Introduction
Diﬀerential geometric side
Algebro-geometric side

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

14, June, 2018 at ShanghaiTech University

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

Today’s goal: rough statement without definitions
Contents

Motivation(M, ω): a C ∞ -symplectic manifold.
Question: Can we construct a Hausdorﬀ moduli space of bihol. classesof Kähler manifolds (X , L) of fixed symplectic type (M, ω).
We expect there should be a structure of complex analytic space on themoduli space, as the notion of complex spaces are designed to describe
their own deformation theory.

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

Today’s goal: rough statement without definitions
Contents

Motivation(M, ω): a C ∞ -symplectic manifold.
Question: Can we construct a Hausdorﬀ moduli space of bihol. classesof Kähler manifolds (X , L) of fixed symplectic type (M, ω).
We expect there should be a structure of complex analytic space on themoduli space, as the notion of complex spaces are designed to describe
their own deformation theory.
We are mainly concerned with the cases ±ω ∈ c1 (M, ω) or c1 (M, ω) = 0.

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

Today’s goal: rough statement without definitions
Contents

Motivation(M, ω): a C ∞ -symplectic manifold.
Question: Can we construct a Hausdorﬀ moduli space of bihol. classesof Kähler manifolds (X , L) of fixed symplectic type (M, ω).
We expect there should be a structure of complex analytic space on themoduli space, as the notion of complex spaces are designed to describe
their own deformation theory.
We are mainly concerned with the cases ±ω ∈ c1 (M, ω) or c1 (M, ω) = 0.
c1 (M, ω) < 0 general type: Yes, we can.
c1 (M, ω) = 0 Calabi-Yau: Yes, we can.
c1 (M, ω) > 0 Fano:

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

Today’s goal: rough statement without definitions
Contents

Motivation(M, ω): a C ∞ -symplectic manifold.
Question: Can we construct a Hausdorﬀ moduli space of bihol. classesof Kähler manifolds (X , L) of fixed symplectic type (M, ω).
We expect there should be a structure of complex analytic space on themoduli space, as the notion of complex spaces are designed to describe
their own deformation theory.
We are mainly concerned with the cases ±ω ∈ c1 (M, ω) or c1 (M, ω) = 0.
c1 (M, ω) < 0 general type: Yes, we can.
c1 (M, ω) = 0 Calabi-Yau: Yes, we can.
c1 (M, ω) > 0 Fano: Naively, it doesn’t exist.

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

Today’s goal: rough statement without definitions
Contents

Pathology: Naive moduli doesn’t exist.

We can find many examples of Fano manifolds X and families of Fano
manifolds π : X → ∆ with the following property.
Trivial away from the origin: π −1 (∆∗ ) ∼= ∆∗ × X
The central fibre X0 = π −1 (0) is a diﬀerent Fano X0 ≇ X .
The biholomorphism class of X0 cannot be separated from that of X ! :(Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

Today’s goal: rough statement without definitions
Contents

Pathology: Naive moduli doesn’t exist.

We can find many examples of Fano manifolds X and families of Fano
manifolds π : X → ∆ with the following property.
Trivial away from the origin: π −1 (∆∗ ) ∼= ∆∗ × X
The central fibre X0 = π −1 (0) is a diﬀerent Fano X0 ≇ X .
The biholomorphism class of X0 cannot be separated from that of X ! :(Metrics in general have good chemistry with Hausdorﬀness.
What about assuming the existence of some ‘canonical metrics’ on Fano
manifolds in order to ensure separatedness?

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

Today’s goal: rough statement without definitions
Contents

Today’s goal: rough statement without definitions

The goal of this talk:Complex analytic moduli space, I. ’17
Let KRGH (n) be the set of n-dimensional Fano manifolds withKähler-Ricci solitons. Then we can make this set KRGH (n) into a
Hausdorﬀ complex analytic space in a canonical way in some sense.
Kähler-Ricci soliton is a special metric on a Fano manifold (withrespect to the anti-canonical polarization −KX ), which generalizes
Kähler-Einstein metric in view of Kähler-Ricci flow.

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

1

Diﬀerential geometric sideKähler-Ricci soliton
Moment map picture2

Algebro-geometric side
K-stability
Complex analytic moduli

Eiji Inoue （瑛二, 井上）

Today’s goal: rough statement without definitions
Contents

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

1. Diﬀerential geometric side - ‘real world’Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map pictureKähler-Ricci soliton

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

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

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

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′ ,Eiji Inoue （瑛二, 井上）

ϕ∗ g1 = g2 .

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

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 .

Theorem (Reductivity, Tian-Zhu ’02)
If (g , ξ ′ ) is a Kähler-Ricci soliton on a Fano manifold X , then Aut(X , ξ ′ )is a maximal reductive subgroup of Aut(X ) and Isom(X , g ) is its
maximal compact subgroup.
Reductivity

We can apply GIT (locally).

We should consider the pair (X , ξ ′ ) rather than mere X .
(in order to construct their moduli space. )Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

modified Futaki invariant(X , ξ ′ ) is a Fano manifold X with a holomorphic vector field ξ ′ .
ω is a Kähler form in 2πc1 (X ).

∫ θ′ n ∫ n√¯ ξ′ withθξ′ is a potential of ξ ′ : Lξ′ ω = −1∂ ∂θe ξω = Xω .
X√
¯ = Ric(ω) − ω.
h is a Ricci potential: −1∂ ∂hDefine Futξ′ : η(X ) → C by
Futξ′ (v ′ ) := −∫v ′ (h − θξ′ )e θξ′ ω n .

X

Independent of the choice of ω.
If there exists a KR soliton (g , ξ ′ ), Futξ′ should vanish.

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

K-optimal vector

If there exists a KR soliton (g , ξ ′ ), Futξ′ should vanish.
X ↶ TC ∼= (C∗ )k : torus action on a Fano manifold X .
N: the 1-psg lattice of T C .
NR := N ⊗ R ⊂ η(X ) by ξ 7→ ξ ′ := Jξ +

√

−1ξ.

Proposition (Tian-Zhu ’02)
For any X ↶ T C , where X does not necessarily admit any KR solitons,
there exists a unique vector ξ ∈ NR with Futξ′ |NR .
We call this unique vector the K-optimal vector of X ↶ T C .

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

Moment map picture

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

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 .

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

Proposition (I. ’17)
The map¯ ξ − n) + (□θ¯ ξ − ξ ′ θξ − θξ )sξ : JT → C ∞ (M) : J 7→ (s(gJ ) + □θ
defines a moment map of (JT , Ωξ ) ↶ HamT . If c1 (M, ω) > 0, thenintegrable complex structures J with sξ (J) = 0 precisely correspond toKähler-Ricci solitons gJ on Fano manifolds (M, J).
We have the following immediate cororally.
For f ∈ t ⊂ Lie(HamT ) = CT∞ (M)/R, the modified Futaki invariant
Futξ′ (Xf′ ) equals to∫⟨sξ , f ⟩ξ =sξ f e θ ξ ω n ,X

which shows the T -equivariant deformation invariance of the modified
Futaki invariant restricted to Lie(T ) = NR .
Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

Summary of this section

In summary,
Input: a Fano T C -manifold X⇓
Intermediate: a C ∞ -symplectic T R -manifold (M, ω) with ω ∈ 2πc1 (M, ω)⇓
Output: the K-optimal vector ξ ∈ NR satisfying Futξ′ |NR ≡ 0.

The K-optimal vector ξ ∈ NR is T -equivaraint deformation invariant!

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

From the uniqueness of KR-soliton, the topological space(sξ |JTint )−1 (0) / HamTξξ

can be naturally identified with the set consisting of biholomorphismclasses, not Tξ -equivariant biholomorphism classes, of Fano manifolds
admitting KR-solitons of symplectic type (M, ω, ξ).

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

From the uniqueness of KR-soliton, the topological space(sξ |JTint )−1 (0) / HamTξξ

can be naturally identified with the set consisting of biholomorphismclasses, not Tξ -equivariant biholomorphism classes, of Fano manifolds
admitting KR-solitons of symplectic type (M, ω, ξ).
We should construct a structure of complex analytic space on this space.
But it seems diﬃcult to deal with holomorphy in this real (non-complex)
geometric viewpoint....

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric sideKähler-Ricci soliton
Moment map picture

From the uniqueness of KR-soliton, the topological space(sξ |JTint )−1 (0) / HamTξξ

can be naturally identified with the set consisting of biholomorphismclasses, not Tξ -equivariant biholomorphism classes, of Fano manifolds
admitting KR-solitons of symplectic type (M, ω, ξ).
We should construct a structure of complex analytic space on this space.
But it seems diﬃcult to deal with holomorphy in this real (non-complex)
geometric viewpoint....
⇓Use K-stability !

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

2. Algebro-geometric side - ‘virtual world’Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

K-stability

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

From now on, the K-optimal vector ξ will be implicitly involved in ourformulations, which is encoded in the torus action X ↶ T C = T .
A special degeneration for a Q-Fano T -variety (X , −KX ) ↶ T is a
T × C∗ -equivariant family of Q-Fano varieties π : X → C ( T acts on C
trivially) endowed with a T -equivariant isomorphism X × C∗ ∼= π −1 (C∗ )∗over C ⊂ C.
We assume −KX is Q-Cartier.
The Donaldson-Futaki invariant DFT (π) of a special degeneration
π : X → C for X ↶ T with respect to the K-optimal vector ξ ∈ NR isgiven byd
DFT (π) := Futξ′ ( λ(t))
dtwhere λ : C∗ → AutT (X0 ) is the 1-psg on the central fibre X0 generated
by the C∗ -action on π : X → C.
There is also an algebraic expression of DFT .
Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

A Q-Fano T -variety X is said to be
K-semistable if for any special degeneration π : X → C for X ↶ T ,DFT (π) ≥ 0.
K-polystable if it is K-semistable and DFT (π) = 0 implies there is a
T -equivariant isomorphism X ∼= X × C.
K-stable if it is K-polystable and Aut0T (X ) = T .
Note here we use DFT modified by the K-optimal vector ξ, which is
diﬀerent from the usual Donaldson-Futaki invariant when ξ ̸= 0.
Theorem (Datar-Sékelyhidi ’15, Chen-Sun-Wang ’15 + BW ’14)
A Fano T -manifold X admits a Kähler-Ricci soliton (g , ξ ′ ) withξ ′ ∈ Lie(T ) ⊂ η(X ) if and only if X ↶ T is K-polystable.
(⇒ A Fano manifold X admits a KR soliton iﬀ there exists some torus action
X ↶ T so that it is K-polystable. )Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

Complex analytic moduli

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

Equivariant formulation is essential
The existence of ‘canonical metrics’ is still not suﬃcient :(There exists an isotrivial degeneration π : X → ∆ of some KE-Fano
manifold X with the following property.
Trivial away from the origin π −1 (∆∗ ) ∼= ∆∗ × X .
The central fibre X0 ≇ X admits Kähler-Ricci soliton with ξ0′ ̸= 0.
This implies that, without our equivariant formulation, X0 will not be
separated from X (in the “naive moduli space of Fano manifolds withKähler-Ricci solitons”, which algebraic geometers might naively imagine),though both of them admit ‘canonical metrics’.
This is not Tξ0′ -equivariant family.
We can show that our equivariant formulation will exclude such examples.

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

Definition (Moduli stack KT (M, ω))
The moduli stack of K-semistable Fano T -manifolds, denoted by
KT (M, ω), is a category with
Object a T -equivariant family X → S of K-semistable Fano
T -manifolds over a complex space S where the underlying
C ∞ -symplectic structure of fibres are (M, ω) ↶ T R ,Morphism (f , ϕ) : (S, X ) → (S ′ , X ′ ) where ϕ is a fibrewise
isomorphism over f .
We have a forgetful functor KT (M, ω) → Can : (X → S) 7→ S, whichmakes KT (M, ω) into a stack over Can and actually Artin analytic.
A morphism from CanS to KT (M, ω) correspond to a T -equivariant
family X → S of K-semistable Fano T -manifolds.
A morphism from KT (M, ω) to CanM gives a ‘functorial’ way toassign a morphism S → M to each family X → S in KT .
Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

Definition of moduli space in terms of the moduli stack

Definition
A complex space K with a morphism KT (M, ω) → CanK is called themoduli space of KT (M, ω) if for any morphism KT (M, ω) → CanS we
have a unique holomorphic morphism K → S completing the following
commutative diagram.
KT (M, ω)KS

The moduli space is unique (up to bihol.) if it exists.

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

Main theorem

Theorem (I. ’17)
The moduli space KT (M, ω) → K exists for any (M, ω) ↶ T withω ∈ c1 (M, ω) (possibly empty).
A morphism from a ‘point’ Canpt to KT (M, ω) corresponds to abiholomorphism class of K-semistable Fano T -manifold.
On the other hand, points of K corresponds to biholomorphism classes ofK-polystable Fano T -manifolds (= Fano manifolds admitting KR-soliton).

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

Our moduli spaces look like GIT
Our moduli space KT (M, ω) → K is (étale) locally isomorphic to the
GIT quotient [HT1 (X , Θ)/AutT (X )] → HT1 (X , Θ) AutT (X ).
Actually, we construct the moduli space KT (M, ω) → K by gluing them
together where X runs all K-polystable Fano T -manifolds.
Here is an incomplete picture of [C2 /C∗ ] → C2
on C2 by (z1 , z2 ).t = (z1 .t, z2 .t −1 ).

Eiji Inoue （瑛二, 井上）C∗ = C where C∗ acts

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

Consistency
Proposition (I. ’17)
The following are naturally identified as topological spaces.
The moduli space K of KT (M, ω).
−1
The symplectic reduction sξ,int(0)/HamT for the K-optimal vector ξ.

The space KRGH,T (M, ω) = { bihol. classes of Fano manifolds
admitting KR solitons, Tξ -equivariantly diﬀeo. to (M, ω, T ) }endowed with the Gromov-Hausdorﬀ topology.
To show this, we use the uniqueness of the moduli space K ( funny! :) ).
In the proof, we also show that the space KRGH (n) consisting of
n-dimensional Fano manifolds admitting KR solitons is a finite disjoint
union of some KRGH,Ti (Mi , ωi ) with dimR Mi = 2n.
Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons

Introduction
Diﬀerential geometric side
Algebro-geometric side

K-stability
Complex analytic moduli

Thank You !

Eiji Inoue （瑛二, 井上）

Moduli space of Fano manifolds admitting Kähler-Ricci solitons