-estimate on $F$ provides a control on
the $C^{k-2,\beta}$-norm of $\Delta F$. By definition $\Delta F=\Delta^{2}G$,
since $F_{H}$ is harmonic. Hence the $C^{k,\beta}$-norm of $F$ control the
$C^{k-2,\beta}$ norm of $\Delta^{2}G$. The operator $\Delta$ is selfajoint,
hence $\Delta G$ is orthogonal to the kernel of $\Delta$. In particular
elliptic Schauder estimates provide a control on the $C^{k,\beta}$-norm of
$\Delta G$. And since $G$ was chosen orthogonal to $\ker\Delta$, we deduce a
$C^{k+2,\beta}$ control on $G$, by elliptic estimates. In conclusion, there
exists a constant $c^{\prime}>0$ (independent of $F$), such that

$\|G\|_{C^{k+2,\beta}}\leq c^{\prime}\|F\|_{C^{k,\beta}}.$

The decomposition $F=F_{H}+\Delta G=F_{H}+dd^{*}G+d^{*}dG$ has three terms.
The last term is orthogonal to ${\mathscr{F}}^{c}(k,\beta)$, since it is
coclosed, hence orthogonal to closed forms. The first two terms are closed
hence

$\Pi(F)=F_{H}+dd^{*}F.$

The projection $\Pi_{\alpha}$ preserves the finite dimensional space of
Harmonic forms. Thus, the restriction of $\Pi_{\alpha}$ to harmonic forms is a
continuous linear map. Now
$\Pi_{\alpha}(F_{H})+dd^{*}G\in{\mathscr{F}}^{c}_{\alpha}(k,\beta)$ is a
closed form with cohomology class in ${\mathbb{R}}\alpha$. It follows that

$\Pi_{\alpha}(F)=\Pi_{\alpha}(F_{H})+dd^{*}G.$

Finally, the $C^{k+2,\beta}$-norm of $G$ controls the $C^{k,\beta}$-norm of
$dd^{*}G$ and the proposition follows. ∎

###### Proof of Theorem 2.16.1.

The map of $F\mapsto\nabla\phi(F)$ is a polynomial of order $3$ in the
coefficients of $F$, hence it is locally Lipschitzian with respect to the
$(k,\beta)$-Hölder norm. By Lemma 2.16.2, the map
$F\mapsto\Pi_{\alpha}\nabla\phi(F)$ is locally Lipschitzian as well. The
standard proof of existence of solution of an evolution equation rests on the
existence of fixed points of a functional

$\Psi:{\mathcal{B}}\to{\mathcal{B}}$

where ${\mathcal{B}}$ is the Banach space of continuous maps
$C^{0}([-\varepsilon,\varepsilon],{\mathscr{F}}^{c}_{\alpha}(k,\beta))$,
endowed with the norm

$\|F\|_{\infty}=\sup_{t\in[-\varepsilon,\varepsilon]}\|F_{t}\|_{k,\beta}.$

More precisely, $\Psi$ is defined for each $F\in{\mathcal{B}}$, by

$\Psi(F)_{t}=F_{0}+\int_{0}^{t}\Pi_{\alpha}\nabla\phi(F_{s})ds$

For $\varepsilon>0$ sufficiently small, the functional $\Psi$ is a locally
$\frac{1}{2}$-contractant map of Banach space for
$({\mathcal{B}},\|\cdot\|_{\infty})$ and the Banach fixed point theorem
applies. ∎

######  Corollary 2.16.3.

For a smooth differential form $F\in{\mathscr{F}}^{c}_{\alpha}$ and $k\geq 1$,
there exists $\varepsilon>0$ and a family of $\overrightarrow{V}$-valued
differential $1$-forms $F_{t}$, defined for $t\in[-\varepsilon,\varepsilon]$,
with the properties that

  * •

$F_{t}$ is closed with cohomology class $\alpha$ for each $t$;

  * •

$F_{0}=F$;

  * •

for each $t\in[-\varepsilon,\varepsilon]$, the $1$-form $F_{t}$ is of class
$C^{k}$ on $M$;

  * •

the map $t\mapsto F_{t}$ is differentialble with respect to the $C^{k}$-norm;

  * •

the family $F_{t}$ is a solution of the modified moment map flow.

In addition the family $F_{t}$, with the above properies, is unique.

###### Proof.

We apply Theorem 2.16.1 with $k-1$ and $k$ derivatives, which is possible
since the initial condition $F$ is smooth. We obtain two constants
$\varepsilon_{k-1}$ and $\varepsilon_{k}>0$ for each number of derivatives and
choose $0<\varepsilon<\min(\varepsilon_{k},\varepsilon_{k-1})$. We obtain a
flow line $F_{t}$ for $t\in[-\varepsilon,\varepsilon]$, with Hölder regularity
$(k,\beta)$ for each $t$. In particular, $F_{t}$ has $C^{k}$ regularity on
$M$, for $t$ fixed.

Conversely, if we are given a family with the property of the corollary and
$\varepsilon$ chosen as above, the $C^{k}$ regularity of $F_{t}$ implies a
Hölder regularity with parameters $(k-1,\beta)$. Then the uniqueness follows
from Theorem 2.16.1. ∎

######  Remark 2.16.4.

At this stage, the long times existence of the modified hyperKähler flow is
not clear. The $L^{2}$-bound seems to be an interesting tool, however it is
not obvious how to get better regularity estimates in this context.

###  2.17. Renormalized flow

We derive the evolution equation of a real blowup of the modified moment map
flow at the origin of ${\mathscr{F}}^{c}_{\alpha}$, also called the degenerate
fixed point of the flow.

Let $F_{t}\in{\mathscr{F}}^{c}_{\alpha}$ be a smooth solution of the modified
moment map flow defined for $t$ in a given interval, say $[0,T)$, with
$[F_{0}]=\alpha$. Then we can define a function $\tau:[0,T)\to{\mathbb{R}}$,
with $\tau(0)=1$ and

$[F_{t}]=\alpha_{t}=\tau(t)\alpha.$

If $\tau(t)$ does not vanish, we can consider the flow

$G_{t}=\frac{1}{\tau(t)}F_{t}.$

Then

$\frac{\partial}{\partial t}G_{t}=\frac{1}{\tau(t)}\frac{\partial
F_{t}}{\partial t}-\frac{\tau^{\prime}(t)}{\tau(t)^{2}}F_{t}.$

Now $\nabla\phi(F_{t})=\nabla\phi(\tau(t)G_{t})=\tau(t)^{3}\nabla\phi(G_{t})$,
and we have the identity

$\frac{\partial}{\partial
t}G_{t}=-\tau(t)^{2}\Pi_{\alpha}\nabla\phi(G_{t})-\frac{\tau^{\prime}(t)}{\tau(t)}G_{t}$

Using the fact that $[G_{t}]=\alpha$, we find the evolution equation of $\tau$

$\tau^{\prime}(t)\alpha=-\tau(t)^{3}[\Pi_{\alpha}\nabla\phi(G_{t})].$

In conclusion, the evolution equation for $(G,\tau)$ is given by

$\displaystyle\frac{\partial G}{\partial t}$
$\displaystyle=\tau^{2}\left(\frac{[\Pi_{\alpha}\nabla\phi(G)]}{\alpha}G-\Pi_{\alpha}\nabla\phi(G)\right)$
$\displaystyle\frac{\partial\tau}{\partial t}$
$\displaystyle=-\tau^{3}\frac{[\Pi_{\alpha}\nabla\phi(G)]}{\alpha}.$

The renormalized equation may degenerate if $\tau(t)$ approaches $0$. Indeed,
any pair $(G,\tau)$, where $\tau\equiv 0$ and $G\in{\mathscr{F}}^{c}$ is
solution of the renormalized equation. This corresponds to degenerate fixed
points $F=0$ of the modified moment map flow. On the contrary, $\tau\neq 0$
correspond to non degenerate fixed points of the modified moment map flow.

######  Remark 2.17.1.

It would be interesting to find an algebraic criterion, perhaps related to
some geometric stability property, in the spirit of _geometric invariant
theory_ , that rules out the convergence of a flow line toward a degenerate
fixed point.

###  2.18. Proofs of the main theorems for the smooth case

In this section, we collect our results to prove the main statement of the
introduction.

###### Proof of Theorem A.

The hyperKähler structure of ${\mathscr{F}}$ is described at §2.12. The fact
that the ${\mathbb{T}}$-action on ${\mathscr{F}}$ admits an explicit
hyperKähler moment map is explained at §2.13. ∎

###### Proof of Theorem B.

This is a restatement of Proposition 2.13.2. ∎

###### Proof of Theorem C.

By Proposition 2.15.1, a fixed point $F$ of the modified moment map flow
belongs to ${\mathscr{F}}^{c}_{\alpha}$ and satisfies ${\bm{\mu}}(F)=0$. Then
$[F]=\tau\alpha$ for some $\tau\in{\mathbb{R}}$. If $\tau\neq 0$,
$G=\tau^{-1}F$ has cohomology class $\alpha$. It follows that there exists
$f\in{\mathscr{M}}$ such that $G={\mathscr{D}}f$, with $\alpha(f)=\alpha$ and
$f$ is symplectic.

If $\tau=0$, then $F$ is an exact form. Hence $F=du$, where
$u:M\to\overrightarrow{V}$ is a smooth function. The condition
${\bm{\mu}}(F)=0$ implies that $u^{*}\omega_{V}$ is a selfdual form. But
$\omega_{V}$ is exact, which forces $u^{*}\omega_{V}=0$. This imples that $u$
is constant, hence $F=0$. ∎

###### Proof of Theorem D.

This is a restatement of Corollary 2.16.3. ∎

##  3\. Polyhedral symplectic geometry

The moment map geometry of the space of the space smooth differentials on a
torus was dealt with at §2. In this section, we show that the moment map
geometry for the smooth setting, admits a polyhedral analogue.

###  3.1. Triangulation and polyhedral maps

Using the notation of §2, the manifold $M=V/\Gamma$ is a quotient torus of
real dimension $4$, where $V$ is an affine space. Furthermore, its underlying
vector space $\overrightarrow{V}$ is equipped with an identification with the
space of quaternions ${\mathbb{H}}$. All the structures and notation defined
on $M$ at §2.12 shall we used without further warning.

A _Euclidean $k$-simplex_ of $V$ is by definition the convex hull of $k+1$
affinely independent points of $V$. A subset $\sigma$ of $M$ is called a
_Euclidean simplex_ of $M$, if there exists a Euclidean simplex
$\tilde{\sigma}$ of $V$, such that the canonical projection $p:V\to M$
restrict as a diffeomorphism
$p:\tilde{\sigma}\stackrel{{\scriptstyle\sim}}{{\to}}\sigma$.

######  Definition 3.1.1.

A _Euclidean triangulation_ of a quotient torus $M=V/\Gamma$ is a simplicial
decomposition of $M$, where each cell is a Euclidean simplex of $M$.

In the sequel, ${\mathscr{T}}$ always denotes a Euclidean triangulation of
$M$. A map $f:\sigma\to{\mathbb{R}}$ (resp. $f:\sigma\to V$), where $\sigma$
is a Euclidean simplex of $M$, is said to be affine if:

  * •

There exists a Euclidean simplex $\tilde{\sigma}$ of $V$ such that that $p$
restricts as a diffeomorphism $p:\tilde{\sigma}\to\sigma$.

  * •

The lift $f\circ p:\tilde{\sigma}\to{\mathbb{R}}$ (resp. $f\circ
p:\tilde{\sigma}\to V$) is an affine map.

More generally we shall use the following terminology:

######  Definition 3.1.2.

A map $f:M\to{\mathbb{R}}$ (resp. $f:M\to V$) is called polyhedral with
respect to a Euclidean triangulation ${\mathscr{T}}$,

  1. (1)

The map $f$ is continuous.

  2. (2)

The restriction of the map $f$ to every Euclidean simplex of the triangulation
is affine.

We say that a map $f:M\to M$ is polyhedral if it is continuous and for every
simplex $\sigma$ of ${\mathscr{T}}$, the restriction $f:\sigma\to M$ admits a
polyhedral lift $\tilde{f}:\sigma\to V$.

The moduli space of polyhedral maps $f:M\to M$, with respect to some fixed
Euclidean triangulation ${\mathscr{T}}$, replaces from now on the space of
smooth maps of $M$, and is still denoted

${\mathscr{M}}={\mathscr{M}}({\mathscr{T}})=\\{f:M\to M,\mbox{ $f$ is
polyhedral with respect to ${\mathscr{T}}$}\\}.$

###  3.2. Differential for polyhedral maps

The differential ${\mathscr{D}}f$ of a polyhedral map $f\in{\mathscr{M}}$ does
not make sense everywhere, since $f$ is only piecewise smooth on $M$. More
precisely ${\mathscr{D}}f$ is defined at every point $x$ contained in the
interior of a $4$-cell of ${\mathscr{T}}$, that is the complement of the
$3$-skeleton $M^{(3)}$ of ${\mathscr{T}}$, denoted

$M^{\prime}=M\setminus M^{(3)}.$

Furthermore, the $\overrightarrow{V}$-valued $1$-form ${\mathscr{D}}f$ is
locally constant on $M^{\prime}$. Thus it is natural to consider the space
${\mathscr{F}}$ of locally constant $\overrightarrow{V}$-valued differential
$1$-forms on $M^{\prime}$:

${\mathscr{F}}=\\{F\in\Omega^{1}(M^{\prime},\overrightarrow{V}),\quad\nabla
F=0\\},$

so that ${\mathscr{D}}$ can be understood as linear map between the vector
spaces:

${\mathscr{D}}:{\mathscr{M}}\to{\mathscr{F}}.$

More generally, we can consider the space of polyhedral $k$-forms on $M$,
denoted

$\Omega^{k}_{pol}({\mathscr{T}}),$

defined as the space of polyhedral functions
$M\to\Lambda^{k}\overrightarrow{V}^{*}$ and the space of locally constant
$k$-forms on $M^{\prime}$, denoted

$\Omega^{k}_{loc}({\mathscr{T}}),$

as the space of locally constant functions
$M^{\prime}\to\Lambda^{k}\overrightarrow{V}^{*}$ defined on the complement of
the $3$-skeleton of $M$.

In conclusion, the exterior differential defines a linear map

$d:\Omega^{k}_{pol}({\mathscr{T}})\to\Omega^{k+1}_{loc}({\mathscr{T}}),$

which is extended canonically as a linear map

$d:\Omega^{k}_{pol}({\mathscr{T}})\otimes\overrightarrow{V}\to\Omega^{k+1}_{loc}({\mathscr{T}})\otimes\overrightarrow{V},$

and the differential ${\mathscr{D}}$ defines a map

${\mathscr{D}}:{\mathscr{M}}=\Omega^{0}_{pol}({\mathscr{T}})\otimes\overrightarrow{V}\to{\mathscr{F}}=\Omega^{1}_{loc}({\mathscr{T}})\otimes\overrightarrow{V}.$

###  3.3. Whitney cohomology

Let $\gamma=(\gamma_{\sigma})$ be a collection of smooth differential
$k$-forms for each $4$-cell $\sigma$ of ${\mathscr{T}}$. Assume that
$\sigma_{1}$ and $\sigma_{2}$ are cells of ${\mathscr{T}}$ intersecting along
a cell $\sigma_{12}=\sigma_{1}\cap\sigma_{2}$. We can consider the pullback
$\gamma_{1}$ of $\gamma_{\sigma_{1}}$ on along $\sigma_{12}$. Similarly, we
can consider the pullback $\gamma_{2}$ of $\gamma_{\sigma_{2}}$ along
$\sigma_{12}$. In general $\gamma_{1}$ and $\gamma_{2}$ do not agree. This
leads to the following definition

######  Definition 3.3.1.

A familly of smooth differential $k$-forms on $4$-cells
$\gamma=(\gamma_{\sigma})$ is called a Whitney form, if for every pair of
$4$-cells $\sigma_{1},\sigma_{2}$ of ${\mathscr{T}}$, the two possible
pullbacks of $\gamma$ along the intersection
$\sigma_{12}=\sigma_{1}\cap\sigma_{2}$ agree.

######  Remark 3.3.2.

A familly of differential forms $(\gamma_{\sigma})$ that satisfies the Whitney
condition is naturally defined along every cell, not only $4$-cells.

A differential form $\gamma\in\Omega_{loc}^{k}({\mathscr{T}})$ is locally
constant on the interior of each $4$-cell $\sigma$ of ${\mathscr{T}}$. Hence,
$\gamma$ can be extended up to the boundary so that $\gamma$ can be understood
as a family $(\gamma_{\sigma})$ of smooth forms on $4$-cells. If the Whitney
condition is met, we say that $\gamma$ is a locally constant Whitney form. The
space of locally constant Whitney forms is denoted
$\Omega^{k}_{Whi}({\mathscr{T}})\subset\Omega^{k}_{loc}({\mathscr{T}})$.

We are going to see that locally constant Whitney forms are an analogue of
smooth closed forms. By definition, the usual exterior differential defines a
map

$d:\Omega^{k}_{pol}({\mathscr{T}})\to\Omega^{k+1}_{loc}({\mathscr{T}})$

with image contained in the space of Whitney forms
$\Omega^{k}_{Whi}({\mathscr{T}})$, according to the next proposition:

######  Proposition 3.3.3.

For every $\gamma\in\Omega_{pol}^{k}$, the differential $d\gamma$ belong to
$\Omega_{Whi}^{k+1}$.

###### Proof.

By definition, the pullback of a polyhedral form on any cell $\sigma$ is
affine. In particular its differential is constant. Let $\iota:\sigma\to M$ be
the canonical inclusion. Then $\iota^{*}d\gamma=d\iota^{*}\gamma$, does not
depend on any choice, so that the form $d\gamma$ satisfies the Whitney
condition. ∎

A Whitney form $\gamma\in\Omega^{k}_{Whi}\otimes\overrightarrow{V}$ defines an
integral

$c\mapsto\int_{c}\gamma,$

where $c$ is a $k$-cycle in $M$. In fact the integral depends only on the
homology class $[c]\in H_{k}(M,{\mathbb{R}})$, by Stokes theorem. Furthermore,
the integrals defined by $\gamma$ and $\gamma+du$ agree for every polyhedral
form $u\in\Omega^{k-1}_{pol}\otimes\overrightarrow{V}$. Thus we have defined a
linear map

$\frac{\Omega^{k}_{Whi}\otimes\overrightarrow{V}}{d\Omega^{k-1}_{pol}\otimes\overrightarrow{V}}\to
H_{k}(M,{\mathbb{R}})^{*}\otimes\overrightarrow{V}$

and it turns out that this map is an isomorphism. In particular, using the
duality $H^{k}(M,{\mathbb{R}})\simeq H_{k}(M,{\mathbb{R}})^{*}$, this proves
the following proposition:

######  Proposition 3.3.4.

There are canonical isomorphisms

$H^{k}(M,{\mathbb{R}})\simeq\frac{\Omega^{k}_{Whi}({\mathscr{T}})}{d\Omega^{k-1}_{pol}({\mathscr{T}})}.$

and

$H^{k}(M,\overrightarrow{V})\simeq\frac{\Omega^{k}_{Whi}({\mathscr{T}})\otimes\overrightarrow{V}}{d\Omega^{k-1}_{pol}({\mathscr{T}})\otimes\overrightarrow{V}}.$

In particular, any Whitney form $F$ in ${\mathscr{F}}$ defines a cohomology
class

$[F]\in H^{1}(M,\overrightarrow{V}).$

Since $H^{1}(M,{\mathbb{R}})$ is identified to $\overrightarrow{V}^{*}$ for a
torus, it follows that $[F]$ is understood as element

$[F]\in
H^{1}(M,{\mathbb{R}})\otimes\overrightarrow{V}\simeq\overrightarrow{V}^{*}\otimes\overrightarrow{V}\simeq\mathrm{End}(\overrightarrow{V}).$

###  3.4. Cohomology class of a map

For every polyhedral map $f\in{\mathscr{M}}$, the differential
${\mathscr{D}}f\in{\mathscr{F}}$ is a Whitney form. In other words

$\mathrm{Im}\;{\mathscr{D}}\subset\Omega^{1}_{Whi}({\mathscr{T}})\otimes\overrightarrow{V}\subset\Omega^{1}_{loc}({\mathscr{T}})\otimes\overrightarrow{V}={\mathscr{F}}.$

In particular, we can define, as in the smooth setting

$\alpha:{\mathscr{M}}\to H^{1}(M,\overrightarrow{V}),$

by

$\alpha(f)=[{\mathscr{D}}f].$

As in the smooth case, we have the following result:

######  Proposition 3.4.1.

For a Whitney form $F\in{\mathscr{F}}$, the following properties are
equivalent

  1. (1)

The cohomology class $[F]\in
H^{1}(M,\overrightarrow{V})=\overrightarrow{V}^{*}\otimes\overrightarrow{V}=\mathrm{End}_{\mathbb{R}}(\overrightarrow{V})$
belongs to the image of $\Gamma^{*}\otimes_{\mathbb{Z}}\Gamma$ in
$\overrightarrow{V}^{*}\otimes\overrightarrow{V}$.

  2. (2)

There exists $f\in{\mathscr{M}}$ such that $F={\mathscr{D}}f$. In particular
$\alpha(f)=[F]$.

###  3.5. HyperKähler structure in the polyhedral case

The space of locally constant $1$-forms ${\mathscr{F}}$ is endowed with a
HyperKähler structure and a toric action. The construction is similar to the
smooth case explained at Section §2.12. We recall the ingredient of the
construction: the affine space $V$, is complex $2$-dimensional and endowed
with a Hermitian structure deduced from ${\mathbb{H}}$. This provides a flat
Kähler structure $(M,g_{M},i,\omega_{M})$ on the torus $M=V/\Gamma$. In
addition, the vector bundle

$T^{*}M\otimes_{\mathbb{R}}\overrightarrow{V}\to M$

admits a fiberbise Euclidean metric $g$. Given $F_{1},F_{2}\in{\mathscr{F}}$,
it makes sense to define

${\mathcal{G}}(F_{1},F_{2})=\int_{M}g(F_{1},F_{2})\sigma_{M}$

exactly as in the smooth case. The fact that the integrand is not well defined
along the $3$ skeleton is not a problem, since its measure is $0$. In fact
this integral can be written as a finite sum:

${\mathcal{G}}(F_{1},F_{2})=\sum_{\sigma_{i}}g(F_{1}(\sigma_{i}),F_{2}(\sigma_{i}))\mathrm{vol}(\sigma_{i})$

where:

  * •

The sum is taken over all $4$-simplices $\sigma_{i}$ of ${\mathscr{T}}$.

  * •

The notation $F(\sigma_{i})$ refers to the (constant) value taken by $F$ on
the interior of $\sigma_{i}$.

  * •

$\mathrm{vol}(\sigma_{i})$ is the volume of $\sigma_{i}$ with respect to the
Riemannian metric $g_{M}$.

Similarly, we define a paring for polyhedral functions on $M$ or locally
constant functions on $M^{\prime}$ by

$\langle\\!\langle u,v\rangle\\!\rangle=\int_{M}uv\sigma_{M}.$

The identification between $\overrightarrow{V}$ and the quaternions
${\mathbb{H}}$ (cf. §2.12 for details) provides three almost complex
structures ${\mathcal{I}},{\mathcal{J}}$ and ${\mathcal{K}}$ acting on
${\mathscr{F}}$. As in the smooth case, we deduce three Kähler forms
$\Omega_{I}$, $\Omega_{J}$ and $\Omega_{K}$ on ${\mathscr{F}}$, associated to
each complex structures ${\mathcal{I}}$, ${\mathcal{J}}$ and ${\mathcal{K}}$.

The involutions of ${\mathscr{F}}$, denoted ${\mathscr{R}}_{I}$,
${\mathscr{R}}_{J}$ and ${\mathscr{R}}_{K}$ are defined as in the smooth
setting. For example, the almost complex structure $I$ defined on
$\overrightarrow{V}$ provides an almost complex structure on the bundle
$\Lambda^{1}M\otimes\vec{V}\to M$, given by

$I\cdot F=-F\circ I.$

Then, ${\mathcal{I}}$ is the almost complex structure on
${\mathscr{F}}=\Omega^{1}_{loc}({\mathscr{T}})$ defined by

$({\mathcal{I}}\cdot F)_{x}=I\cdot F_{x}.$

Then we define ${\mathscr{R}}_{I}F=i{\mathcal{I}}\cdot F$.

###  3.6. Hamiltonian torus action in the polyhedral case

We introduce the torus

${\mathbb{T}}\subset\Omega^{0}_{loc}({\mathscr{T}})\otimes_{\mathbb{R}}{\mathbb{C}},$

as the space of locally constant functions on $M^{\prime}$ with values in
$S^{1}\subset{\mathbb{C}}$.

The moduli space ${\mathscr{F}}$ admits a canonical action of ${\mathbb{T}}$,
induced by the pointwise standard complex multiplication of
$\overrightarrow{V}$. More precisely, for $\lambda\in{\mathbb{T}}$ and
$F\in{\mathscr{F}}$, we define $\lambda\cdot F\in{\mathscr{F}}$ by the
condition

$(\lambda\cdot F)_{x}=u(x)F_{x}$

for every $x\in M^{\prime}$. Roughly stated, ${\mathbb{T}}$ is identified to
$(S^{1})^{k}$, where $k$ is the number of $4$-simplices in ${\mathscr{T}}$.
The Lie algebra $\mathfrak{t}$ of the group ${\mathbb{T}}$ is identified to
the space of locally constant real-valued functions on $M^{\prime}$, that is

$\mathfrak{t}=\Omega^{0}_{loc}({\mathscr{T}}),$

with exponential map $\exp:\mathfrak{t}\to{\mathbb{T}}$, given by

$\exp(\zeta)=e^{i\zeta}.$

Then the infinitesimal action of the Lie algebra $\mathfrak{t}$ on
${\mathscr{F}}$, is given for each $\zeta\in\mathfrak{t}$ by the vector Field

$X_{\zeta}(F)=\zeta iF.$

As in the smooth case, we have the following result:

######  Theorem 3.6.1.

Given a Hermitian affine space $V$ with an identification between the
underlying vector space $\overrightarrow{V}$ and ${\mathbb{H}}$, the space
${\mathscr{F}}$ admits a natural hyperKähler structure

$({\mathscr{F}},{\mathcal{G}},{\mathcal{I}},{\mathcal{J}},{\mathcal{K}}),$

and a action of the torus ${\mathbb{T}}$, which preserves the hyperKähler
structure. Furthermore, the torus action is Hamiltonian with respect to the
three symplectic forms $\Omega_{I}$, $\Omega_{J}$ and $\Omega_{K}$ with moment
maps

$\mu_{\bullet}:{\mathscr{F}}\to\mathfrak{t}$

given by

$\mu_{\bullet}(F)=-\frac{F^{*}\omega_{V}\wedge\hat{\omega}_{\bullet}}{\sigma_{M}}\quad\in\Omega^{0}_{loc}({\mathscr{T}})=\mathfrak{t},$

for $\bullet=I,J$ and $K$. In other words, we have the identities:

$d\langle\\!\langle\mu_{\bullet},\zeta\rangle\\!\rangle=-\iota_{X_{\zeta}}\Omega_{\bullet}.$

###### Proof.

The proof is formally the same as in the smooth setting. ∎

###  3.7. Application to symplectic polyhedral maps

The vanishing locus of the hyperKähler moment map

${\bm{\mu}}:{\mathscr{F}}\to\mathfrak{t}^{3}$

is closely related to the group of smooth symplectic diffeomorphisms of $M$,
as proved in Proposition 2.13.2. The situation in the polyhedral setting is
analogous. To see this, we first define the notion of _polyhedral symplectic_
map:

######  Definition 3.7.1.

A polyhedral map $f\in{\mathscr{M}}({\mathscr{T}})$, such that

$f^{*}\omega_{M}=\omega_{M}$

along the complement of the $3$-skeleton of ${\mathscr{T}}$, is called a
symplectic polyhedral map.

######  Remark 3.7.2.

The restriction of a symplectic polyhedral map $f$ to a $4$-cell $\sigma$ of
${\mathscr{T}}$ is a symplectic affine map. In particular, the corresponding
linear map is invertible, so that $f:\sigma\to M$ is an embedding. The
property $f^{*}\omega_{M}=\omega_{M}$, along each face implies that
$f^{*}[\omega_{M}]=[\omega_{M}]$ which implies that $f$ has degree $1$. It
seems from this perspective that $f$ is quite close to be a homeomorphism.
Yet, $f$ could be ramified along the $3$-skeleton of ${\mathscr{T}}$, thus is
not clear to us whether or not $f$ it is a homeomorphism.

######  Corollary 3.7.3.

Let $f\in{\mathscr{M}}$ be a polyhedral map such that
$f^{*}[\omega_{M}]=[\omega_{M}]$. If $f$ satisfies the equation

${\bm{\mu}}({\mathscr{D}}f)=0.$

Then $f$ is a symplectic polyhedral map.

###### Proof.

The proof is almost identical to the one in the smooth case. Let
$f\in{\mathscr{M}}$ be a polyhedral map that satisfies the hypothesis of the
theorem. First, we notice that $f^{*}\omega_{M}-\omega_{M}$ is a locally
constant form which satisfies the Whitney condition. By assumption, the
cohomology class of this form vanishes as an element of
$\Omega^{2}_{Whi}/d\Omega^{1}_{pol}\simeq H^{2}(M,{\mathbb{R}})$. Thus, there
exists a polyhedral $1$-form $\Gamma$, such that

$d\gamma=f^{*}\omega_{M}-\omega_{M}.$

The condition ${\bm{\mu}}({\mathscr{D}}f)=0$ insures that $f^{*}\omega_{M}$ is
a selfdual form on $M^{\prime}$. It follows that $d\gamma$ is a locally
constant selfdual form defined on $M^{\prime}$. Thus

$\int_{M}\|d\gamma\|^{2}_{g}\sigma_{M}=\int_{M}(d\gamma)\wedge(d\gamma)=\int
d(\gamma\wedge d\gamma).$

But $\gamma\wedge d\gamma$ satisfies the Whitney condition so that the Stokes
theorem applies. Thus the integral vanishes and $d\gamma=0$. ∎

###  3.8. Modified moment map flow

We proceed exactly as in the smooth case and introduce a modified hyperKähler
moment map. For this purpose, we consider the functional

$\phi:{\mathscr{F}}\to{\mathbb{R}},$

given by

$\phi(F)=\|{\bm{\mu}}(F)\|^{2},$

and its downward gradient flow

$\frac{\partial F}{\partial t}=-\nabla\phi(F),$

which is the classical hyperKähler flow. However, this flow does not preserve
neither the closed closed forms nor their cohomology classes. To fix this, we
consider the space ${\mathscr{F}}^{c}$ of Whitney forms in ${\mathscr{F}}$.
More explicitely

${\mathscr{F}}^{c}=\Omega_{Whi}^{1}({\mathscr{T}})\otimes\overrightarrow{V}.$

Given a cohomology class $\alpha\in H^{1}(M,\overrightarrow{V})$, the space
${\mathscr{F}}^{c}_{\alpha}$ is the space of Whitney forms
$F\in{\mathscr{F}}^{c}$ whose cohomology class satisfies
$[F]\in{\mathbb{R}}\alpha$:

${\mathscr{F}}^{c}_{\alpha}=\\{F\in{\mathscr{F}}^{c},\exists\tau\in{\mathbb{R}},[F]=\tau\alpha\\}.$

The orthogonal projection of ${\mathscr{F}}$ onto
${\mathscr{F}}^{c}_{\alpha}$, with respect to the metric ${\mathcal{G}}$ is
denoted

$\Pi_{\alpha}:{\mathscr{F}}\to{\mathscr{F}}^{c}_{\alpha}.$

Then, we consider the modified moment map flow on
${\mathscr{F}}^{c}_{\alpha}$:

$\frac{\partial F}{\partial t}=-\Pi_{\alpha}\nabla\phi(F).$ (3.1)

This is an ordinary differential equation, where the RHS is a homogeneous
polynomial expression of degree $3$ in $F$. In particular, the existence of
solutions for short time is insured by the Cauchy-Lipschitz theorem. Most
results proved in the smooth setting are proved formally in the same manner
for this finite dimensional version, in particular Propositions 2.15.1, 2.15.2
hold for the flow (3.1). Hence we obtain:

######  Theorem 3.8.1.

The fixed points of the modified moment map flow are the zeroes of the
hyperKähler moment map ${\bm{\mu}}$, that belong to
${\mathscr{F}}^{c}_{\alpha}$. Furthermore,

$\frac{\partial}{\partial
t}\|F_{t}\|^{2}_{{\mathcal{G}}}=-4\|{\bm{\mu}}(F_{t})\|^{2}$

along the flow, so that the norm $\|F\|_{\mathcal{G}}$ can only decrease.

######  Corollary 3.8.2.

Flow lines of the modified moment map flow are bounded in
${\mathscr{F}}^{c}_{\alpha}$. In particular, every maximal solution $F_{t}$ of
the flow, defined for $t\in[t_{0},T)$, satisfies $T=+\infty$. Furthermore, the
limiting orbits

${\mathscr{O}}(F_{t})=\bigcap_{t^{\prime}\geq t_{0}}\overline{\\{F_{t},t\geq
t^{\prime}\\}}$

are compact sets that consist of zeroes of ${\bm{\mu}}$.

###### Proof.

The fact that $\|F_{t}\|_{\mathcal{G}}$ is bounded along the flow insures long
time existence of the flow, by a classical result of ODE theory. The
compactness of the limiting orbits is also clear from the boundedness of the
flow.

The fact that the limiting orbits are zeroes of the hyperKähler moment map is
perhaps less standard, and rests on special features of the downward gradient
flow along ${\mathscr{F}}^{c}_{\alpha}$. We are going to prove that if
$F_{\infty}$ belongs to the limiting orbit, then ${\bm{\mu}}(F_{\infty})=0$.
If this is not the case, ${\bm{\mu}}(F_{\infty})\neq 0$. We can choose an open
neighborhood $U$ of $F_{\infty}$, sufficiently small, so that

$\|{\bm{\mu}}(F)\|^{2}_{\mathcal{G}}\geq\frac{1}{2}\|{\bm{\mu}}(F_{\infty})\|^{2}_{\mathcal{G}}=c>0$

for every $F\in U$. Furthermore, by the Cauchy-Lipschitz theorem, there exists
open neighborhoods $V\subset W$ of $F_{\infty}$, and $\delta\in(0,1)$,
sufficiently small, such that every flow starting at time $t=0$ from initial
condition $F\in V$ exist for a time interval $t\in[-\delta,\delta]$ and is
contained in $W$.

We consider the open set $U^{\prime}=U\cap V$. Since $F_{\infty}$ belongs to
the limiting orbit of $F_{t}$, we can choose $t_{k}\in[t_{0},+\infty)$ to be
an increasing sequence, such that $t_{0}=t_{0}$, $t_{k+1}\geq t_{k}+1$ and
$\lim F_{t_{k}}=F_{\infty}$. For $k$ sufficiently large, all the terms
$F_{t_{k}}$ belong to $U^{\prime}$. Up to extraction, we may assume that all
the terms $F_{t_{k}}$ belong to $U^{\prime}$, for $k\geq 0$. Hence, we can
write

$\displaystyle\|F_{t_{k}+\delta}\|^{2}-\|F_{t_{0}}\|^{2}$
$\displaystyle=\int_{t_{0}}^{t_{k}+\delta}\frac{\partial}{\partial
t}\|F_{t}\|^{2}dt$
$\displaystyle=-4\int_{t_{0}}^{t_{k}+\delta}\|{\bm{\mu}}(F_{t})\|^{2}dt$
$\displaystyle\leq-8\delta kc.$

This shows that $\|F_{t_{k}+\delta}\|^{2}$ is negative for $k$ sufficiently
large, which is a contradiction. ∎

###  3.9. Stability near the regular locus

The vanishing locus of the moment map

${\bm{\mu}}:{\mathscr{F}}^{c}_{\alpha}\to\mathfrak{t}^{3}$

is the space of fixed points and limiting orbits of the modified moment map
flow. This space is a cone, and it is not clear at this stage whether it is
regular in some sense.

######  Definition 3.9.1.

A point $F\in{\mathscr{F}}^{c}_{\alpha}\cap{\bm{\mu}}^{-1}(0)$ is called
regular, if the map ${\bm{\mu}}:{\mathscr{F}}^{c}_{\alpha}\to\mathfrak{t}^{3}$
has constant rank on a neighborhood of $F$ in ${\mathscr{F}}^{c}_{\alpha}$.

In particular, the space ${\mathscr{F}}^{c}_{\alpha}\cap{\bm{\mu}}^{-1}(0)$ is
a submanifold in a neighborhood of a regular point. However, it is not even
clear whether the moduli space admits _any_ regular point, although we suspect
that this condition is _generic_. We could not prove this claim so far. It is
fairly likely that this issue could be tackled using polyhedral approximation
of the smooth setting. Indeed, such an approach was successful in the
analogous situation of polyhedral Lagrangian tori (cf. [5] and [8]). At this
stage, we can merely show that the modified moment maps flow is well behavec
in a neighborhood of regular points of the moduli space:

######  Theorem 3.9.2.

Let $F\in{\mathscr{F}}^{c}_{\alpha}\cap{\bm{\mu}}^{-1}(0)$ be a regular point
of the moduli space. Then, the modified moment map flow is stable in a
neighborhood of $F$ and converges exponentially fast.

###### Proof.

Recall that $\nabla\phi(F)=\sum W_{\bullet}(F)$ by Equation (2.15). By Formula
(2.14),

$DW_{\bullet}|_{F}\cdot\dot{F}=-(D\mu_{\bullet}|_{F}\cdot\dot{F}){\mathscr{R}}_{\bullet}F-\mu_{\bullet}(F){\mathscr{R}}_{\bullet}\dot{F}.$

By assumption, $\mu_{\bullet}(F)=0$ and by Formula (2.8)
$D\mu_{\bullet}|_{F}\cdot\dot{F}=-g({\mathscr{R}}_{\bullet}F,\dot{F})$, so
that

$DW_{\bullet}|_{F}\cdot\dot{F}=g({\mathscr{R}}_{\bullet}F,\dot{F}){\mathscr{R}}_{\bullet}F.$

Therefore

${\mathcal{G}}(DW_{\bullet}|_{F}\cdot\dot{F},\dot{F})=\int_{M}g({\mathscr{R}}_{\bullet}F,\dot{F})^{2}\sigma_{M}.$

It follows that the Hessian of $\phi$ at $F$ is given by

$H_{F}(\dot{F},\dot{F})=\int_{M}\left(g({\mathscr{R}}_{I}F,\dot{F})^{2}+g({\mathscr{R}}_{J}F,\dot{F})^{2}+g({\mathscr{R}}_{K}F,\dot{F})^{2}\right)\sigma_{M}$

The integrand vanishes identically if, and only if, the vector $\dot{F}$
belongs to the tangent space of the moduli space ${\bm{\mu}}^{-1}(0)$ in
${\mathscr{F}}^{c}_{\alpha}$. We deduce that the Hessian is positive in
directions transverse to the moduli space. From this stage, it is a classical
result that the moduli space is stable under the flow in a neighborhood of $F$
and that the convergence is exponential. ∎

###  3.10. Convergence

The regularity assumption of Theorem 3.9.2 cannot be generally insured and the
strong conclusions of exponential convergence do not hold for every flow line
of the modified moment map flow. In spite of that, flow lines converge anyway,
according to the following result:

######  Theorem 3.10.1.

Every solution $F_{t}$ of the modified moment map flow, defined for some
interval $t\in[t_{0},+\infty)$, admits a limit

$F_{\infty}=\lim_{t\to+\infty}F_{t}\quad\in{\mathscr{F}}^{c}_{\alpha}$

solution of the equation ${\bm{\mu}}(F_{\infty})=0$.

One can define an extended flow map

$\Psi:[0,+\infty]\times{\mathscr{F}}^{c}_{\alpha}\to{\mathscr{F}}^{c}_{\alpha}$

given as follows: for $F\in{\mathscr{F}}^{c}_{\alpha}$ and $t\in[0,+\infty]$,
we put

$\Psi(t,F)=F_{t},$

where $F_{t}$ is the flow line with initial condition $F_{0}=F$. If
$t=+\infty$, it is understood that $F_{\infty}=\lim_{t\to\infty}F_{t}$. Then
$\Psi$ defines a continuous retraction of ${\mathscr{F}}_{\alpha}^{c}$ onto
${\mathscr{F}}^{c}_{\alpha}\cap{\bm{\mu}}^{-1}(0)$.

######  Corollary 3.10.2.

Let $f\in{\mathscr{M}}$ be a polyhedral map of the torus $M$, such that $f$
preserve the symplectic class:

$f^{*}[\omega_{M}]=[\omega_{M}].$

We put $\alpha=\alpha(f)$ and consider the corresponding modified moment map
flow. Let $F_{t}$, be a flow line, defined for $t\in[0,+\infty)$, with initial
condition $F_{0}={\mathscr{D}}f$. Then $F_{t}$ converges toward some
$F_{\infty}\in{\mathscr{F}}^{c}_{\alpha}\cap{\bm{\mu}}^{-1}(0)$. Furthermore

  1. (1)

If $[F_{\infty}]=0$, then $F_{\infty}=0$, which is the differential of any
constant map $f_{\infty}\in{\mathscr{M}}$.

  2. (2)

If $[F_{\infty}]=\tau\alpha\neq 0$, then $[\tau^{-1}F]=\alpha$, and there
exists a map $f_{\infty}\in{\mathscr{M}}$ such that
${\mathscr{D}}f_{\infty}=\tau^{-1}F_{\infty}$, and $f_{\infty}$ is a
symplectic polyhedral map.

###### Proof.

By Theorem 3.10.1, the flow $F_{t}$ converge with limit
$F_{\infty}\in{\mathscr{F}}^{c}_{\alpha}$, such that
${\bm{\mu}}(F_{\infty})=0$.

Assume that $[F_{\infty}]=0$, which implies that $F_{\infty}=d\gamma$, for
some $\gamma\in\Omega^{0}_{pol}\otimes\overrightarrow{V}$. The condition
${\bm{\mu}}(F_{\infty})$ implies that $d\gamma$ is selfdual. However selfdual
exact form must vanish (cf. proof of Corollary 3.7.3). Hence $F_{\infty}=0$,
which is the differential ${\mathscr{D}}f_{\infty}$ of some constant map:

If $[F_{\infty}]=\tau\alpha\neq 0$, then
$[\tau^{-1}F_{\infty}]=\alpha=\alpha(f)$. It follows that
$\tau^{-1}F_{\infty}$ is the differential ${\mathscr{D}}f_{\infty}$ of a
polyhedral map homotopic to $f$. In particular
$f^{*}_{\infty}[\omega_{M}]=[\omega_{M}]$ and
${\bm{\mu}}({\mathscr{D}}f_{\infty})={\bm{\mu}}(\tau^{-1}F_{\infty})=\tau^{-2}{\bm{\mu}}(F_{\infty})=0$.
We conclude by Corollary 3.7.3 that $f_{\infty}$ is a symplectic polyhedral
map. ∎

The proof of Theorem 3.10.1 relies on a result due to Lojasiewicz (cf. [6,
Lemma 2.2] or Proposition 1 p. 67 of [7], or Proposition 6.8 in [1]).

######  Lemma 3.10.3 (Lojasiewicz gradient inequality).

Let $\phi$ be a real analytic function on an open set
$W\subset{\mathbb{R}}^{k}$ and $x\in W$, a critival point of $\phi$, such that
$\phi(x)=0$. Then, there exists a neighborhood $U$ of $x$, some constants
$c>0$ and $\beta\in(0,1)$, such that

$\|\nabla\phi(y)\|\geq c|\phi(y)|^{\beta}$

for every $y\in W$, where $\|\cdot\|$ is an Euclidean norm on
${\mathbb{R}}^{k}$ and $\nabla$ is the gradient with respect to the Euclidean
inner product.

###### Proof of Theorem 3.10.1.

The function $\phi:{\mathscr{F}}^{c}_{\alpha}\to{\mathbb{R}}$ is polynomial of
degree $4$. In particular, $\phi$ is analytic and Lojasiewicz gradient
inequality applies (cf Lemma 3.10.3). Let $F_{0}\in{\mathscr{F}}^{c}_{\alpha}$
with flow line $F_{t}$, defined for $t\in[0,+\infty)$, and ${\mathscr{O}}$ its
limiting orbit. By Corollary 3.7.3, we know that

${\mathscr{O}}\subset\phi^{-1}(0)$

and, using the fact that ${\mathscr{O}}$ is compact,

$\lim_{t\to+\infty}\phi(F_{t})=0.$

By Proposition 2.15.2, the norm $\|F_{t}\|_{\mathcal{G}}$ decreases along the
flow. Put $K=\|F_{0}\|$. Then the flow line $\\{F_{t},t\geq 0\\}$ and its
limiting orbit ${\mathscr{O}}$ are contained in the interior of the closed
ball $\overline{B}(0,2K)\subset{\mathscr{F}}^{c}_{\alpha}$ of radius $2K$
centered at $0$.

For each $F\in\overline{B}(0,2K)$, there exists an open neighborhood $U_{F}$
of $F$ in ${\mathscr{F}}^{c}_{\alpha}$ and constants $c_{F}>0$ and
$\beta_{F}\in(0,1)$, which satisfy Lemma 3.10.3 for the function
$\phi:{\mathscr{F}}^{c}_{\alpha}\to{\mathbb{R}}$. In other words, for every
$F\in U_{F}$

$\|\nabla\phi(F)\|\geq c_{F}\phi(F)^{\beta_{F}}$

By compactness of $\overline{B}(0,2K)$, we can extract a finite cover
$U_{1},\cdots,U_{k}$ of the ball $\overline{B}(0,2K)$ by open sets of the
family $U_{F}$. We put

$U=\bigcup U_{i},\quad\beta=\max\beta_{i}\quad\mbox{ and }\quad c=\min c_{i},$

where $\beta_{i}$ and $c_{i}$ are the constants corresponding to the open sets
$U_{i}$. Then, for every $F\in U$ such that $\phi(F)\leq 1$, we have
$\phi(F)^{\beta_{i}}\geq\phi(F)^{\beta}$. Thus, for every $F\in U$ such that
$\phi(F)\leq 1$, we have

$\|\nabla\phi(F)\|_{\mathcal{G}}\geq c\phi(F)^{\beta}.$

Now, for every initial condition $F^{\prime}_{0}\in B(0,2K)$, the flow line
satisfies $F^{\prime}_{t}\in\overline{B}(0,2K)\subset U$ for every $t\geq 0$,
since the norm decreases along the flow. Since $\lim\phi(F^{\prime}_{t})=0$,
we have $\phi(F^{\prime}_{t})\leq 1$ for every $t$ sufficiently large, so that

$\|\nabla\phi(F^{\prime}_{t})\|_{\mathcal{G}}\geq
c\phi(F^{\prime}_{t})^{\beta}.$ (3.2)

Using the fact that

$\frac{\partial}{\partial
t}\phi(F_{t})=d\phi|_{F_{t}}\cdot\frac{\partial}{\partial
t}F_{t}={\mathcal{G}}(\nabla\phi(F_{t}),\nabla\phi(F_{t}))=-\|\nabla\phi(F_{t})\|^{2}_{\mathcal{G}},$

we compute

$-\frac{\partial}{\partial
t}\phi(F^{\prime}_{t})^{1-\beta}=(1-\beta)\phi(F^{\prime}_{t})^{-\beta}\|_{\mathcal{G}}\nabla\phi(F^{\prime}_{t})\|^{2}.$

For $t$ sufficiently large, we deduce by (3.2), the differential inequality

$-\frac{\partial}{\partial
t}\phi(F^{\prime}_{t})^{1-\beta}\geq(1-\beta)c\|\nabla\phi(F^{\prime}_{t})\|_{\mathcal{G}}.$

Hence, for $t_{1}>t_{0}$ sufficiently large, we obtain by integrating the
above inequality

$\phi(F^{\prime}_{t_{0}})^{1-\beta}-\phi(F^{\prime}_{t_{1}})^{1-\beta}=-\int_{t_{0}}^{t_{1}}\frac{\partial}{\partial
t}\phi(F^{\prime}_{t})^{1-\beta}dt\geq(1-\beta)c\int_{t_{0}}^{t_{1}}\|\nabla\phi(\zeta^{\prime}_{t})\|_{\mathcal{G}}dt.$

Put $C=\frac{1}{(1-\beta)c}$ so that we have

$\displaystyle d(F^{\prime}_{t_{0}},F^{\prime}_{t_{1}})$
$\displaystyle\leq\int_{t_{0}}^{t_{1}}\left\|\frac{\partial}{\partial
t}F^{\prime}_{t}\right\|_{\mathcal{G}}dt$
$\displaystyle\leq\int_{t_{0}}^{t_{1}}\|\nabla\phi(F^{\prime}_{t})\|_{\mathcal{G}}dt$
(3.3) $\displaystyle\Rightarrow d(F^{\prime}_{t_{0}},F^{\prime}_{t_{1}})$
$\displaystyle\leq
C\left(\phi(F^{\prime}_{t_{0}})^{1-\beta}-\phi(F^{\prime}_{t_{1}})^{1-\beta}\right).$
(3.4)

Since $\lim\phi(F^{\prime}_{t})^{1-\beta}=0$, the function
$t\mapsto\phi(F^{\prime}_{t})^{1-\beta}$ satisfies the Cauchy criterion, and
so does $t\mapsto F^{\prime}_{t}$ by Inequality (3.4). In particular
$F^{\prime}_{t}$ converges as $t$ goes to infinity. This shows that the flow
$F^{\prime}_{t}$ converges as $t$ goes to infinity for every
$F^{\prime}_{0}\in\overline{B}(0,2K)$. In fact, we could have chosen any
constant $K>0$ in our argument, so that the flow converges for any initial
condition $F^{\prime}_{0}\in{\mathscr{F}}^{c}_{\alpha}$. Thus the flow extends
as a map
$[0,+\infty]\times{\mathscr{F}}^{c}_{\alpha}\to{\mathscr{F}}^{c}_{\alpha}$ by
setting $\Psi(+\infty,F_{0})=F_{\infty}$, where $F_{\infty}$ is the limit of
the flow.

Since the flow $\Psi$ is continuous on
$[0,+\infty)\times{\mathscr{F}}^{c}_{\alpha}$, it is sufficient to prove that
the limit $F_{\infty}$ of $F_{t}$ depends continously on $F_{0}$ to prove that
the retraction map is continuous. To do this, first observe that passing to
the limit in Inequality (3.4) provides a control between $F^{\prime}_{t}$ and
its limit $F^{\prime}_{\infty}$. More precisely, for all $F^{\prime}_{0}\in
B(0,2K)$ and $T>0$ sufficiently large, such that $\phi(F_{t}^{\prime})\leq 1$,
we have

$d(F_{t}^{\prime},F^{\prime}_{\infty})\leq C\phi(F^{\prime}_{t})^{1-\beta}$
(3.5)

for every $t\geq T$, where $T$ is chosen sufficiently large..

In conclusion, starting from $F_{0}\in{\mathscr{F}}^{c}_{\alpha}$ with
$\|F_{0}\|=K$ as before, we pick $T>0$ sufficiently large so that
$\phi(F_{T})\leq\frac{1}{2}$ and $C\phi(F_{T})^{1-\beta}\leq\varepsilon$.
Since the flow
$\varphi_{T}:{\mathscr{F}}^{c}_{\alpha}\to{\mathscr{F}}^{c}_{\alpha}$ is
continuous, there exists a neigborhood $U^{\prime}\subset U\cap B(0,2K)$ of
$F_{0}$, such that for every $F\in\varphi_{T}(U^{\prime})$, we have

$d(F,F_{T})\leq\varepsilon,\quad\phi(F)\leq 1$

and

$C\phi(F)^{1-\beta}\leq 2\varepsilon.$

We deduce that for every $F^{\prime}_{0}\in U^{\prime}$, with corresponding
flow $F^{\prime}_{t}$

$d(F_{\infty},F^{\prime}_{\infty})\leq
d(F_{\infty},F_{T})+d(F_{T},F^{\prime}_{T})+d(F^{\prime}_{T},F^{\prime}_{\infty})\leq\varepsilon+\varepsilon+2\varepsilon=4\varepsilon.$

We conclude that the retraction map is continuous. ∎

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