# Foldable fans, cscK surfaces and local K-moduli

Carl Tipler  Univ Brest, UMR CNRS 6205, Laboratoire de Mathématiques de
Bretagne Atlantique, France<EMAIL_ADDRESS>
###### Abstract.

We study the moduli space of constant scalar curvature Kähler surfaces around
the toric ones. To this aim, we introduce the class of foldable surfaces :
smooth toric surfaces whose lattice automorphism group contain a non trivial
cyclic subgroup. We classify such surfaces and show that they all admit a
constant scalar curvature Kähler metric (cscK metric). We then study the
moduli space of polarised cscK surfaces around a point given by a foldable
surface, and show that it is locally modeled on a finite quotient of a toric
affine variety with terminal singularities.

##  1\. Introduction

The construction of moduli spaces of varieties is a central problem in complex
geometry. Pioneered by Riemann, the case of moduli spaces of curves is now
fairly well understood. In higher dimension however, the situation becomes
much more delicate. Indeed, iterated blow-ups (see e.g. [25, Example 4.4]) or
the presence of non-discrete automorphisms induce non-separatedness in moduli
considerations. It is then remarkable that despite their very simple
combinatorial description, toric surfaces are subject to those two issues, and
typically correspond to pathological points in the moduli space of surfaces.
In this paper, we will show that when one restricts to constant scalar
curvature Kähler (cscK) surfaces, the moduli space enjoys a very nice
structure near its toric points.

Our motivation to restrict to cscK surfaces comes from the Yau–Tian–Donaldson
conjecture (YTD conjecture, see [53, 48, 13]) which predicts that the
existence of a constant scalar curvature Kähler metric (cscK metric for short)
on a given polarised Kähler manifold should be equivalent to (a uniform
version of) K-polystability. In the Fano case, the proof of this conjecture
led to the recent construction of the moduli space of K-polystable Fano
varieties by Odaka and Li–Wang–Xu [38, 30, 31] (see [48, 4, 9, 49] for the YTD
conjecture and also [52, Part II] and reference therein for a historical
survey on the related moduli space). Moreover, from Odaka’s work [36, 37], in
the canonically polarised case, K-stability is equivalent to having semi-log
canonical singularities, a class that was successfully introduced to construct
the so-called KSBA moduli space (see [25] for a survey on moduli of varieties
of general type). While a general algebraic construction of a moduli space for
K-polystable varieties seems out of reach at the moment, on the differential
geometric side, Dervan and Naumann recently built a separated coarse moduli
space for compact polarised cscK manifolds [12], extending Fujiki and
Schumacher’s construction that dealt with the case of discrete automorphism
groups [15] (see also Inoue’s work on the moduli space of Fano manifolds with
Kähler–Ricci solitons [22]).

However, it seems to the author that the Fano case, and its log or pair
analogues, are the only sources of examples for those moduli spaces, when
non–discrete automorphism groups are allowed (see also the discussion in [12,
Example 4.16] that may lead to further examples via moduli of polystable
bundles). Restricting to surfaces, moduli spaces of Kähler–Einstein Del Pezzos
were explicitly constructed in [39]. Our focus will then be on the moduli
space of polarised cscK surfaces, and its geometry around points corresponding
to the toric ones. Given that they satisfy the YTD conjecture by Donaldson’s
work [13], toric surfaces draw a lot of attention (see e.g. [14, 29, 27, 50,
41, 54, 2, 55, 8, 42, 1]), and they appear as natural candidates to test the
general machinery on (compare also with [23, 40, 19]). As the full
classification of cscK toric surfaces is yet to be carried out, we will
consider a subclass whose fans carry further symmetries, which we introduce
now.

######  Definition 1.1.

Let $N$ be a rank two lattice and $\Sigma$ a complete smooth fan in
$N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$. The fan $\Sigma$ is called
foldable if its lattice automorphism group $\mathrm{Aut}(N,\Sigma)$ contains a
non-trivial cyclic subgroup. The associated toric surfaces will be called
foldable surfaces.

In order to understand this class of surfaces amongst the toric ones, we show
that all crystallographic groups arise as lattice automorphism groups of two
dimensional smooth fans (see Section 3.1):

######  Proposition 1.2.

Let $N$ be a rank two lattice and $\Sigma$ a complete smooth fan in
$N_{\mathbb{R}}$. Then $\mathrm{Aut}(N,\Sigma)$ is isomorphic to one of the
groups in the following set

$\\{C_{1},C_{2},C_{3},C_{4},C_{6},D_{1},D_{2},D_{3},D_{4},D_{6}\\}.$

Moreover, any group in the above list is isomorphic to the lattice
automorphism group of some complete two dimensional smooth fan.

In Proposition 1.2, we denote the cyclic group of order $p$ by $C_{p}$ and the
dihedral group of order $2p$ by $D_{p}$. We make a distinction between $C_{2}$
and $D_{1}$ by assuming that $C_{2}$ acts via $-\mathrm{Id}$ on $N$ while
$D_{1}$ acts through a reflection (see Section 3.1). Then, from Proposition
1.2, a complete and smooth two dimensional fan $\Sigma$ in $N_{\mathbb{R}}$ is
foldable if and only if $\mathrm{Aut}(N,\Sigma)$ is not isomorphic to $C_{1}$
or $D_{1}$. The associated class of toric surfaces then provides a wide class
of examples of cscK surfaces (see Proposition 3.13) :

######  Proposition 1.3.

Let $X$ be a foldable surface. Then $X$ admits a cscK metric in some Kähler
class.

This result relies on the classification of foldable surfaces (Section 3.2)
and an application of Arezzo–Pacard–Singer and Székelyhidi’s blow-up theorem
for extremal Kähler metrics [3, 47]. Similar arguments were used in [50] to
show that any toric surface admits an iterated toric blow-up with a cscK
metric.

######  Remark 1.4.

To our knowledge, all known examples of toric cscK surfaces are foldable (see
e.g. [50] for a family of examples with unbounded Picard rank, and [51] for a
complete classification of polarised cscK toric surfaces up to Picard rank
$4$). It would be interesting to produce examples of non foldable toric cscK
surfaces.

Our main result shows that the moduli space of polarised cscK surfaces, which
is known to be a complex space [15, 12], is quite well behaved around a
foldable one, as it is locally modeled on a finite quotient of a toric
terminal singularity (see Section 4.3 for a definition of this class of
singularities that originated in the MMP).

######  Theorem 1.5.

Let $(X,[\omega])$ be a polarised cscK foldable surface with fan $\Sigma$ in
$N_{\mathbb{R}}$, and let $G=\mathrm{Aut}(N,\Sigma)$. Denote by
$[X]\in\mathscr{M}$ the corresponding point in the moduli space $\mathscr{M}$
of polarised cscK surfaces. Then, there are :

  1. (i)

A Gorenstein toric affine $G$-variety $W$ with at worst terminal
singularities,

  2. (ii)

Open neighborhoods $\mathscr{W}$ and $\mathscr{U}$ of respectively (the image
of) the torus fixed point $x\in\mathscr{W}\subset W/G$ and of
$[X]\in\mathscr{U}\subset\mathscr{M}$,

such that $(\mathscr{U},[X])$ and $(\mathscr{W},x)$ are isomorphic as pointed
complex spaces.

This result sheds some light on the singularities that may appear on the
moduli space of polarised cscK surfaces. Note that by [5, Theorem 4], the good
moduli space of smooth K-polystable Fano manifolds of fixed dimension and
volume has klt type singularities (see Section 4.3 for definitions). Also, a
combination of Braun, Greb, Langlois and Moraga’s work on GIT quotients of klt
type singularities [5, Theorem 1], together with the construction of Dervan
and Naumann’s moduli space for cscK polarised manifolds [12, Section 3],
directly implies that the moduli space of polarised cscK surfaces has klt type
singularities around its toric points (see Proposition 2.3). Hence, Theorem
1.5 provides a refinement of Proposition 2.3, showing that, up to a finite
covering, the toric structure is locally preserved on the moduli space, and
that the singularities are terminal, at a foldable point. On the other hand,
one may not expect such a nice structure at toric boundary points of a (still
hypothetical) compactification $\overline{\mathscr{M}}$. Indeed, the local
structure of the K-moduli space of Fano varieties was studied around singular
toric varieties in [23, 40, 19], where much wilder phenomena were observed,
mainly due to the existence of obstructed deformations.

######  Remark 1.6.

It would be interesting to understand whether the final quotient $W/G$ in
Theorem 1.5 still is terminal or not. Note however that it is a non-trivial
problem to settle when a finite quotient of a terminal singularity is terminal
(see [26, Remark page 161]).

######  Remark 1.7.

Due to cohomology vanishings for toric surfaces, the germ of the moduli space
of polarised cscK surfaces at a toric point does not depend on the chosen cscK
polarisation. See however [43] for local wall-crossing type phenomena when the
polarisation is pushed away from the cscK locus of the Kähler cone.

######  Remark 1.8.

Theorem 1.5 provides another instance where canonical Kähler metrics, or
K-polystability, turns useful in moduli problems. We would like to mention two
other approaches to produce moduli spaces for (or around) toric varieties. In
[34], a notion of analytic stack is introduced to produce moduli spaces of
integrable complex structures modulo diffeomorphisms. Those spaces carry more
information, as they classify a wider class of varieties, but are typically
non-separated. In another direction, leaving the classical setting to the non-
commutative one, quantum toric varieties admit moduli spaces which are
orbifolds, see [24].

We proceed now to an overview of the proof of Theorem 1.5. If $(X,[\omega])$
is a cscK foldable surface, relying on [45, 12] and an observation in [41],
the deformation theory of $(X,[\omega])$ is entirely encoded by the set of
unobstructed polystable points in $H^{1}(X,TX)$ under the action of
$\mathrm{Aut}(X)$, the latter being reductive by Matsushima and Lichnerowicz’s
theorem [33, 32]. Nill’s study of toric varieties with reductive automorphism
groups [35] then shows that either $X$ is isomorphic to
$\mathbb{C}\mathbb{P}^{2}$ or
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$ in which case it is
rigid, or $\mathrm{Aut}^{0}(X)\simeq T$, where $T$ is the torus of $X$. Then,
by Ilten’s work [21], $H^{2}(X,TX)=0$, so that locally, the moduli space we
are after is given by (a neighborhood of the origin in) the GIT quotient
$H^{1}(X,TX)//\mathrm{Aut}(X)$, where $\mathrm{Aut}(X)\simeq T\rtimes G$.
Thanks again to [21], the deformation theory of toric surfaces is explicitly
described in terms of their fan, and the affine toric variety
$W:=H^{1}(X,TX)//T$ in Theorem 1.5 can be explicitly computed. The proof of
Theorem 1.5 then goes as follows. By a classification result (see Lemma 4.3),
$X$ can be obtained by successive $C_{p}$-equivariant blow-ups of some
“minimal” models in a list of $6$ toric surfaces. We then use the
combinatorial description of toric terminal singularities (see e.g. [10,
Section 11.4]) to prove the result for surfaces in that list. Then, using
convex geometry, we show that the desired properties for the local moduli
space are preserved after the $C_{p}$-equivariant blow-ups, and hold around
$[X]$.

######  Remark 1.9.

In Section 4.4, we produce an example of a toric surface $X$ such that
$\mathrm{Aut}(X)\simeq T$ and the quotient $H^{1}(X,TX)//T$ is not
$\mathbb{Q}$-Gorenstein. We do not know whether this surface carries a cscK
metric. If not, the expectation is that the theory should be extended to
smooth pairs $(X,D)$ where $D\subset X$ is a snc divisor, considering singular
or Poincaré type cscK metrics, such as in [1].

The paper is organised as follows. In Section 2, we settle the necessary
background material on cscK manifolds. Then, Section 3 is devoted to the
classification of foldable surfaces and the proofs of Proposition 1.2 and
Proposition 1.3. In Section 4, we carry over the construction of the local
moduli spaces, and prove Theorem 1.5. Finally, in Section 5.1, we provide maps
between the local moduli spaces and speculate about their Weil–Petersson
geometry, while in Section 5.2 we discuss the higher dimensional case.

### Notations

We will use the notations from [10]. For a toric variety $X$,
$\mathrm{Aut}(X)$ stands for its automorphism group, and
$\mathrm{Aut}^{0}(X)\subset\mathrm{Aut}(X)$ the connected component of the
identity. We denote $T$ the torus of $X$, $N$ the lattice of its one parameter
subgroups, with dual lattice $M$ and pairing $\langle m,u\rangle$, for
$(m,u)\in M\times N$. The fan of $X$ will be denoted $\Sigma$ (or
$\Sigma_{X}$) and letters $\tau,\sigma$ will be used for cones in $\Sigma$.
For $0\leq j\leq\mathrm{dim}(X)$, $\Sigma(j)$ is the set of $j$-dimensional
cones in $\Sigma$. For $\mathbb{K}\in\\{\mathbb{Q},\mathbb{R},\mathbb{C}\\}$,
we let $N_{\mathbb{K}}:=N\otimes_{\mathbb{Z}}\mathbb{K}$, and similarly
$M_{\mathbb{K}}=M\otimes_{\mathbb{Z}}\mathbb{K}$. Finally, $X_{\Sigma}$ may be
used to refer to the toric variety associated to $\Sigma$.

### Acknowledgments

The author would like to thank Ruadhaí Dervan and Cristiano Spotti for
answering his questions on moduli of cscK manifolds and for several helpful
comments, as well as Ronan Terpereau for stimulating discussions on the topic.
The author is partially supported by the grants MARGE ANR-21-CE40-0011 and
BRIDGES ANR–FAPESP ANR-21-CE40-0017.

##  2\. Background on cscK metrics and their moduli

Let $(X,\Omega)$ be a compact Kähler manifold with Kähler class $\Omega$. We
will give a very brief overview on extremal metrics and their deformations,
and refer the reader to [18, 46] for a more comprehensive treatment.

###  2.1. Extremal Kähler metrics

Extremal Kähler metrics were introduced by Calabi [6] and provide canonical
representatives of Kähler classes. They are defined as the critical points of
the so-called Calabi functional, that assign to each Kähler metric in $\Omega$
the $L^{2}$-norm of its scalar curvature. They include as special cases of
interest cscK metrics and thus Kähler–Einstein metrics. The following
obstruction to the existence of a cscK metric is due to Matsushima and
Lichnerowicz [33, 32].

######  Theorem 2.1 ([33, 32]).

Assume that $X$ carries a cscK metric. Then the automorphism group of $X$ is
reductive.

Later on, Futaki discovered another obstruction to the existence of a cscK
metric on $(X,\Omega)$ : the vanishing of the so-called Futaki invariant [16,
17]. Moreover, from [7], an extremal Kähler metric in the Kähler class
$\Omega$ is cscK precisely when the Futaki invariant $\mathrm{Fut}_{\Omega}$
vanishes. Together with Arezzo–Pacard–Singer and Székelyhidi’s results on
blow-ups of extremal Kähler metrics, we deduce the following existence result
for toric surfaces :

######  Theorem 2.2 ([3, 47]).

Let $(X,\Omega)$ be a smooth polarised toric surface with cscK metric
$\omega\in\Omega$. Let $Z\subset X$ be a set of torus fixed points, and
$G\subset\mathrm{Aut}(X)$ a finite subgroup such that :

  * (i)

The set $Z$ is $G$-invariant,

  * (ii)

The class $\Omega$ is $G$-invariant,

  * (iii)

The adjoint action of $G$ on $\mathrm{Lie}(\mathrm{Aut}(X))$ has no fixed
point but zero.

Then, there is $\varepsilon_{0}>0$ such that
$(\mathrm{Bl}_{Z}(X),\Omega_{\varepsilon})$ carries a cscK metric in the class
$\Omega_{\varepsilon}:=\pi^{*}\Omega-\varepsilon c_{1}(E)$ for
$\varepsilon\in(0,\varepsilon_{0})$, where $\pi:\mathrm{Bl}_{Z}(X)\to X$
stands for the blow-down map and $E=\sum_{z\in Z}E_{z}$ is the exceptional
locus of $\pi$.

###### Proof.

From [3, 47], there is $\varepsilon_{0}$ such that
$(\mathrm{Bl}_{Z}(X),\Omega_{\varepsilon})$ admits an extremal Kähler metric
for $\varepsilon\in(0,\varepsilon_{0})$. We only need to show the vanishing of
the Futaki invariant $\mathrm{Fut}_{\Omega_{\varepsilon}}$. This will follow
from its equivariance, as already used in [44, Proposition 2.1]. Denote
$\tilde{X}=\mathrm{Bl}_{Z}(X)$. By $(i)$ and $(ii)$, $G$ lifts to a subgroup
of $\mathrm{Aut}(\tilde{X})$, such that $\Omega_{\varepsilon}$ is
$G$-invariant. Then, by equivariance of
$\mathrm{Fut}_{\Omega_{\varepsilon}}\in(\mathrm{Lie}(\mathrm{Aut}(\tilde{X})))^{*}$
(see e.g. [17, Chapter 3] or [28, Section 3.1]), together with $(iii)$, we
deduce $\mathrm{Fut}_{\Omega_{\varepsilon}}=0$, which concludes the proof. ∎

###  2.2. Local moduli of polarised cscK manifolds

A moduli space $\mathscr{M}$ for polarised cscK manifolds has been constructed
in [12], generalising the results in [15] by taking care of non-discrete
automorphisms by mean of [45, 22]. This space is Hausdorff and endowed with a
complex space structure. It is a coarse moduli space in the following sense :
its points are in bijective correspondence with isomorphism classes of
polarised cscK manifolds and for any complex analytic family
$(\mathscr{X},\mathscr{L})\to\mathscr{S}$ of polarised cscK manifolds
($\mathscr{L}$ stands for the relative polarisation of
$\mathscr{X}\to\mathscr{S}$), there is an induced map
$\mathscr{S}_{\mathrm{red}}\to\mathscr{M}$ where $\mathscr{S}_{\mathrm{red}}$
is the reduction of $\mathscr{S}$. In what follows, we will restrict to the
connected components corresponding to the moduli space of polarised cscK
surfaces, still denoted by $\mathscr{M}$. Roughly, $\mathscr{M}$ is
constructed by patching together local moduli spaces
$(\mathscr{M}_{X})_{X\in\mathfrak{S}}$ parameterised by $\mathfrak{S}$, the
set of polarised cscK surfaces, each $\mathscr{M}_{X}$ being obtained as some
open set in an analytic GIT quotient $W_{X}//G$. Our case of interest is when
$X$ is a cscK toric surface. We then extract from [12] the following
proposition :

######  Proposition 2.3.

Let $(X,\Omega)$ be a polarised cscK toric surface. Then the local moduli
space $\mathscr{M}_{X}$ is given by an open neighborhood of $\pi(0)$ in the
GIT quotient

$\pi:H^{1}(X,TX)\to H^{1}(X,TX)//\mathrm{Aut}(X).$

In particular, its singularities are of klt type.

By GIT quotient in this affine setting, we mean the good categorical quotient

$H^{1}(X,TX)//\mathrm{Aut}(X):=\mathrm{Spec}(R^{\mathrm{Aut}(X)}),$

where $R$ stands for the ring of regular functions on $H^{1}(X,TX)$. We refer
to Section 4.3 for the definition of klt type singularities.

###### Proof.

From [12, Section 3], $\mathscr{M}_{X}$ is given by an open neighborhood of
$0$ in the analytic GIT quotient $W_{X}//G$, where $G$ stands for the
complexification of the isometry group of the cscK metric in $\Omega$, and
where $W_{X}\subset\tilde{H}^{1}$ is the smallest $G$-invariant Stein space
containing small integrable infinitesimal polarised deformations of
$(X,\Omega)$ (see [45] for the construction of the space $\tilde{H}^{1}$). As
noticed in [41, Lemma 2.10], by Bott–Steenbrink–Danilov’s vanishing of
$h^{0,1}(X)$ and $h^{0,2}(X)$ (see [10, Theorem 9.3.2]), the vector space
$\tilde{H}^{1}$ is $G$-equivariantly isomorphic to $H^{1}(X,TX)$. Also, for a
toric surface, $\mathrm{Aut}(X)$ is a linear algebraic group, so that
$G\simeq\mathrm{Aut}(X)$, as the Lie algebra of $\mathrm{Aut}(X)$ has no
parallel vector field (see [18, Chapter 2]). Finally, $H^{2}(X,TX)=0$ by [21,
Corollary 1.5], so that any small element of $H^{1}(X,TX)$ is integrable.
Altogether, we obtain the first part of Proposition 2.3. The statement about
klt type singularities follows directly from [5, Theorem 1]. ∎

##  3\. Foldable toric surfaces

Let $N$ be a rank two lattice and $\Sigma$ be a fan in $N_{\mathbb{R}}$, with
associated toric surface $X$. We will assume $\Sigma$ to be smooth, that is
each cone $\sigma\in\Sigma$ is generated by elements in $N$ that form part of
a $\mathbb{Z}$-basis, and to be complete, i.e.

$\bigcup_{\sigma\in\Sigma}\sigma=N_{\mathbb{R}}.$

Hence, $X$ is a smooth and compact toric surface.

###  3.1. Lattice automorphisms and fans

We will be interested in automorphism groups of smooth toric surfaces. Denote
by $T:=N\otimes_{\mathbb{Z}}\mathbb{C}^{*}$ the torus of $X$, and by
$\mathrm{Aut}(N,\Sigma)$ the group of lattice automorphisms of $\Sigma$.
Recall that $\mathrm{Aut}(N,\Sigma)$ is the subgroup of
$\mathrm{GL}_{\mathbb{Z}}(N)$ consisting in elements
$g\in\mathrm{GL}_{\mathbb{Z}}(N)$ such that the induced isomorphisms
$g\in\mathrm{GL}_{\mathbb{R}}(N_{\mathbb{R}})$ send bijectively $\Sigma$ to
$\Sigma$. By a result of Demazure [11, Proposition 11], the automorphism group
$\mathrm{Aut}(X)$ of $X$ is a linear algebraic group isomorphic to

$\mathrm{Aut}^{0}(X)\rtimes G,$

where $G$ is a quotient of $\mathrm{Aut}(N,\Sigma)$. We will later on be
interested in the GIT quotient of $H^{1}(X,TX)$ by $\mathrm{Aut}(X)$, assuming
$X$ to admit a cscK metric. A combination of Matsushima and Lichnerowicz’s
obstruction together with Nill’s work on toric varieties with reductive
automorphism group [35] implies :

######  Proposition 3.1.

Assume that $X$ admits a cscK metric. Then either

$X\in\\{\mathbb{P}^{2},\mathbb{P}^{1}\times\mathbb{P}^{1}\\},$

or

$\mathrm{Aut}(X)\simeq T\rtimes\mathrm{Aut}(N,\Sigma).$

###### Proof.

By Matsushima and Lichnerowicz theorem [33, 32], $\mathrm{Aut}(X)$ is
reductive. The result then follows from Nill’s result [35, Theorem 1.8]
together with Demazure’s structure theorem [11, Proposition 11]. ∎

As $\mathbb{P}^{2}$ and $\mathbb{P}^{1}\times\mathbb{P}^{1}$ are rigid, we
will focus now on the case

$\mathrm{Aut}(X)\simeq T\rtimes\mathrm{Aut}(N,\Sigma),$

and characterise the possible finite groups that arise as lattice automorphism
groups of rank two complete fans.

######  Lemma 3.2.

Let $g\in\mathrm{Aut}(N,\Sigma)$, and denote by $m\in\mathbb{N}$ the order of
$g$. Then

$m\in\\{1,2,3,4,6\\}.$

Moreover, the complex linear extension of $g$ to $N_{\mathbb{C}}$ is
conjugated (in $\mathrm{GL}(N_{\mathbb{C}})\simeq\mathrm{GL}_{2}(\mathbb{C})$)
to the following :

  1. (1)

If $m=1$, then $g\sim\mathrm{Id}$,

  2. (2)

If $m=2$ and $\det(g)=1$, then $g\sim-\mathrm{Id}$,

  3. (3)

If $m=2$ and $\det(g)=-1$, then $g\sim\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]$,

  4. (4)

If $m=3$, then $g\sim\left[\begin{array}[]{cc}j&0\\\
0&j^{2}\end{array}\right]$,

  5. (5)

If $m=4$, then $g\sim\left[\begin{array}[]{cc}i&0\\\ 0&-i\end{array}\right]$,

  6. (6)

If $m=6$, then $g\sim\left[\begin{array}[]{cc}-j&0\\\
0&-j^{2}\end{array}\right]$,

where we denoted by $j=e^{i\frac{2\pi}{3}}$ and by $\sim$ the equivalence
relation given by conjugation, and where we used an isomorphism
$N_{\mathbb{C}}\simeq\mathbb{C}^{2}$.

The proof is an elementary exercise in linear algebra. We include it for
convenience of the reader.

###### Proof.

Fix and isomorphism $N\simeq\mathbb{Z}^{2}$ and identify $g$ with an element
of $\mathrm{GL}_{2}(\mathbb{Z})$. The characteristic polynomial of

$g=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]$

then reads

$\chi(Y)=Y^{2}-(a+d)Y+(ad-bc).$

On the other hand, Jordan’s normal form for the $\mathbb{C}$-linear extension
of $g$ implies that it is conjugated (in $\mathrm{M}_{2}(\mathbb{C})$) to

$\left[\begin{array}[]{cc}\alpha&\delta\\\ 0&\beta\end{array}\right]$

for $(\alpha,\beta)\in\mathbb{C}^{2}$ and $\delta\in\\{0,1\\}$. As $g$ is both
invertible and of finite order, we deduce that $\delta=0$ and $g$ is
conjugated to

$\left[\begin{array}[]{cc}\alpha&0\\\ 0&\beta\end{array}\right].$

Moreover, $\alpha^{m}=\beta^{m}=1$, so that $\alpha$ and $\beta$ are $m$-th
roots of unity in $\mathbb{C}$. On the other hand, by conjugation invariance,
we also have

$\det(g)=\alpha\beta=ad-bc\in\\{-1,+1\\}$

as $g\in\mathrm{GL}_{2}(\mathbb{Z})$ and

$\mathrm{trace}(g)=\alpha+\beta=a+d\in\mathbb{Z}.$

If $\alpha$ and $\beta$ belong to $\\{-1,+1\\}$, that is if $m=2$, then we are
done. If not, at least one of $\alpha$ or $\beta$ is not real, and thus
$\alpha$ and $\beta$ must be complex conjugated roots of $\chi(Y)$. Hence

$\mathrm{trace}(g)=\alpha+\overline{\alpha}\in\mathbb{Z},$

but also

$|\alpha+\overline{\alpha}|<2$

so that

$\alpha+\overline{\alpha}\in\\{-1,0,1\\}.$

Hence

$\alpha\in\\{\pm i,\pm j,\pm j^{2}\\},$

from which the result follows easily. ∎

Recall that we denote by $C_{p}$ the cyclic group of order $p$ and by $D_{p}$
the dihedral group of order $p$. We then have :

######  Proposition 3.3.

The group $\mathrm{Aut}(N,\Sigma)$ is isomorphic to one of the groups in the
following set

$\\{C_{1},C_{2},C_{3},C_{4},C_{6},D_{1},D_{2},D_{3},D_{4},D_{6}\\}.$

###### Proof.

As $\mathrm{Aut}(N,\Sigma)$ is finite, we can fix an
$\mathrm{Aut}(N,\Sigma)$-invariant euclidean metric on $N_{\mathbb{R}}$ (again
considering the $\mathbb{R}$-linear extension of $\mathrm{Aut}(N,\Sigma)$).
Then, by the classification of finite subgroups of the group of orthogonal
transformations of the plane, we deduce that $\mathrm{Aut}(N,\Sigma)$ is
isomorphic to $C_{p}$ or $D_{p}$, for some $p\in\mathbb{N}^{*}$. As those
groups admit elements of order $p$, by Lemma 3.2, the result follows. ∎

######  Remark 3.4.

This proposition simply recovers the well known classification of
crystallographic groups in dimension $2$.

We will now show that all the above listed groups arise as lattice
automorphism groups of rank two complete smooth fans. Note that $C_{2}$ and
$D_{1}$ will be distinguished by the fact that their generator is
$-\mathrm{Id}$ in the first case and a reflection in the second case. Consider
then the following fans in $\mathbb{Z}^{2}$, where we denote by
$\\{e_{1},e_{2}\\}$ the standard basis of $\mathbb{Z}^{2}$.

######  Example 3.5.

Let $\mathbb{F}_{n}$ be the $n$-th Hirzebruch surface, that is the total space
of the fibration

$\mathbb{P}(\mathscr{O}_{\mathbb{P}^{1}}\oplus\mathscr{O}_{\mathbb{P}^{1}}(n))\rightarrow\mathbb{C}\mathbb{P}^{1}.$

Note that
$\mathbb{F}_{0}=\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$. Then,
up to isomorphism, the fan $\Sigma_{\mathbb{F}_{n}}$ of $\mathbb{F}_{n}$ is
described by

$\Sigma_{\mathbb{F}_{n}}(1)=\\{\mathbb{R}_{+}\cdot(0,-1),\>\mathbb{R}_{+}\cdot(1,0),\>\mathbb{R}_{+}\cdot(0,1),\>\mathbb{R}_{+}\cdot(-1,-n)\\}.$

As an example, $\mathbb{F}_{2}$ admits the fan description of Figure 1, that
we will denote by $\Sigma_{1}^{\prime}$. In this figure, and the one that
follow, the dots represent the lattice points, and we only represent the ray
generators of the complete two dimensional fan.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{2}}$$\scriptstyle{e_{1}}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 1. Fan $\Sigma_{1}^{\prime}$ of $\mathbb{F}_{2}$

######  Example 3.6.

Here are the fans $\Sigma_{\mathbb{P}^{2}}=\Sigma_{3}^{\prime}$ and
$\Sigma_{\mathbb{P}^{1}\times\mathbb{P}^{1}}=\Sigma_{4}^{\prime}$ of
$\mathbb{P}^{2}$ and $\mathbb{P}^{1}\times\mathbb{P}^{1}$ respectively (the
numbering of the fans is motivated by Proposition 3.8 below).

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{2}}$$\scriptstyle{e_{1}}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 2. Fan $\Sigma_{3}^{\prime}$ of $\mathbb{P}_{2}$
$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{2}}$$\scriptstyle{e_{1}}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 3. Fan $\Sigma_{4}^{\prime}$ of
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$

######  Example 3.7.

Recall that blowing-up a smooth toric surface along a torus fixed point
produces a new smooth toric surface. By the orbit cone correspondence, such a
fixed point corresponds to a two dimensional cone

$\sigma=\mathbb{R}_{+}\cdot e_{i}+\mathbb{R}_{+}\cdot e_{i+1}$

in the fan of the blown-up surface, and the fan of the resulting surface is
obtained by adding the ray generated by $e_{i}+e_{i+1}$ to the set of rays of
the initial surface (see [10, Chapter 3, Section 3]). By iterated blow-ups, we
obtain the following fans.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 4. Fan $\Sigma_{2}^{\prime}$ : iterated blow-up of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$

The following is a two points blow-up of $\mathbb{P}^{1}\times\mathbb{P}^{1}$
and at the same time a three points blow-up of $\mathbb{C}\mathbb{P}^{2}$.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 5. Fan $\Sigma_{6}^{\prime}$ : blow-up of $\mathbb{P}^{2}$ along its
three fixed points.

Here are further examples to complete our classification of lattice
automorphism groups of rank two complete smooth fans. The first one is a
single blow-up of $\mathbb{F}_{2}$, that cancels the symmetry of
$\Sigma_{1}^{\prime}=\Sigma_{\mathbb{F}_{2}}$.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{2}}$$\scriptstyle{e_{1}}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 6. Fan $\Sigma_{1}$ : one-point blow-up of $\mathbb{F}_{2}$

The next one is obtained from $\Sigma_{2}^{\prime}$ by blowing-up in a
$C_{2}$-equivariant way (recall that $C_{2}=\langle-\mathrm{Id}\rangle$).

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 7. Fan $\Sigma_{2}$.

The third one is obtained from the fan of $\mathbb{P}^{2}$ by three successive
blow-ups of three distinct fixed points, in a symmetric way.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 8. Fan $\Sigma_{3}$

The next one is obtained by two successive blow-ups of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ at four fixed points.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 9. Fan $\Sigma_{4}$

The last one is obtained from $\Sigma_{6}^{\prime}$ by two successive blow-ups
at six fixed points.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 10. Fan $\Sigma_{6}$.

######  Proposition 3.8.

Let $p\in\\{1,2,3,4,6\\}$. Then we have

$\mathrm{Aut}(\mathbb{Z}^{2},\Sigma_{p})\simeq C_{p}$

and

$\mathrm{Aut}(\mathbb{Z}^{2},\Sigma_{p}^{\prime})\simeq D_{p}.$

###### Proof.

The proof is straightforward, although a bit tedious, so we will only give a
sketch of it. Fix an orientation of $\mathbb{R}^{2}$, and pick a two
dimensional cone $\sigma$ of $\Sigma_{p}$ (the proof goes the same for
$\Sigma_{p}^{\prime}$). Let $g\in\mathrm{Aut}(\mathbb{Z}^{2},\Sigma_{p})$.
Then $g\cdot\sigma\in\Sigma_{p}(2)$. Moreover, $g$ maps the two facets of
$\sigma$ to the two facets of $g\cdot\sigma$. As the fan $\Sigma_{p}$ is
explicitly described, for each possible $\tau=g\cdot\sigma\in\Sigma_{p}(2)$,
we deduce the possible matrix coefficients for $g$ :

$g=\left[\begin{array}[]{cc}a_{\tau}&b_{\tau}\\\
c_{\tau}&d_{\tau}\end{array}\right]\in\mathrm{GL}_{2}(\mathbb{Z}).$

For any possible $\tau$, one can then explicitly compute the images of the
rays of $\Sigma_{p}$ by $g$. They should all belong to $\Sigma_{p}(1)$, and
this enables to select the allowed $\tau$’s for $g$ to be a lattice
automorphism of $\Sigma_{p}$. The result then follows easily. ∎

######  Remark 3.9.

One can also show similarly that for any Hirzebruch surface $\mathbb{F}_{n}$
with $n\geq 1$, and with fan $\Sigma_{\mathbb{F}_{n}}$, one has
$\mathrm{Aut}(N,\Sigma_{\mathbb{F}_{n}})\simeq D_{1}$.

###  3.2. Classification of foldable surfaces

We first recall the definition :

######  Definition 3.10.

We will say that a smooth and complete two dimensional fan is foldable if its
lattice automorphism group contains a non-trivial cyclic group. A foldable
surface is a toric surface whose fan is foldable.

We now proceed to a partial classification of foldable toric surfaces. Notice
that if $G\subset\mathrm{Aut}(N,\Sigma)$, then we can find a $G$-action on
$X$, faithfull as soon as $\mathrm{Aut}(X)\simeq
T\rtimes\mathrm{Aut}(N,\Sigma)$.

######  Definition 3.11.

A $G$-equivariant blow-up of $X$ is a blow-up of $X$ along a $G$-orbit of
torus fixed points. We will say that a smooth toric surface $\tilde{X}$ is
obtained from $X$ by successive $G$-equivariant blow-ups if there is a
sequence of $G$-equivariant blow-ups :

$\tilde{X}=X_{k}\to X_{k-1}\to\ldots\to X_{1}\to X_{0}=X.$

Denote by $\mathrm{Bl}_{p_{1},p_{2},p_{3}}(\mathbb{P}^{2})$ the toric surface
associated to the fan of Figure $5$, that is the blow-up of $\mathbb{P}^{2}$
at its three torus fixed points.

######  Proposition 3.12.

Let $X$ be a foldable surface with fan $\Sigma$. Assume that
$\mathrm{Aut}(N,\Sigma)$ contains a subgroup isomorphic to $C_{p}$, $p\geq 2$.

  * •

If $p\in\\{2,4\\}$, then $X$ is obtained from
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ by successive $C_{p}$-equivariant blow-
ups,

  * •

If $p=3$, then $X$ is obtained from $\mathbb{P}^{2}$ by successive
$C_{p}$-equivariant blow-ups,

  * •

If $p=6$, then $X$ is obtained from
$\mathrm{Bl}_{p_{1},p_{2},p_{3}}(\mathbb{P}^{2})$ by successive
$C_{p}$-equivariant blow-ups.

###### Proof.

Through its representation

$C_{p}\to\mathrm{GL}_{\mathbb{R}}(N_{\mathbb{R}}),$

the subgroup $C_{p}$ of $\mathrm{Aut}(N,\Sigma)$ acts on $N_{\mathbb{R}}$ by
rotations (once an invariant euclidean metric is fixed), hence has no fixed
points. We deduce that the action of $C_{p}$ on $\Sigma(1)$ is free, and then
$p$ divides $|\Sigma(1)|$, the number of rays of $\Sigma$. If $|\Sigma(1)|=p$,
then from the classification of toric surfaces (see e.g. [10, Chapter 10]) we
must have

$p\in\\{3,4,6\\}.$

In that situation, if $p=3$, then $X\simeq\mathbb{P}^{2}$. If $p=4$, then
$X\simeq\mathbb{F}_{n}$ for some $n\in\mathbb{N}$. By Remark 3.9, we see that
$X\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$. We will see shortly that if
$p=6$, then $X\simeq\mathrm{Bl}_{p_{1},p_{2},p_{3}}(\mathbb{P}^{2})$.

So we may assume that $|\Sigma(1)|\geq 5$, and that $p$ divides $|\Sigma(1)|$.
By the classification of toric surfaces, we know that $X$ is obtained by
successive blow-ups from $\mathbb{F}_{n}$, $n\geq 0$, or from
$\mathbb{P}^{2}$. Then, there is a ray $\rho_{i}$ in $\Sigma(1)$ generated by
the sum of two primitive elements

$u_{i}=u_{i-1}+u_{i+1}$

generating adjacent rays $\rho_{i-1}$ and $\rho_{i+1}$ in $\Sigma(1)$ (that is
both $\rho_{i}+\rho_{i-1}$ and $\rho_{i}+\rho_{i+1}$ belong to $\sigma(2)$).
By linearity, for any $g\in C_{p}$ we have

$g\cdot u_{i}=g\cdot u_{i-1}+g\cdot u_{i+1},$

hence we can contract the $C_{p}$-orbit of the torus invariant divisor
$D_{\rho_{i}}$ associated to $\rho_{i}$ and obtain a smooth toric surface
whose fan still admits $C_{p}$ as a subgroup of its lattice symmetry group. By
induction, we obtain the result. ∎

From the classification, we obtain the following proposition.

######  Proposition 3.13.

Let $X$ be a foldable surface. Then $X$ admits a cscK metric.

###### Proof.

This is simply an induction on the number of $C_{p}$-equivariant blow-ups in
Proposition 3.12, using Theorem 2.2 at each step, and the fact that
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, $\mathbb{P}^{2}$ and
$\mathrm{Bl}_{p_{1},p_{2},p_{3}}(\mathbb{P}^{2})$ admit cscK metrics in
$C_{p}$-equivariant classes (for
$\mathrm{Bl}_{p_{1},p_{2},p_{3}}(\mathbb{P}^{2})$, it follows again from
Theorem 2.2 applied to $\mathbb{P}^{2}$). ∎

##  4\. The local cscK moduli around a foldable surface

Let $X$ be a smooth and complete toric surface with fan $\Sigma$ and one
parameter subgroups lattice $N$. We will denote by $(u_{i})_{0\leq i\leq r}$
the primitive elements of $N$ that generate the rays

$\rho_{i}=\mathbb{R}_{+}\cdot u_{i}\in\Sigma(1),$

labeled in the counterclockwise order, so that for each $i$, $(u_{i},u_{i+1})$
is a positively oriented $\mathbb{Z}$-basis of $N$, and
$\rho_{i}+\rho_{i+1}\in\Sigma(2)$, with the convention $u_{0}=u_{r}$.

###  4.1. Deformation theory of toric surfaces

The action of the torus $T$ on $X$ naturally induces a representation of $T$
on

$V:=H^{1}(X,TX).$

We will denote by

$V=\bigoplus_{m\in M}V_{m}$

the associated weight space decomposition. From [21, Corollary 1.5], for each
weight $m\in M=\mathrm{Hom}_{\mathbb{Z}}(N,\mathbb{Z})$ :

(1)
$\mathrm{dim}(V_{m})=\sharp\displaystyle\left\\{\rho_{i}\in\Sigma(1)\>\left|\begin{array}[]{c}\langle
m,u_{i}\rangle=-1\\\ \langle m,u_{i\pm
1}\rangle<0\end{array}\right.\right\\}.$

######  Example 4.1.

We will provide explicit examples of weight space decompositions for
$V=H^{1}(X,TX)$, when $X$ is the smooth toric surface associated to a foldable
fan. First, $\mathbb{P}^{1}\times\mathbb{P}^{1}$, $\mathbb{P}^{2}$ and
$\mathrm{Bl}_{p_{1},p_{2},p_{3}}(\mathbb{P}^{2})$ are rigid, as can be checked
directly using Formula (1). In those cases, $V=0$. Then, denote by $Y_{2}$ the
foldable toric surface associated to the fan $\Sigma_{2}^{\prime}$ of Figure
4. We also introduce the foldable toric surfaces $Y_{4}$ and $Y_{3}$
associated to the following fans :

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 11. Fan of $Y_{4}$.
$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 12. Fan of $Y_{3}$

Set $V_{i}=H^{1}(Y_{i},TY_{i})$. Testing the conditions in Formula (1) for
each ray generator gives

(2) $\begin{array}[]{ccc}V_{2}&=&V^{1}_{+e_{2}^{*}}\oplus
V^{1}_{-e_{2}^{*}}\oplus V^{1}_{-e_{1}^{*}+e_{2}^{*}}\oplus
V^{1}_{e_{1}^{*}-e_{2}^{*}}\\\ &&\\\ V_{3}&=&V^{2}_{-e_{2}^{*}}\oplus
V^{2}_{e_{1}^{*}}\oplus V^{2}_{-e_{1}^{*}+e_{2}^{*}}\\\ &&\\\
V_{4}&=&V^{1}_{+e_{1}^{*}}\oplus V^{1}_{-e_{1}^{*}}\oplus
V^{1}_{+e_{2}^{*}}\oplus V^{1}_{-e_{2}^{*}}\\\ &&\end{array}$

where $V^{j}_{m}$ stands for a $j$-dimensional weight $m$ representation of
$T$.

######  Remark 4.2.

One can check by unraveling the isomorphisms used in [21, Section 1] that the
$C_{3}$-action on $Y_{3}$ induces an action on the coordinates $\chi_{j}^{m}$
of $V_{3}$, with two orbits given by

$\\{\chi_{j}^{-e_{2}^{*}},\chi_{j}^{e_{1}^{*}},\chi_{j}^{-e_{1}^{*}+e_{2}^{*}}\\},\>j\in\\{1,2\\}$

where $\chi_{j}^{m}$, for $1\leq j\leq\mathrm{dim}(V_{m})$, stand for
generators of the weight $m$ space $V_{m}$. Similarly, the $D_{2}$ (resp.
$D_{4}$) action on $Y_{2}$ (resp. $Y_{4}$) induces a transitive action on the
coordinates of $V_{2}$ (resp. $V_{4}$).

###  4.2. The toric GIT quotient

We will assume from now on that $X$ carries a cscK metric, so by Proposition
3.1, we have

$\mathrm{Aut}(X)\simeq T\rtimes\mathrm{Aut}(N,\Sigma).$

We will set

$G:=\mathrm{Aut}(N,\Sigma).$

From Proposition 2.3, the local moduli space $\mathscr{M}_{X}$ of polarised
cscK surfaces around $[X]\in\mathscr{M}$ is given by a neighborhood of the
origin in the GIT quotient of $V$ by $T\rtimes G$. Assume now that $X$ is
foldable. To prove Theorem 1.5, it is then enough to show that the GIT (or
categorical) quotient

$W:=V//T$

is toric, Gorenstein and terminal. We will end this section by showing that
$W$ is indeed an affine toric variety. We first need the following lemma,
where by $C_{p}\subset G$ we mean that there is an injection of $C_{p}$ in
$G$.

######  Lemma 4.3.

One of the following holds :

  * •

If $X$ is rigid, then
$X\in\\{\mathbb{P}^{2},\mathbb{P}^{1}\times\mathbb{P}^{1},\mathrm{Bl}_{p_{1},p_{2},p_{3}}(\mathbb{P}^{2})\\}$.

  * •

If $X$ is not rigid and $C_{2}\subset G$, $X\in\\{Y_{2},Y_{4}\\}$ or is
obtained from $Y_{2}$ or $Y_{4}$ by successive $C_{2}$-equivariant blow-ups.

  * •

If $X$ is not rigid and $C_{3}\subset G$, $X=Y_{3}$ or is obtained from
$Y_{3}$ by successive $C_{3}$-equivariant blow-ups.

Recall that $Y_{2}$, $Y_{3}$ and $Y_{4}$ were defined in Example 4.1.

###### Proof.

Note that by Example 4.1, $\mathbb{P}^{2}$,
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathrm{Bl}_{p_{1},p_{2},p_{3}}(\mathbb{P}^{2})$ are rigid. If $X$ is not one
of those three foldable surfaces, then from Proposition 3.12, either it
belongs to $\\{Y_{2},Y_{3},Y_{4}\\}$, or is obtained from $Y_{j}$ by
successive $C_{p}$-equivariant blow-ups, where $p=2$ for $j\in\\{2,4\\}$ and
$p=3$ for $j=3$. From [21, Corollary 1.6], $H^{1}(Y_{j},TY_{j})$ injects in
$H^{1}(X,TX)$. Hence from the computations of Example 4.1, we see that $X$ is
not rigid, and the result follows. ∎

######  Proposition 4.4.

The GIT quotient $W$ inherits the structure of a normal toric affine variety.

Although the proof is straightforward, we should warn the reader that our
setting is slightly different from the standard quotient construction of toric
varieties (see e.g. [10, Chapter 5]), as the weight spaces $V_{m}$ may have
dimension greater than $1$, and the quotient map $\tilde{\sigma}\to\sigma$
(see the proof below) may send several rays to the same one. This detailed
proof will also settle the necessary notations for the following section.

###### Proof.

If $V=0$, then there is nothing to prove. We then assume that $X$ is not
rigid. From Lemma 4.3, $X$ is obtained from $Y_{j}$ by successive
$C_{p}$-equivariant blow-ups, where $j\in\\{2,3,4\\}$, for suitable
$p\in\\{2,3\\}$.

Denote by $d=\mathrm{dim}(V)$ and by $\tilde{N}\simeq\mathbb{Z}^{d}$ the
lattice of one parameter subgroups of $\tilde{T}\simeq(\mathbb{C}^{*})^{d}$
acting on $V\simeq\mathbb{C}^{d}$ by multiplication on each coordinate :

$\forall(t,x)\in\tilde{T}\times V,\>t\cdot x=(t_{i}x_{i})_{1\leq i\leq d}.$

The $T$-module structure of $V$ is then equivalent to the injection
$T\to\tilde{T}$, or the lattice monomorphism

$B:N\to\tilde{N}$

defined by

$B(u)=(\langle m_{i},u\rangle)_{1\leq i\leq d}$

where the $m_{i}$’s run through all the weights in the weight space
decomposition of $V$ (with multiplicities when $\mathrm{dim}(V_{m})\geq 2$).
The fact that $B$ is injective comes again from the injection $V_{j}\subset V$
(cf [21, Corollary 1.6]) and the explicit computation of $V_{j}$ in Example
4.1. We then consider the quotient

$N^{\prime}:=\tilde{N}/N,$

where we identify $N$ with $B(N)$ by abuse of notation. We claim that
$N^{\prime}$ is again a lattice, that is $N$ is saturated in $N^{\prime}$. If
$X=Y_{j}$, for $j\in\\{2,3,4\\}$, then this can be checked directly using the
description of $V_{j}$ in Example 4.1. If $X$ is a blow-up of $Y_{j}$, then
$V_{j}\subset V$, so that $B$ can be written $B=(B_{j},B_{+})$, where

$B_{j}:N\to\tilde{N}_{j}$

is the injection corresponding to the weight space decomposition of $V_{j}$,
with

$\tilde{N}_{j}\simeq\mathbb{Z}^{d_{j}},$

for $d_{j}=\mathrm{dim}(V_{j})$. Then, the fact that $B_{j}(N)$ is saturated
in $\tilde{N}_{j}\subset\tilde{N}$ implies that $B(N)$ is saturated in
$\tilde{N}$. Thus, we have a short exact sequence of lattices

(3) $0\longrightarrow N\stackrel{{\scriptstyle
B}}{{\longrightarrow}}\tilde{N}\stackrel{{\scriptstyle
A}}{{\longrightarrow}}N^{\prime}\longrightarrow 0$

where

$A:\tilde{N}\to\tilde{N}/N$

denote the quotient map.

Introduce $(\tilde{e}_{i})_{1\leq i\leq d}$ the basis of $V$ dual to the
coordinates $\chi^{m_{i}}$ of the weight spaces $V_{m_{i}}$. It corresponds to
a $\mathbb{Z}$-basis of $\tilde{N}$, still denoted $(\tilde{e}_{i})_{1\leq
i\leq d}$. The cone

$\tilde{\sigma}=\sum_{i=1}^{d}\mathbb{R}_{+}\cdot\tilde{e}_{i}\subset\tilde{N}_{\mathbb{R}}$

satisfies

$\mathrm{Spec}(\mathbb{C}[\tilde{\sigma}^{\vee}\cap\tilde{M}])=V,$

where

$\tilde{M}=\mathrm{Hom}_{\mathbb{Z}}(\tilde{N},\mathbb{Z}).$

Consider the cone of $N^{\prime}_{\mathbb{R}}$ :

$\sigma=A(\tilde{\sigma})=\sum_{j=i}^{d}\mathbb{R}_{+}\cdot A(\tilde{e}_{i})$

(we will still denote by $A$ and $B$ their $\mathbb{R}$-linear extensions).
Our goal is then to show that $W$ is isomorphic to the affine toric variety
defined by $\sigma$. We first need to prove that $\sigma$ has the required
properties to define such a variety. The cone $\sigma$ clearly is rational,
convex and polyhedral. We now prove that $\sigma$ is strictly convex, that is

$-\sigma\cap\sigma=\\{0\\}.$

So let $v\in-\sigma\cap\sigma$. Then there is
$(u_{+},u_{-})\in(\tilde{\sigma})^{2}$ such that $A(u_{\pm})=\pm v$ . We
deduce that

$u_{+}+u_{-}\in\ker(A)\cap\tilde{\sigma}=\mathrm{Im}(B)\cap\tilde{\sigma}.$

We will then show that

(4) $\mathrm{Im}(B)\cap\tilde{\sigma}=\\{0\\},$

which implies $u_{+}=u_{-}=0$ by strict convexity of $\tilde{\sigma}$. Notice
that it is enough to show (4) for $X=Y_{j}$. Indeed, using the decomposition
$B=(B_{j},B_{+})$ as before, if

$u=B(x)\in\tilde{\sigma},$

then

$u_{j}=B_{j}(x)\in\tilde{\sigma}_{j},$

where $\tilde{\sigma}_{j}\subset(\tilde{N}_{j})_{\mathbb{R}}$ is the cone
corresponding to $V_{j}$. If

(5) $\mathrm{Im}(B_{j})\cap\tilde{\sigma}_{j}=\\{0\\},$

by injectivity of $B_{j}$, we deduce $x=0$ and then $u=0$. Remains to prove
(5), which follows again from the explicit descrition of the weights of the
$T$ action on $V_{j}$. For example, if $j=2$, and if

$B_{2}(x_{1},x_{2})=(x_{2},-x_{2},-x_{1}+x_{2},x_{1}-x_{2})\in\tilde{\sigma}_{2},$

then by definition of $\tilde{\sigma}_{2}$ we deduce

$\left\\{\begin{array}[]{ccc}x_{2}&\geq&0\\\ -x_{2}&\geq&0\\\
-x_{1}+x_{2}&\geq&0\\\ x_{1}-x_{2}&\geq&0\end{array}\right.$

and thus $x_{1}=x_{2}=0$. The cases $j\in\\{3,4\\}$ are similar. We just
proved that $\sigma$ is a strongly convex rational polyhedral cone, and thus
defines an affine toric variety.

We now claim that

$W\simeq\mathrm{Spec}(\mathbb{C}[\sigma^{\vee}\cap M^{\prime}])$

with

$M^{\prime}=\mathrm{Hom}_{\mathbb{Z}}(N^{\prime},\mathbb{Z}).$

It is equivalent to show

$\mathbb{C}[\sigma^{\vee}\cap M^{\prime}]\simeq\mathbb{C}[\chi^{m_{i}}]^{T}.$

The latter equality follows easily from the definitions. Indeed, considering
the dual sequence to (3),

(6) $0\longrightarrow M^{\prime}\stackrel{{\scriptstyle
A^{*}}}{{\longrightarrow}}\tilde{M}\stackrel{{\scriptstyle
B^{*}}}{{\longrightarrow}}M\longrightarrow 0$

for $m^{\prime}\in M^{\prime}$, if $\tilde{m}=A^{*}(m^{\prime})$, then

$\begin{array}[]{ccc}\chi^{m^{\prime}}\in\sigma^{\vee}&\Longleftrightarrow&\forall
i\in[\\![1,d]\\!],\>\langle m^{\prime},A(\tilde{e}_{i})\rangle\geq 0\\\ &&\\\
&\Longleftrightarrow&\forall i\in[\\![1,d]\\!],\>\langle
A^{*}(m^{\prime}),\tilde{e}_{i}\rangle\geq 0\\\ &&\\\
&\Longleftrightarrow&\left\\{\begin{array}[]{c}B^{*}(\tilde{m})=0\\\ \forall
i\in[\\![1,d]\\!],\>\langle\tilde{m},\tilde{e}_{i}\rangle\geq
0\end{array}\right.\\\ &&\\\
&\Longleftrightarrow&\chi^{\tilde{m}}\in\mathbb{C}[\chi^{m_{i}}]^{T}.\end{array}$

This proves the claim, and the result follows. ∎

###  4.3. Singularities

We keep the notations from the previous section (see in particular the proof
of Proposition 4.4). Our goal here is to conclude the proof of Theorem 1.5, by
showing that $W$ is Gorenstein and terminal. Let us first recall the
definitions of those notions, and their combinatorial characterisation in the
toric case.

######  Definition 4.5.

Let $Z$ be a normal toric variety111Being normal is not required in the most
general definition of Gorenstein singularities. However, we will only deal
with normal toric varieties, that are rational [10, Theorem 11.4.2], hence
Cohen–Macaulay, which is usually required to define Gorenstein singularities..
Then $Z$ is Gorenstein if the canonical divisor $K_{Z}$ is Cartier
($\mathbb{Q}$-Gorenstein if $K_{Z}$ is $\mathbb{Q}$-Cartier). In that case,
$Z$ has terminal singularities if there is a resolution of singularities
$\pi:\tilde{Z}\to Z$ such that if we set

$K_{\tilde{Z}}=\pi^{*}K_{Z}+\sum_{i}a_{i}E_{i},$

where the $E_{i}$’s are distinct irreducible divisors, then for all $i$, we
have $a_{i}>0$.

One can check that the above definition doesn’t depend on the choice of
resolution. Terminal singularities play an important role in the minimal model
program, being the singularities of minimal models. Their logarithmic version
turned out very useful as well. Recall that a log pair is a normal variety $Z$
together with an effective $\mathbb{Q}$-divisor $D$ with coefficients in
$[0,1]\cap\mathbb{Q}$. A log resolution for a log pair $(Z,D)$ is a resolution
of singularities $\pi:\tilde{Z}\to Z$ such that the exceptional locus
$\mathrm{Exc}(\pi)$ of $\pi$ is of pure codimension $1$ and the divisor
$\pi_{*}^{-1}D+\mathrm{Exc}(\pi)$ has simple normal crossing.

######  Definition 4.6.

Let $(Z,D)$ be a log pair such that $K_{Z}+D$ is $\mathbb{Q}$-Cartier (the
pair is then called $\mathbb{Q}$-Gorenstein). In that case, $(Z,D)$ has klt
singularities if there is a log-resolution $\pi:\tilde{Z}\to Z$ such that if
we set

$K_{\tilde{Z}}=\pi^{*}(K_{Z}+D)+\sum_{i}a_{i}E_{i},$

where the $E_{i}$’s are distinct irreducible divisors, then for all $i$, we
have $a_{i}>-1$. Following [5], we will say that a normal variety $Z$ is of
klt type if there exists an effective $\mathbb{Q}$-divisor $D$ such that
$(Z,D)$ is a $\mathbb{Q}$-Gorenstein log pair with klt singularities.

At that stage, from [5, Theorem 1], we know that $W$ is of klt type. Also, by
[10, Corollary 11.4.25 ], if $W$ is Gorenstein, then it has log terminal
singularities, meaning that the pair $(W,0)$ is klt. We will give a direct
proof that it is actually Gorenstein with terminal singularities, using the
following characterisation from [10, Proposition 11.4.12] (where we use the
notation $u_{\rho}$ for the primitive generator of a ray $\rho$ in
$\sigma(1)$).

######  Proposition 4.7.

Consider $W=\mathrm{Spec}(\mathbb{C}[\sigma^{\vee}\cap M^{\prime}])$ the
affine toric variety associated to the rational strictly convex polyhedral
cone $\sigma$. Then

  * •

$W$ is Gorenstein if and only if there exists $m\in M^{\prime}$ such that for
all $\rho\in\sigma(1)$, $\langle m,u_{\rho}\rangle=1$.

  * •

In that case, $W$ has terminal singularities if and only if the only lattice
points in

$\Pi_{\sigma}:=\left\\{\sum_{\rho\in\sigma(1)}\lambda_{\rho}u_{\rho}\>|\>\sum_{\rho\in\sigma(1)}\lambda_{\rho}\leq
1,\>0\leq\lambda_{\rho}\leq 1\right\\}$

are given by its vertices.

From the discussion in Section 4.2, the following proposition concludes the
proof of Theorem 1.5.

######  Proposition 4.8.

The affine toric variety $W$ is Gorenstein and has terminal singularities.

###### Proof.

We will exclude the rigid cases, and as in the proof of Proposition 4.4,
assume that $X$ is obtained from $Y_{j}$ by successive $C_{p}$-equivariant
blow-ups, where $j\in\\{2,3,4\\}$, for suitable $p\in\\{2,3\\}$.

We use the characterisation in Proposition 4.7, and first need to describe the
rays of $\sigma$ and their primitive generators. Note that by construction,
the rays of $\sigma$ belong to the set $\\{\mathbb{R}_{+}\cdot
A(\tilde{e}_{i}),\>1\leq i\leq d\\}$. Let $\rho_{i}=\mathbb{R}_{+}\cdot
A(\tilde{e}_{i})$ be such a ray. We claim that $A(\tilde{e}_{i})$ is primitive
in $N^{\prime}$. The argument is similar to the one used to prove strict
convexity in the proof of Proposition 4.4. Suppose by contradiction that there
is $a\in\mathbb{N}$, $a\geq 2$, and $\tilde{e}\in\tilde{\sigma}\cap\tilde{N}$
such that $A(\tilde{e}_{i})=aA(\tilde{e})$. Notice that there is $x\in N$ such
that

(7) $B(x)=(B_{j}(x),B_{+}(x))=\tilde{e}_{i}-a\tilde{e}.$

By injectivity of $B_{j}$, if $B_{j}(x)=0$, then $x=0$ hence
$\tilde{e}_{i}=a\tilde{e}$. This is absurd as $\tilde{e}_{i}$ is primitive. So
we may assume $B_{j}(x)\neq 0$. Similarly, using Equation (5), we may assume
as well that $B_{j}(x)\notin-\tilde{\sigma}_{j}$ and
$B_{j}(x)\notin\tilde{\sigma}_{j}$. Hence $B_{j}(x)$ must have at least one
positive and one negative coordinate in the basis $(\tilde{e}_{k})_{1\leq
k\leq d_{j}}$. As $\tilde{e}\in\tilde{\sigma}$, we can write

$a\tilde{e}=\sum_{k=1}^{d}a_{k}\tilde{e}_{k}$

with $a_{k}=0$ or $a_{k}\geq 2$. Then, from Equation (7), $B_{j}(x)$ has
exactly one coordinate equal to $1$, while its other non-zero coordinates are
all less or equal to $-2$. A case by case analysis using the description of
$V_{j}$ in Equation (2) (cf Example 4.1) shows that this is impossible (for
$V_{3}$, use the fact that the weight spaces are $2$-dimensional).

We will then prove that there is $m^{\prime}\in M^{\prime}$ such that

$\forall i\in[\\![1,d]\\!]\>,\>\langle m^{\prime},A(\tilde{e}_{i})\rangle=1,$

which implies that $W$ is Gorenstein. So let $m^{\prime}\in M^{\prime}$. Set

$\tilde{m}=A^{*}(m^{\prime})\in\ker(B^{*}).$

Then,

$\begin{array}[]{ccc}\forall i\in[\\![1,d]\\!]\>,\>\langle
m^{\prime},A(\tilde{e}_{i})\rangle=1&\Longleftrightarrow&\forall
i\in[\\![1,d]\\!]\>,\>\langle\tilde{m},\tilde{e}_{i}\rangle=1\\\ &&\\\
&\Longleftrightarrow&\tilde{m}=(1,\ldots,1)\end{array}$

where we used the basis $(\tilde{e}_{i})$ to produce the coordinates of
$\tilde{m}$. Hence, it is equivalent to show that

$(1,\ldots,1)\in\ker(B^{*}),$

which by definition of $B$ is equivalent to

$\sum_{i=1}^{d}m_{i}=0\in M$

where $m_{i}$’s are the weights describing the $T$-action on $V$. This is
where we use that $X$ is foldable. The $G$-action on $N$ naturally induces a
$G$-action on $M$, and on the weights $(m_{i})$’s. As $G$ contains a non-
trivial cyclic group, there is an element $g\in\mathrm{Aut}(N)$ of order $p$
with no fixed point but $0$. This element generates a cyclic group that acts
freely on $\\{m_{i},\>1\leq i\leq d\\}$. Hence

$\sum_{i=1}^{d}m_{i}=\sum_{i^{\prime}}\sum_{k=0}^{p-1}g^{k}\cdot
m_{i}^{\prime}$

where we picked a single element $m_{i}^{\prime}$ in each orbit under this
action. As $g\neq\mathrm{Id}$, we have

$\mathrm{Id}+g+\cdots+g^{p-1}=0$

so that for each orbit

$\sum_{k=0}^{p-1}g^{k}\cdot m_{i}^{\prime}=0.$

We then deduce the existence of the required $m^{\prime}$, and that $W$ is
Gorenstein.

We proceed to the proof of the fact that $W$ has terminal singularities. Let

$\Pi_{\sigma}:=\left\\{\sum_{\rho\in\sigma(1)}\lambda_{\rho}u_{\rho}\>|\>\sum_{\rho\in\sigma(1)}\lambda_{\rho}\leq
1,\>0\leq\lambda_{\rho}\leq 1\right\\}.$

As described above, we may fix a subset of

$\\{A(\tilde{e}_{i}),\>1\leq i\leq d\\}$

as a set of ray generators for $\sigma$. Let $u^{\prime}\in\Pi_{\sigma}\cap
N^{\prime}$ given by

$u^{\prime}=\sum_{i=1}^{d}\lambda_{i}A(\tilde{e}_{i}),$

with

$0\leq\lambda_{i}\leq 1,$

and

$\lambda_{1}+\ldots+\lambda_{d}\leq 1,$

assuming $\lambda_{i}=0$ when $A(\tilde{e}_{i})$ is not in our fixed chosen
set of ray generators. Assume that there is $i_{0}$ with $\lambda_{i_{0}}\neq
0$. We need to show that $\lambda_{i_{0}}=1$ and $\lambda_{i}=0$ for $i\neq
i_{0}$. By assumptions on the $(\lambda_{i})$’s, it is enough to show that
$\lambda_{i}\in\mathbb{Z}$ for all $i$. As $u^{\prime}\in N^{\prime}$, there
exists $\tilde{u}\in\tilde{N}$ such that

$A(\tilde{u})=A(\sum_{i=1}^{d}\lambda_{i}\tilde{e}_{i}),$

and then there is $x\in N_{\mathbb{R}}$ such that

$\tilde{u}-\sum_{i=1}^{d}\lambda_{i}\tilde{e}_{i}=B(x)=(B_{j}(x),B_{+}(x)).$

By injectivity of $B_{j}$ again, we only need to consider the case when
$B_{j}(x)\neq 0$ and $1\leq i_{0}\leq d_{j}$ (as if $\lambda_{i}=0$ for all
$i\leq d_{j}$ then $x\in N$ and $B(x)\in\tilde{N}$). Hence, we reduced
ourselves to study the case when $X=Y_{j}$. We will use again the explicit
descriptions of the $V_{j}$’s from Example 4.1. For $V_{3}$, we find the
system

$\left\\{\begin{array}[]{ccc}\tilde{u}_{1}-\lambda_{1}&=&-x_{2}\\\
\tilde{u}_{2}-\lambda_{2}&=&-x_{2}\\\ \tilde{u}_{3}-\lambda_{3}&=&x_{1}\\\
\tilde{u}_{4}-\lambda_{4}&=&x_{1}\\\
\tilde{u}_{5}-\lambda_{5}&=&-x_{1}+x_{2}\\\
\tilde{u}_{6}-\lambda_{6}&=&-x_{1}+x_{2}\end{array}\right.$

from which we obtain

$(\lambda_{1}+\lambda_{3}+\lambda_{5},\lambda_{1}+\lambda_{3}+\lambda_{6},\lambda_{1}+\lambda_{4}+\lambda_{6},\lambda_{1}+\lambda_{4}+\lambda_{5})\in\mathbb{Z}^{4}$

and

$(\lambda_{2}+\lambda_{3}+\lambda_{5},\lambda_{2}+\lambda_{3}+\lambda_{6},\lambda_{2}+\lambda_{4}+\lambda_{6},\lambda_{2}+\lambda_{4}+\lambda_{5})\in\mathbb{Z}^{4}.$

Together with

$0\leq\lambda_{1}+\ldots+\lambda_{6}\leq 1,$

we deduce that all those linear combinations of the $\lambda_{i}$’s equal $0$
or $1$, with at least one that is non zero. Using $\lambda_{i}\geq 0$, we then
deduce that $\lambda_{i_{0}}=1$ and the result follows in that case. The cases
$V_{2}$ and $V_{4}$ are similar, so we will only treat $V_{4}$. In that case,
one checks that

$\left\\{\begin{array}[]{ccc}A(\tilde{e}_{1})&=&A(\tilde{e}_{2})\\\
A(\tilde{e}_{3})&=&A(\tilde{e}_{4})\end{array}\right.$

so we can pick the ray generators $(A(\tilde{e}_{1}),A(\tilde{e}_{3}))$ for
$\sigma(1)$. The description of $V_{4}$ then implies

$\left\\{\begin{array}[]{ccc}\tilde{u}_{1}-\lambda_{1}&=&x_{1}\\\
\tilde{u}_{2}&=&-x_{1}\\\ \tilde{u}_{3}-\lambda_{3}&=&x_{2}\\\
\tilde{u}_{4}&=&-x_{2}\end{array}\right.$

hence $x\in N$ and the result follows. ∎

###  4.4. Examples

We first provide two examples of local moduli spaces and then an example of a
quotient $H^{1}(X,TX)//T$ that is not $\mathbb{Q}$-Gorenstein, for $X$ a
smooth toric surface.

####  4.4.1. A smooth example : $Y_{4}$

We consider the toric variety $W_{4}$ which is the quotient of $V_{4}$ by $T$
(recall the definition of $Y_{4}$ in Example 4.1, and that
$V_{4}=H^{1}(Y_{4},TY_{4})$). As seen in the proof of Proposition 4.8, the
generators of $\sigma$ in that case can be taken to be $A(\tilde{e}_{1})$ and
$A(\tilde{e}_{3})$, so that $\sigma$ is isomorphic to the cone

$\mathbb{R}_{+}\cdot(1,0)+\mathbb{R}_{+}\cdot(0,1)\subset\mathbb{R}^{2}.$

Then,

$W_{4}\simeq\mathbb{C}^{2},$

and the $G$-action on $V_{4}$ (see Remark 4.2), descends to a $D_{1}$-action
on $W_{4}\simeq\mathbb{C}^{2}$ generated by a reflection

$(x,y)\mapsto(y,x).$

Hence, we conclude that the local moduli space is modeled on

$W_{4}//G\simeq\mathbb{C}^{2}.$

####  4.4.2. A singular example : $Y_{3}$

Let’s consider now $W_{3}$, given by the GIT quotient of $V_{3}$ by $T$. We
can pick isomorphisms $N\simeq\mathbb{Z}^{2},$ $\tilde{N}\simeq\mathbb{Z}^{6}$
and

$N^{\prime}=\tilde{N}/N\simeq\mathbb{Z}^{4}$

such that the map $B:N\to\tilde{N}$ is given by

$B=\left[\begin{array}[]{cc}0&-1\\\ 0&-1\\\ 1&0\\\ 1&0\\\ -1&1\\\
-1&1\end{array}\right]$

and the map $A:\tilde{N}\to N^{\prime}$ by

$A=\left[\begin{array}[]{cccccc}-1&1&0&0&0&0\\\ 0&0&-1&1&0&0\\\ 1&0&1&0&1&0\\\
1&0&1&0&0&1\end{array}\right].$

Hence, $W_{3}$ is the toric variety associated to the cone

$\sigma=\sum_{i=1}^{6}\mathbb{R}_{+}\cdot e_{i}^{\prime}\subset\mathbb{R}^{4}$

where $(e_{i}^{\prime})_{1\leq i\leq 6}$ are given by the columns of the
matrix $A$. Using Sage Math, we find

$W_{3}\simeq\mathrm{Spec}(\mathbb{C}[z_{0},\ldots,z_{7}]/I_{3})$

with $I_{3}$ the ideal whose set of generators is given by

$\begin{array}[]{ccc}I_{3}&=&\langle-
z_{1}z_{4}+z_{0}z_{7},-z_{1}z_{5}+z_{0}z_{6},-z_{4}z_{6}+z_{5}z_{7},\\\
&&z_{0}z_{4}-z_{3}z_{5},-z_{2}z_{4}+z_{3}z_{7},-z_{1}z_{4}+z_{3}z_{6},\\\
&&z_{0}z_{2}-z_{1}z_{3},z_{2}z_{6}-z_{1}z_{7},-z_{1}z_{4}+z_{2}z_{5}\rangle.\end{array}$

This toric affine variety is singular (it is not even simplicial). The
$G$-action descends to a $D_{3}$-action on $W_{3}$. This action is induced by
a representation

$D_{3}\to\mathrm{GL}_{4}(\mathbb{Z}),$

and a set of generators for this action is given by

$\left[\begin{array}[]{cccc}0&0&-1&1\\\ 1&0&0&0\\\ 0&0&1&0\\\
0&1&1&0\end{array}\right]\mathrm{and}\left[\begin{array}[]{cccc}-1&0&0&0\\\
0&-1&0&0\\\ 1&1&0&1\\\ 1&1&1&0\end{array}\right]$

where the first matrix represents an element of order $3$ and the second one a
reflection. The final quotient $W_{3}//D_{3}$ provides a singular example of
local moduli space.

###  4.5. A non–Gorenstein example

We produce here an example of a toric surface $X$ with $\mathrm{Aut}(X)\simeq
T$, and $H^{1}(X,TX)//T$ a non $\mathbb{Q}$-Gorenstein toric variety. For
this, simply blow-up $Y_{4}$ in a single point to produce the fan on the
following page :

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 13. Fan of $X$.

As $X$ is blown-up from $Y_{4}$, we have $\mathrm{Aut}^{0}(X)\simeq T$. One
can also check that we actually have $\mathrm{Aut}(X)\simeq T$. Moreover, the
arguments in Section 4.2 go through, and we find that if $X$ were cscK, then
its local moduli space would be modeled on the GIT quotient of $H^{1}(X,TX)$
by $T$. A direct computation again, using the method in Section 4.1, provides
the weight space decomposition

$H^{1}(X,TX)=V^{1}_{+e_{1}^{*}}\oplus V^{2}_{-e_{1}^{*}}\oplus
V^{1}_{+e_{2}^{*}}\oplus V^{1}_{-e_{2}^{*}}\oplus
V^{1}_{e_{1}^{*}-e_{2}^{*}}.$

Given that the sum of the weights that appear in this decomposition doesn’t
vanish, the quotient $H^{1}(X,TX)//T$ is not $\mathbb{Q}$-Gorenstein (see the
proof of Proposition 4.7).

##  5\. Discussion and perspectives

###  5.1. Relations between local moduli and Weil–Petersson metrics

We will discuss in this section the fact that the local moduli spaces we
considered are related by toric fibrations. Assume that $\pi:X\to X_{0}$ is a
$G$-equivariant blow-up between two foldable toric surfaces. We keep the
notations from previous sections, using the subscript $0$ to refer the the
spaces associated to $X_{0}$. From [21, Corollary 1.6], the corresponding
space $V_{0}$ injects in $V$. It is a straightforward exercise to check that
we have the following commutative diagram :

$\begin{array}[]{ccccccccc}0&\longrightarrow&N&\stackrel{{\scriptstyle
B}}{{\longrightarrow}}&\tilde{N}&\stackrel{{\scriptstyle
A}}{{\longrightarrow}}&N^{\prime}&\longrightarrow&0\\\
&&\downarrow&&\downarrow&&\downarrow&&\\\
0&\longrightarrow&N&\stackrel{{\scriptstyle
B_{0}}}{{\longrightarrow}}&\tilde{N}_{0}&\stackrel{{\scriptstyle
A_{0}}}{{\longrightarrow}}&N_{0}^{\prime}&\longrightarrow&0\end{array}$

where the first vertical arrow is the identity and the last two are
surjective. By construction, one sees that the surjective map $N^{\prime}\to
N_{0}^{\prime}$ is compatible with $\sigma$ and $\sigma_{0}$, and induces a
toric locally trivial fibration $W\to W_{0}$ whose fiber is itself an affine
toric variety. The whole construction is $G$-equivariant, and provides maps
between the associated local moduli spaces (up to shrinking the neighborhoods
we considered).

The cscK metric on $X$ lives in the class $\pi^{*}[\omega_{0}]-\varepsilon
E_{i}$, where $\omega_{0}$ is cscK on $X_{0}$, $\varepsilon$ is small and the
$E_{i}$’s stand for the exceptional divisors of the blow-up. It would be
interesting to understand the behaviour of the associated Weil–Petersson
metrics $(\Omega^{WP}_{\varepsilon})_{0<\varepsilon<\varepsilon_{0}}$ on the
local moduli spaces $\mathscr{W}_{\varepsilon}\subset W/G$ as constructed in
[12] when $\varepsilon$ goes to zero. It seems natural to expect that the
volume of the fibers of the fibration $W/G\to W_{0}/G$ go to zero, so that
$\mathscr{W}_{\varepsilon}$ would converge in Gromov–Hausdorff sense to
$\mathscr{W}_{0}$.

###  5.2. Higher dimensional case

Many features that hold for toric surfaces fail in higher dimension. First,
even when it is reductive, the identity component of the automorphism group of
a non-rigid toric variety will not a priori be isomorphic to the torus (see
[35]). Then, from dimension $3$, toric varieties may be obstructed (see [20]).
Finally, the toric MMP (that produces the classification of toric surfaces)
produces singular varieties in general. So our main ingredients to prove
Theorem 1.5 do not generalise, a priori, in higher dimension.

Nevertheless, it would be interesting to study what goes through the new
difficulties. We expect the right extension of the notion of foldable fans in
higher dimension to be fans whose lattice automorphism group admit a subgroup
that acts with no fixed point. This raises several questions :

  1. (1)

Do all crystallographic groups arise as lattice automorphism groups of smooth
complete fans?

  2. (2)

Do all foldable toric varieties admit a cscK metric?

  3. (3)

Are foldable toric varieties unobstructed?

  4. (4)

What are the singularities of the moduli space of cscK metrics around cscK
foldable toric varieties?

## References

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