Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)

Complex analytic Moduli space of Fano manifolds
admitting Kähler-Ricci solitons
Eiji Inoue （瑛二, 井上）

14, July, 2018 at Hayama

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Moduli problem(M, ω): a C ∞ -symplectic manifold.
Question: Can we construct a Hausdorﬀ (global) moduli space of bihol.
classes of complex manifolds (X , c) of fixed symplectic type (M, ω)?
Here c ∈ H 2 (X , R) is a cohomology class, called polarization.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Moduli problem(M, ω): a C ∞ -symplectic manifold.
Question: Can we construct a Hausdorﬀ (global) moduli space of bihol.
classes of complex manifolds (X , c) of fixed symplectic type (M, ω)?
Here c ∈ H 2 (X , R) is a cohomology class, called polarization.
X with a polarization c is of fixed symplectic type (M, ω) if∃f : X −−−→ M s.t. f ∗ ω is a Kähler form in c,
diﬀeo.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Moduli problem(M, ω): a C ∞ -symplectic manifold.
Question: Can we construct a Hausdorﬀ (global) moduli space of bihol.
classes of complex manifolds (X , c) of fixed symplectic type (M, ω)?
Here c ∈ H 2 (X , R) is a cohomology class, called polarization.
X with a polarization c is of fixed symplectic type (M, ω) if∃f : X −−−→ M s.t. f ∗ ω is a Kähler form in c,
diﬀeo.

(X , c) and (X ′ , c ′ ) are biholomorphic if∃ϕ : X −−−→ X ′ s.t. c = ϕ∗ c ′ .
bihol.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Complex structure on the moduli spacee.g. The moduli space of all K3 surfaces is a 20-dimensional complex
manifold but for Hausdorﬀness. The moduli space of polarized K3
surafces is a 19-dimensional Hausdorﬀ complex space.

As local semi-moduli (constructed by Kuranishi after Kodaira-Spencer) of
deformation of complex manifolds are complex spaces, it is natural toexpect: the global moduli space should also admit a natural structure ofcomplex analytic space.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Problematic Fano
Main interest: ±ω ∈ 2πc1 (M, ω) (c = ±c1 (X )) or c1 (M, ω) = 0.
c1 (M, ω) < 0 general type: ∃ Hausdorﬀ moduli.
c1 (M, ω) = 0 Calabi-Yau: ∃ Hausdorﬀ moduli (under polarization).
c1 (M, ω) > 0 Fano:Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Problematic Fano
Main interest: ±ω ∈ 2πc1 (M, ω) (c = ±c1 (X )) or c1 (M, ω) = 0.
c1 (M, ω) < 0 general type: ∃ Hausdorﬀ moduli.
c1 (M, ω) = 0 Calabi-Yau: ∃ Hausdorﬀ moduli (under polarization).
c1 (M, ω) > 0 Fano: Naively, no such moduli.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Problematic Fano
Main interest: ±ω ∈ 2πc1 (M, ω) (c = ±c1 (X )) or c1 (M, ω) = 0.
c1 (M, ω) < 0 general type: ∃ Hausdorﬀ moduli.
c1 (M, ω) = 0 Calabi-Yau: ∃ Hausdorﬀ moduli (under polarization).
c1 (M, ω) > 0 Fano: Naively, no such moduli.
Pathology: Many Fano manifolds X and families π : X → ∆ with

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Problematic Fano
Main interest: ±ω ∈ 2πc1 (M, ω) (c = ±c1 (X )) or c1 (M, ω) = 0.
c1 (M, ω) < 0 general type: ∃ Hausdorﬀ moduli.
c1 (M, ω) = 0 Calabi-Yau: ∃ Hausdorﬀ moduli (under polarization).
c1 (M, ω) > 0 Fano: Naively, no such moduli.
Pathology: Many Fano manifolds X and families π : X → ∆ with∼ ∆∗ × X ,Trivial away from the origin: π −1 (∆∗ ) =Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Problematic Fano
Main interest: ±ω ∈ 2πc1 (M, ω) (c = ±c1 (X )) or c1 (M, ω) = 0.
c1 (M, ω) < 0 general type: ∃ Hausdorﬀ moduli.
c1 (M, ω) = 0 Calabi-Yau: ∃ Hausdorﬀ moduli (under polarization).
c1 (M, ω) > 0 Fano: Naively, no such moduli.
Pathology: Many Fano manifolds X and families π : X → ∆ with∼ ∆∗ × X ,Trivial away from the origin: π −1 (∆∗ ) =The central fibre X0 = π −1 (0) is a diﬀerent Fano X0 ≇ X .

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Problematic Fano
Main interest: ±ω ∈ 2πc1 (M, ω) (c = ±c1 (X )) or c1 (M, ω) = 0.
c1 (M, ω) < 0 general type: ∃ Hausdorﬀ moduli.
c1 (M, ω) = 0 Calabi-Yau: ∃ Hausdorﬀ moduli (under polarization).
c1 (M, ω) > 0 Fano: Naively, no such moduli.
Pathology: Many Fano manifolds X and families π : X → ∆ with∼ ∆∗ × X ,Trivial away from the origin: π −1 (∆∗ ) =The central fibre X0 = π −1 (0) is a diﬀerent Fano X0 ≇ X .

[X0 ] cannot be separated from [X ]!
:(Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Canonical metric → Hausdorﬀ(∵)◦ oO （Metrics in general have good chemistry with Hausdorﬀness....）

Idea: What about assuming the existence of some ‘canonical metrics’ on
Fano manifolds in order to ensure separatedness?

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
Motivation

Canonical metric → Hausdorﬀ(∵)◦ oO （Metrics in general have good chemistry with Hausdorﬀness....）

Idea: What about assuming the existence of some ‘canonical metrics’ on
Fano manifolds in order to ensure separatedness?
The following is a success story.
Theorem (Odaka ’15, Li-Wang-Xu ’15)
Kähler-Einstein Fano manifolds admit a natural Hausdorﬀ, complex
analytic moduli space.
Remark: YTD-correspondence (proved by Chen-Donaldson-Sun, Tian) on
KE metrics and K-stability is used in the construction of the moduli.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Introduction
The goal of this talkToday’s goal: enlargement

The goal of this talk:
Theorem (I. ’17)
Let KR(M, ω) be the set of Fano manifolds with Kähler-Ricci solitons of
fixed symplectic type (M, ω). We can make this set KR(M, ω) into a
Hausdorﬀ complex analytic space in a canonical way.
Kähler-Ricci soliton is a special metric on a Fano manifold, which
generalizes Kähler-Einstein metric in view of Kähler-Ricci flow.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector

1. Kähler-Ricci soliton, moment map and K-optimal vector

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
Kähler-Ricci soliton

Kähler-Ricci soliton
Definition (Kähler-Ricci soliton)
A Kähler-Ricci soliton on a Fano manifold X is a pair (g , ξ ′ ) of a Kähler
metric g and a holomorphic vector field ξ ′ satisfying the following
equation:Ric(g ) − Lξ′ g = g .
Remember ξ := Im(ξ ′ ) generates a closed torus TξR = exp Rξ ⊂ Aut(X ).

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
Kähler-Ricci soliton

Kähler-Ricci soliton
Definition (Kähler-Ricci soliton)
A Kähler-Ricci soliton on a Fano manifold X is a pair (g , ξ ′ ) of a Kähler
metric g and a holomorphic vector field ξ ′ satisfying the following
equation:Ric(g ) − Lξ′ g = g .
Remember ξ := Im(ξ ′ ) generates a closed torus TξR = exp Rξ ⊂ Aut(X ).
Theorem (Uniqueness, Tian-Zhu ’02)
If (g1 , ξ1′ ) and (g2 , ξ2′ ) are two Kähler-Ricci solitons on a Fano manifold
X , then there exists an element ϕ ∈ Aut0 (X ) such thatϕ∗ ξ1′ = ξ2′ ,ϕ∗ g1 = g2 .

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
Kähler-Ricci soliton

Example: All toric Fano manifolds admit Kähler-Ricci solitons.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
Moment map picture

Infinite dimensional symplectic manifold JT
Fix our notation.
(M, ω): a simply connected C ∞ -symplectic manifold.
(T , ξ): a closed torus acting on (M, ω) and an element of Lie(T ).
∫θξ : dθξ = 2iξ ω with the normalization M θξ e θξ ω n = 0.
JT : the space of T -inv. almost complex srtuctures on (M, ω).
HamT : the group of T -equiv. symplectic diﬀeomorphisms of (M, ω).
JT admits a HamT -invariant smooth symplectic form Ωξ defined by∫Ωξ,J (A, B) :=Tr(JAB) e θξ ω n
M

for A, B ∈ TJ JT .

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
Moment map picture

Moment map sξ : JT → ham∗T
We identify the Lie algebra hamT := Lie(HamT ) with∫∞∞CT ,ξ (M) = {f ∈ CT (M) |f e θξ ω n = 0}.
M

Proposition (I. ’17)
The map Sξ : JT → ham∗T defined by∫ ()¯ gJ θξ − s̄ + □¯ gJ θξ − ξJ′ θξ − θξ • e θξ ω n .
J 7→ ⟨sξ (J), •⟩ξ =s(gJ ) + □M

is a moment map of (JT , Ωξ ) ↶ HamT .
If c1 (M, ω) > 0, an integrable complex structure J satisfies sξ (J) = 0 iﬀ
gJ is a Kähler-Ricci solitons on the Fano manifold (M, J).

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
Moment map picture

We have the following immediate cororally.
For f ∈ t ⊂ hamT = CT∞,ξ (M), an easy caluculation shows that themodified Futaki invariant is equal to∫⟨sξ , f ⟩ξ =sξ f e θ ξ ω n ,Mwhich shows the T -equivariant deformation invariance of the modified
Futaki invariant restricted to t ⊂ H 0 (M, TJ M).
Becaused⟨sξ (Jt ), f ⟩ξ = −Ωξ (LXf Jt , J˙t ) = 0.
dt

Remark
If the set sξ−1 (0) is not empty, then ⟨sξ , •⟩ξ should vanish.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
K-optimal vector

K-optimal vector
The following is a restatement of Tian-Zhu’s result.
Proposition (Tian-Zhu ’02)
For any (M, ω) ↶ T , there exists a unique vector ξ ∈ Lie(T ) with⟨sξ , •⟩ ≡ 0.
Definition
We call this unique vector ξ the K-optimal vector of (M, ω) ↶ T .
We call the action (M, ω) ↶ T is K-optimal if the K-optimal vector
generates T .

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
K-optimal vector

The boundedness of Fano manifolds (Kollár-Miyaoka-Mori)=⇒ Only finitely many non-equivalent K-optimal actions (M, ω) ↶ Ti
with sξi (Ji ) = 0 for some integrable Ji ∈ JTi and the K-optimal ξi .

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
K-optimal vector

The boundedness of Fano manifolds (Kollár-Miyaoka-Mori)=⇒ Only finitely many non-equivalent K-optimal actions (M, ω) ↶ Ti
with sξi (Ji ) = 0 for some integrable Ji ∈ JTi and the K-optimal ξi .
The uniqueness of KR-soliton =⇒ The topological space⨿
K (M, ω) :=(sξi |JTint )−1 (0) / HamTi
ii

can be naturally identified with the set of biholomorphism classes of Fano
manifolds admitting KR-solitons of symplectic type (M, ω). The space
K (M, ω) is Hausdorﬀ as the action JT ↶ HamT is proper.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
KR soliton, moment map and K-optimal vector
K-optimal vector

The boundedness of Fano manifolds (Kollár-Miyaoka-Mori)=⇒ Only finitely many non-equivalent K-optimal actions (M, ω) ↶ Ti
with sξi (Ji ) = 0 for some integrable Ji ∈ JTi and the K-optimal ξi .
The uniqueness of KR-soliton =⇒ The topological space⨿
K (M, ω) :=(sξi |JTint )−1 (0) / HamTi
ii

can be naturally identified with the set of biholomorphism classes of Fano
manifolds admitting KR-solitons of symplectic type (M, ω). The space
K (M, ω) is Hausdorﬀ as the action JT ↶ HamT is proper.
Remark: The space K (M, ω) is naturally identified with the space{J ∈ J int (M, ω) | (M, J) admits a Kähler-Ricci soliton }/bihol.
as sets, but not as topological spaces. Actually, the latter topologicalspace is not Hausdorﬀ in general. The equivariant formulation is essential.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space

2. Structure of the moduli spaceModuli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

Charts
We must construct a structure of complex space on the space K (M, ω).

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

Charts
We must construct a structure of complex space on the space K (M, ω).
1

A homeomorphism ϕ : U → V ⊂ K (M, ω) where V is an open
neighbourhood of a point X = (M, J) ∈ K (M, ω) and U is a small
open neighbourhood of [0] ∈ HT1 i (X , TX ) AutTi (X ).

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

Charts
We must construct a structure of complex space on the space K (M, ω).
1

2

A homeomorphism ϕ : U → V ⊂ K (M, ω) where V is an open
neighbourhood of a point X = (M, J) ∈ K (M, ω) and U is a small
open neighbourhood of [0] ∈ HT1 i (X , TX ) AutTi (X ).
The holomorphy of coordinate changes ϕ−1
U2 ◦ ϕU1 .

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

Charts
We must construct a structure of complex space on the space K (M, ω).
1

2

A homeomorphism ϕ : U → V ⊂ K (M, ω) where V is an open
neighbourhood of a point X = (M, J) ∈ K (M, ω) and U is a small
open neighbourhood of [0] ∈ HT1 i (X , TX ) AutTi (X ).
The holomorphy of coordinate changes ϕ−1
U2 ◦ ϕU1 .

The map sξ is not holomorphic, 0 ∈ ham∗T is not the regular value of sξ ,the Fréchet Lie group HamT is not complex Lie group, etc.
:(

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

How to deal with holomorphy?
Idea: make use of the holomorphy of the semi-universal family X → Ũon a small ball Ũ ⊂ HT1 i (X , TX ).

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

How to deal with holomorphy?
Idea: make use of the holomorphy of the semi-universal family X → Ũon a small ball Ũ ⊂ HT1 i (X , TX ).
Not descends to a family X̄ → U = ŨAut

Aut.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

How to deal with holomorphy?
Idea: make use of the holomorphy of the semi-universal family X → Ũon a small ball Ũ ⊂ HT1 i (X , TX ).
Not descends to a family X̄ → U = ŨAut

Aut.

Descends to a ‘family’ on the quotient stack [ŨAut/Aut].

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

How to deal with holomorphy?
Idea: make use of the holomorphy of the semi-universal family X → Ũon a small ball Ũ ⊂ HT1 i (X , TX ).
Not descends to a family X̄ → U = ŨAut

Aut.

Descends to a ‘family’ on the quotient stack [ŨAut/Aut].
After introducing a suitable moduli stack K(M, ω), we have anatural morphism [ŨAut/Aut] → K(M, ω) between stacks.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

How to deal with holomorphy?
Idea: make use of the holomorphy of the semi-universal family X → Ũon a small ball Ũ ⊂ HT1 i (X , TX ).
Not descends to a family X̄ → U = ŨAut

Aut.

Descends to a ‘family’ on the quotient stack [ŨAut/Aut].
After introducing a suitable moduli stack K(M, ω), we have anatural morphism [ŨAut/Aut] → K(M, ω) between stacks.
Generality on GIT quotient: a natural morphism [ŨAut/Aut] → U
enjoying the following universal property:[ŨAut/Aut]∀U∃!
holomorphic∀S

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ][Ũ2 Aut2 /Aut2 ]K(M, ω)Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ][Ũ2 Aut2 /Aut2 ]K(M, ω)Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ][Ũ2 Aut2 /Aut2 ]K(M, ω)Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ]U1[Ũ2 Aut2 /Aut2 ]K(M, ω)U2

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ]U1[Ũ2 Aut2 /Aut2 ]K(M, ω)U2

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ]U1

U3[Ũ2 Aut2 /Aut2 ]U2

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ]U1

U3[Ũ2 Aut2 /Aut2 ]U2

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ]

U1
3

U3

[Ũ2 Aut2 /Aut2 ]U2

These morphisms are compatible with maps ϕi : Ui → K (M, ω)(i = 1, 2, 3). =⇒ U3 → Ui are biholomorphic onto its open image.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Holomorphic gluing

The story of holomorphic gluing
1Show the morphism [ŨAut/Aut] → K(M, ω) is an open imersion.

2[Ũ3 Aut3 /Aut3 ][Ũ1 Aut1 /Aut1 ]

U1
3

4

U3

[Ũ2 Aut2 /Aut2 ]U2

These morphisms are compatible with maps ϕi : Ui → K (M, ω)(i = 1, 2, 3). =⇒ U3 → Ui are biholomorphic onto its open image.
The coordinate change ϕ−1
2 ◦ ϕ1 between U1 and U2 is holomorphic.

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Universality

The universality of the moduli space

Theorem (I. ’17)
The complex space K (M, ω) is uniquely characterized by the universalityof a natural morphism K(M, ω) → K (M, ω).
K(M, ω)∀K (M, ω)∃!

∀S

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Gormov-Hausdorﬀ topology

Gromov-Hausdorﬀ topology

Proposition (I. ’17)
The space
KRGH (M, ω) = {Fano mfds of type (M, ω) with ∃ KR solitons}/bihol.
endowed with the Gromov-Hausdorﬀ topology is naturally homeomorphic
to K (M, ω).

Moduli space of Fano manifolds admitting Kähler-Ricci solitons (Eiji Inoue)
Structure of the moduli space
Gormov-Hausdorﬀ topology

Thank You !