µ-cscK metric and µK-stability (Eiji Inoue)µ-cscK metric and µK-stability of polarized
manifolds
井上 瑛二 （東大数理）13, July, 2020, Geometric Complex Analysis seminar, Tokyoµ-cscK metric and µK-stability (Eiji Inoue)

Contents:
1

Background: cscK metrics & K-stability

2

Background: Kähler–Ricci solitons & modified K-stability

3

Introduction to µ-cscK metrics – special features

4

How to formulate µK-stability? – equivariant calculus

5Bonus talk: Beyond YTD conjectureµ-cscK metric and µK-stability (Eiji Inoue)
Background: cscK metrics & K-stability

1. Background: cscK metrics & K-stability

µ-cscK metric and µK-stability (Eiji Inoue)
Background: cscK metrics & K-stability

cscK metrics and Kähler–Einstein metrics
For a polarized manifold (X , L), when the Kähler class c1 (L) admits aKähler metric ω with constant scalar curvature (cscK metric)?
Kähler–Einstein metric: When λc1 (L) = 2πc1 (X ) for some λ ∈ R, then ωis cscK iﬀ it satisfies Ric(ω) = λω.
(λ < 0) KX > 0 ⇒ ∃ unique KE metric.
(λ = 0) KX ≡ 0 ⇒ ∃ unique Ricci flat metric in any L.
(λ > 0) KX < 0 ⇒ Futaki invariant is an obstruction.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: cscK metrics & K-stability

Yau–Tian–Donaldson conjecture

Yau–Tian–Donaldson conjecture∃ cscK metrics in c1 (L) ⇐⇒ (X , L) is K-‘poly’stable.
cf. Kobayashi–Hitchin correspondence (Donaldson, Uhlenbeck–Yau’s theorem)

For a normal test configuration (X /C, L) of (X , L), theDonaldson–Futaki invariant is given by
DF (X , L) := (KX̄ /CP 1 .L·n ) −n (KX .L·(n−1) ) ·(n+1)
(L̄
).
n+1
(L·n )The K-(semi)stability of (X , L) is the positivity (non-negativity) of
Donaldson–Futaki invariants. cf. Hilbert-Mumford criterionµ-cscK metric and µK-stability (Eiji Inoue)
Background: cscK metrics & K-stability

Donaldson–Fujiki moment map picture

Donaldson-Fujiki moment map picture(M, ω): C ∞ -symplectic manifold. Scalar curvature gives a moment map
on J (M, ω). Namely, the map S : J (M, ω) → Lie(Ham(M, ω))∨ given
by∫⟨S(J), f ⟩ =(s(gJ ) − s̄)f ω n
M

is a unique moment map for the symplectic structure Ω on J (M, ω):∫ΩJ (A, B) =(JA, B)gJ ω n .
Mµ-cscK metric and µK-stability (Eiji Inoue)
Background: cscK metrics & K-stability

Kempf–Ness theorem: model of YTD conjecture
Let (B, Ω + ν) ⟲ K be a projective manifold with a Hamiltonian actionof compact Lie group K .
G = K c : the complexification
LG := [Ω + ν] ∈ HK2 (B, R) = HG2 (B, R)η ∨ ∈ HC2 × (C, R): the positive generator
Kempf–Ness theorem (+ Hilbert–Mumford criterion)
For b ∈ B,
(Semistability) ν −1 (0) ∩ b.G ̸= ∅ ⇐⇒ for every Λ : C× → G−Λ∗b LG /η ∨ = −⟨ν(Λb (0))c , Λ∗ η⟩ ≥ 0.
(‘Poly’stability) ν −1 (0) ∩ b.G ̸= ∅ ⇐⇒ if moreover Λ∗b LG /η ∨ = 0
only when Λ : C× → Gx .
Semistability is Zariski open condition, while polystability is not so.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: cscK metrics & K-stability

Uniqueness and Existence

Theorem (Berman-Berndtsson)
CscK metrics in c1 (L) are unique modulo Aut0 (X , L).
Theorem (Bando–Mabuchi, Stoppa, Berman-Darvas-Lu, et al.)If the Kähler class c1 (L) admits a cscK metric, then (X , L) is
K-‘poly’stable.
Theorem (Chen-Donaldson-Sun, Tian, (Aubin, Yau, Odaka))
When −KX ∈ R.L, the Kähler class c1 (L) admits a cscK metric (KEmetric) if (and only if) (X , L) is K-‘poly’stable.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: cscK metrics & K-stability

Moduli space of Kähler–Einstein Fano varieties
Theorem (Paul-Tian)
For a G -equivariant family (X , L) → B of polarized schemes, there exists
a G -equivariant line bundle CM(X , L) on B such that×−c1C (f ∗ CM(X , L)) = DF (f ∗ X , f ∗ L).η ∨ ∈ HC2 × (C, Z) ∼= Z.η ∨

for every C× -equivariant morphism f : C → B.
+
Donaldson–Sun, Chen–Donaldson–Sun, Tian⇓
Theorem (Odaka, Li-Wang-Xu)
Q-smoothable Fano varieties with Kähler–Einstein metrics form a proper
algebraic moduli space.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

2. Background: Kähler–Ricci solitons & modified K-stability

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Kähler–Ricci soliton
Examples:Fano manifold X = k-point blow up of CP n (k = 1, . . . , n) does not
admit KE metrics.
A toric Fano manifold admits a KE metric iﬀ the barycenter of Fano
polytope is the origin.
There are infinitely many toric Fano orbifolds with no KE metrics,
while toric Fano orbifolds admitting KE metrics are finite in eachdimension.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Kähler–Ricci soliton
Examples:Fano manifold X = k-point blow up of CP n (k = 1, . . . , n) does not
admit KE metrics.
A toric Fano manifold admits a KE metric iﬀ the barycenter of Fano
polytope is the origin.
There are infinitely many toric Fano orbifolds with no KE metrics,
while toric Fano orbifolds admitting KE metrics are finite in eachdimension.
Kähler–Ricci soliton: Ric(ω) − LJξ ω = λω
cf. normalized Kähler–Ricci flow: Ric(ωt ) − λωt = ω̇tEvery toric Fano orbifold admits a KRs.
Every horospherical Fano manifold admits a KRs, which includes
infinitely many Fano manifolds with ρ(X ) = 1 & no KE metrics.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Tian–Zhu’s volume minimization and modified K-stability
For a Fano manifold X ⟲ T and ξ ∈ t, the modified Futaki invariant
Futξ ∈ t∨ is defined by∫
Futξ (η) := −θη e θξ ω n ,X

where θξ = −2µξ for [ω + µ] ∈ c1T (X ). Independent of ω ∈ c1 (X ).
∃ KRs ⇒ Futξ = 0
Proposition (Tian–Zhu)
Regardless of the existence of KRs, ∃!ξ ∈ t satisfying Futξ = 0.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Tian–Zhu’s volume minimization and modified K-stability
For a Fano manifold X ⟲ T and ξ ∈ t, the modified Futaki invariant
Futξ ∈ t∨ is defined by∫
Futξ (η) := −θη e θξ ω n ,X

where θξ = −2µξ for [ω + µ] ∈ c1T (X ). Independent of ω ∈ c1 (X ).
∃ KRs ⇒ Futξ = 0
Proposition (Tian–Zhu)
Regardless of the existence of KRs, ∃!ξ ∈ t satisfying Futξ = 0.
Modified K-(semi)stability of X with respect to ξ: For a T -equivariant
special degeneration X = (X /C, −KX /C ), the modified Futaki invariantof X is given by∫
Futξ (X ) := −θη e θξ .
X0

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Uniqueness and Existence

Theorem (Tian–Zhu)
Kähler–Ricci solitons on a Fano manifold are unique modulo Aut0 (X )(and up to scaling).
Theorem (Berman–Witt-Nyström)
If a Fano manifold admits a Kähler–Ricci soliton, then X is modified
K-polystable.
Theorem (Datar–Székelyhidi)
A Fano manifold X admits a Kähler–Ricci soliton if (and only if) X is
modified K-polystable.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Moduli space of KRs Fano manifolds
Berman–Witt-Nyström+
Moment map picture for KRs + Uniqueness of polystable degeneration⇓
Theorem (I. ’19, Adv. Math.)
Fano manifolds with KRs form a complex analytic moduli space.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Moduli space of KRs Fano manifolds
Berman–Witt-Nyström+
Moment map picture for KRs + Uniqueness of polystable degeneration⇓
Theorem (I. ’19, Adv. Math.)
Fano manifolds with KRs form a complex analytic moduli space.
↓
Theorem (Dervan–Naumann)
CscK manifolds form a complex analytic moduli space.

µ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Summaryµ-cscK metric and µK-stability (Eiji Inoue)
Background: Kähler–Ricci solitons & modified K-stability

Summaryµ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

3. Introduction to µ-cscK – special featuresµ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special featuresµ-scalar curvature: definition
X ⟲T ∼= (U(1))×k : holomorphic action on a complex (Kähler) manifoldµ-scalar curvature
For λ ∈ R and ξ ∈ t and a T -equivariant Kähler metric ω + µ, we putsξλ (ω) := (s(ω) − ∆µξ ) − (∆µξ + 2|∇µξ |2 ) + 2λµξ¯ ξ ) + (□θ¯ ξ − (Jξ)θξ ) − λθξ .
= (s(ω) + □θ

Definition
A Kähler metric ω is a µλξ -cscK metric if sξλ (ω) is constant.
Independent of the choice of the moment map µ for ω.
µλ0 -cscK metric ⇐⇒ cscK metric.
When λω ∈ 2πc1 (X ),µλξ -cscK metric ⇐⇒ Kähler-Ricci soliton: Ric(ω) − LJξ ω = λω.

µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special featuresµ-scalar curvature: “naturality” of the concept
RecallDonaldson-Fujiki moment map picture(M, ω): C ∞ -symplectic manifold. Scalar curvature gives a moment map
on J (M, ω). Namely, the map S : J (M, ω) → Lie(Ham(M, ω))∨ given
by∫⟨S(J), f ⟩ =(s(gJ ) − s̄)f ω n
M

is a moment map for the symplectic structure Ω on J (M, ω):∫ΩJ (A, B) =(JA, B)gJ ω n .
Mµ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special featuresµ-scalar curvature: “naturality” of the concept∫Puts̄ξλ :=sξλ (gJ ) e θξ ω nM/∫e θξ ω n .

M

Proposition (Moment map picture for µ-cscK, I. ’19, Lahdili ’19)(M, ω) ⟲ T : C ∞ -symplectic manifold. µ-scalar curvature gives amoment map on JT (M, ω). Namely, the map
Sξλ : JT (M, ω) → Lie(HamT (M, ω))∨ given by∫⟨Sξλ (J), f ⟩ =(sξλ (gJ ) − s̄ξλ )f e θξ ω n
M

is a moment map for the symplectic structure Ωξ on JT (M, ω):∫(JA, B)gJ e θξ ω n .
Ωξ,J (A, B) =Mµ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special featuresµ-Futaki invariant and µ-entropy
For ξ ∈ t, the µ-Futaki invariant Futλξ ∈ t∨ is defined by∫/∫Futλξ (η) := −⟨Sξλ (J), θη ⟩ = − (sξλ (ω) − s̄ξλ )θη e θξ ω ne θξ ω n .
X

Independent of ω ∈ [ω] and µ : X → t∨ .
∃ µλξ -cscK metric in [ω] ⇒ Futλξ = 0X

µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special featuresµ-Futaki invariant and µ-entropy
For ξ ∈ t, the µ-Futaki invariant Futλξ ∈ t∨ is defined by∫/∫Futλξ (η) := −⟨Sξλ (J), θη ⟩ = − (sξλ (ω) − s̄ξλ )θη e θξ ω ne θξ ω n .
X

X

Independent of ω ∈ [ω] and µ : X → t∨ .
∃ µλξ -cscK metric in [ω] ⇒ Futλξ = 0∫µ (−2ξ) = −
λ

X

∫
∫
n
¯ )e θξ ω n(s + □θ(n + θξ )e θξ ω n
θξ ω
X ∫
∫ θξ
+
−
λ
λ
log
e
n!
e ξ ωne θξ ω n
X
X
X

Also independent of ω ∈ [ω] and µ : X → t∨ .
Dξ µλ = Futλξµ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Properties of µλ -entropy
Theorem (I. ’19)
(Existence) Critical points of µλ always exist regardless of theexistence of µλξ -cscK metrics in [ω].
(Uniqueness/phase transition) For each X ⟲ T ,{λfreeze := sup λ ∈ R
is always finite (never ±∞).

′}
µλ admits a unique′critical point for every λ ≤ λµ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Properties of µλ -entropy
Theorem (I. ’19)
(Existence) Critical points of µλ always exist regardless of theexistence of µλξ -cscK metrics in [ω].
(Uniqueness/phase transition) For each X ⟲ T ,{λfreeze := sup λ ∈ R′}µλ admits a unique′critical point for every λ ≤ λis always finite (never ±∞).
(Extremal limit) Let ξ λ be the unique critical point of µλ for
λ < λfreeze . Then λξ λ converges to the extremal vector field ξext as
λ tends to −∞.
The extremal vector field ξext is the unique critical point of∫∫∫∫(ŝ(ω) − θ̂ξ )2 ω n −ŝ 2 ω n . (fˆ := f −f ωn /
ωn )
X

X

X

X

µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Behavior of µλ -entropy: typical example
We can explicitly compute µλ of CP 1 ⟲ U(1). For ξ = x.η ∈ u(1),µλ−KCP 1 (ξ) = 2π(1 −xx
2 sinh x) + λ(−1 +) − λ log.
tanh xtanh xx

λfreeze (CP 1 , −KCP 1 ) = 4π.

µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Behavior of µλ -entropy: typical example
We can explicitly compute µλ of CP 1 ⟲ U(1). For ξ = x.η ∈ u(1),µλ−KCP 1 (ξ) = 2π(1 −xx
2 sinh x) + λ(−1 +) − λ log.
tanh xtanh xx

λfreeze (CP 1 , −KCP 1 ) = 4π.
There actually exists a µλξ -cscK metric for exactly two ξ ̸= 0 (and
ξ = 0) when λ > 4π.
As λ → ∞, the family of (non-cscK) µλ -cscK metrics ωλ admits afamily of diﬀeomorphisms fλ : D 2 → C ⊂ CP 1 from a disk of radius√
2 such that fλ∗ ωλ converges to the flat metric. (while fλ does notconverge to a diﬀeomorphism onto C. )

µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special featuresµ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Closedness of framework(Scaling) ω: µλξ -cscK metric ⇒ c −1 ω: µcλcξ -cscK metric.
(Product) (X , ωX ), (Y , ωY ): µλ -cscK metrics with the same λ andwith respect to vector fields ξX , ξY , respectively ⇒(X × Y , ωX ⊕ ωY ): µλ -cscK metric with respect to ξX ⊕ ξY .

µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Closedness of framework(Scaling) ω: µλξ -cscK metric ⇒ c −1 ω: µcλcξ -cscK metric.
(Product) (X , ωX ), (Y , ωY ): µλ -cscK metrics with the same λ andwith respect to vector fields ξX , ξY , respectively ⇒(X × Y , ωX ⊕ ωY ): µλ -cscK metric with respect to ξX ⊕ ξY .
(Perturbation of λ) ∃ µλ -cscK metric in [ω] with λ < λ1 for the first
eigenvalue λ1 of ∆ − ∇µξ ⇒ ∃ µλ̃ -cscK metric in the same [ω] forλ̃ ∈ (λ − ϵ, λ + ϵ).
(Perturbation of Kähler class) We can also perturb Kähler classes
under the above condition.

µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Closedness of framework(Scaling) ω: µλξ -cscK metric ⇒ c −1 ω: µcλcξ -cscK metric.
(Product) (X , ωX ), (Y , ωY ): µλ -cscK metrics with the same λ andwith respect to vector fields ξX , ξY , respectively ⇒(X × Y , ωX ⊕ ωY ): µλ -cscK metric with respect to ξX ⊕ ξY .
(Perturbation of λ) ∃ µλ -cscK metric in [ω] with λ < λ1 for the first
eigenvalue λ1 of ∆ − ∇µξ ⇒ ∃ µλ̃ -cscK metric in the same [ω] forλ̃ ∈ (λ − ϵ, λ + ϵ).
(Perturbation of Kähler class) We can also perturb Kähler classes
under the above condition.
(Propagation) ∃ extremal metric in [ω] ⇒ µλ -cscK metric in thesame [ω] for λ ≪ λfreeze and also for λ ≫ λfreeze .
(Uniqueness) Convexity of weighted Mabuchi functional shows that
µλ -cscK metrics are unique for λ < λfreeze . (Lahdili)µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Calabi ansatz on PΣ (L ⊕ O)
Consider the ruled manifold PΣ (L ⊕ O) for a positive L on an algebraic
curve Σ. Let F denote a fibre and B = {(x, (0 : 1)) | x ∈ Σ} denote thesection at infinity. The Kähler cone is given by{aF + bB | b > 0,

a
degL
>−
}.
b
2

Theorem (I. ’20)
Every Kähler class in the cone {aF + bB | a, b > 0} admits a µλ -cscK
metric for every λ ≥ 0 (for some ξ).
For g (Σ) ≥ 2 and small ba , the Kähler class aF + bB does not admit
extremal metrics. (rel. K-unstable ⇒ no µλ -cscK metrics for λ ≪ 0. )

µ-cscK metric and µK-stability (Eiji Inoue)
Introduction to µ-cscK metric – special features

Calabi ansatz on CP 2 #CP 2 = PCP 1 (O(1) ⊕ O)
The anti-canonical class −KX of X = CP 2 #CP 2 = PCP 1 (O(1) ⊕ O)
admits both KRs and extremal metric (no cscK metrics).
Calabi ansatz: ∃ µλ -cscK metrics for every λ ∈ R (with a negativexλ = ξ λ /η = (6/11) · ξ λ /ξext ) connecting KRs and extremal metric.

We can see 2.9 × 2π < λfreeze < 3 × 2π.

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

4. How to formulate µK-stability? – equivariant calculus

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Remarks
RecallDF (X , L) := (KX̄ /CP 1 .L·n ) −n (KX .L·(n−1) ) ·(n+1)
(L̄
).
n+1
(L·n )

In moduli context, test configurations appear by pulling back theuniversal family U on Hilb along C× -equivariant morphisms
C → Hilb, which is not necessarily normal.
We can define DF also for non-normal (X , L) by using homology
Todd class τ (OX̄ ) = [X̄ ] − 12 κX̄ + · · · ∈ AQ (X̄ ) instead of KX̄ . (cf.
Fulton, Edidin-Graham)
The intersection formula is useful to see the behavior of DF (X , L)along the normalization and resolutions of X .
Recall
µ =−
λ

∫
X

∫
∫
ω+µ
¯
(Ric + □µ)e(ω + µ)e ω+µ
X
∫
∫
e ω+µ
+λ
− λ log
e ω+µ
e ω+µ
X
X
X

µ-cscK metric and µK-stability (Eiji Inoue)µK-stabilityµFutaki invariant of test configuration(X , L): T -equivariant polarized manifold (scheme)
For ξ ∈ t, we define the µ0ξ -Futaki invariant of a T -equivariant test
configuration (X , L) by the following equivariant intersection formula:
)
(
L̄T
L̄T
LT
T LT
.e
)
·
(e
)
−
(κ
.e
)(e
)
Evξ (κT
1
X
X̄ /CP
Fut0ξ (X , L) := 4π∈ R.
(Evξ (e LT ))2
When X is smooth, this is equivalent to:
∫
∫
∫ Ω+Θ
∫ Ω+Θ
ξ
ξ
¯ Θ̃ξ )e Ω+Θξ¯ ξ )e ω+θξ
(Ricrel+□e− X (Ricω + □θ
e
Ω̃
X̄
X ∫
X̄
Ω̃
,
−2
ω+θ
2
ξ)
( Xe
= Ric(Ω̃) − π ∗ Ric(ωCP 1 ) for some metrics Ω̃, ωCP 1 on
where RicrelΩ̃1X̄ , CP .
We can similarly define
Futλξ (X , L) := Fut0ξ (X , L) + λ(equiv. intersection on L̄).

µ-cscK metric and µK-stability (Eiji Inoue)µK-stabilityµK-semistability
A T -polarized manifold is µλξ K-semistable if Futλξ is non-negative for any
test configuration.
Theorem (I. ’20 + Lahdili ’19)
If c1 (L) admits a µλξ -cscK metric, then (X , L) is µλξ K-semistable.
If (X , L) is µλ K-semistable for λ ≪ 0, then (X , L) is relatively
K-semistable.
Recall:If (X , L) admits and extremal metric, then it admits µλ -cscK metrics
for λ ≪ 0 and λ ≫ 0.
Even if (X , L) is µλ K-semistable for λ ≫ 0, (X , L) may not be
relatively K-semistable.

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Generalization of CM line bundle
Theorem (I. ’20)
For λ ∈ R and ξ ∈ t, there exists a characteristic class Dξ µλ assigningDξ µλ (X /B, L) ∈ HG2 (B, R) for each T × G -equivariant family ofpolarized schemes (X /B, L) over smooth G -variety B which enjoys thefollowing:
1

Naturality: f ∗ Dξ µλ (X /B, L) = Dξ µλ (X ′ /B ′ , L′ ) for
X′
B′

2

3

X
f

B

µ-Futaki invariant: Dξ µλ (X /C, L) = Futλξ (X , L).η ∨ for any
T -equivariant test configuration (X , L)
CM line bundle: D0 µλG (X /B, L) = − (L4π·n ) c1G (CM(X /B, L))µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Application

Combining with Chen–Sun’s deep analysis on Kähler–Ricci flow and theanalytic openness of µK-semistable locus established in the previous
work, we can show that µK-semistable locus of a Q-Fano family is Zariski
open on the base. Then we get the following result on algebraicity.
Theorem (I. ’20)
The moduli space of Fano manifolds with KRs is an algebraic space.
I also have a plan for compactifying the moduli space (in progress).

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Idea of construction – the case λ = 0 (to economize space)
Recall the following expression of µλ -entropy:∫
ω+µ¯(Ric + □µ)e0
µ = − X ∫ ω+µ
.
e
X
∫
∫
ω+µ
¯
Both X (Ric + □µ)e
and X e ω+µ are the integration of equivariant
forms. In other words, we can regard these as the pushforward of theequivariant cohomology classes
KXT ⌢ e LT , e LT ∈ ĤT (X , R) :=∞∏HT2k (X , R)

k=0

∼ ∏∞ k ∨ and arealong p : X → pt, which are elements of ĤT (pt, R)∫ = k=0 S t ω+µ¯
theexpansion (at 0 ∈ t) of the functionals X (Ric + □µ)e,
∫ Taylor
ω+µ
on t. For a G -equivariant polarized family (X /B, L), we put
eXµ0X /B,L := 2ππ∗ (κX /B .e L )∈ ĤG (B, R).
π∗ (e L )

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Idea of construction – Sketch of equivariant calculus
1(Diﬀerential at ξ along G ) For ξ ∈ t, we introduce a diﬀerential
operationDξ : HTω×G (B, R) → HG2 (B, R)
for some subring HTω×G (B, R) of ĤT ×G (B, R) where T acts on B
trivially. When G = {1} and B = pt, HTω×G (B, R) is identified withthe ring of real analytic functions on t.

2(Convergence result) For T × G -equivariant polarized family(X /B, L), we can show that µλX /B,L is in HTω×G (B, R), using
Cartan model of equivariant deRham current homology. The
element Dξ µλX /B,L ∈ HG2 (B, R) is what we want!

3(Equivariant Grothendieck-Riemann-Roch) Naturality and theidentification with CM line bundle comes from the equivariant
Grothendieck-Riemann-Roch theorem by Edidin-Graham.

4(Localization formula) Using the equivariant localization formula, we
can see Dξ µλX /C,L = Futλξ (X , L).η ∨ .

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Thank you for listening!

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

5. Bonus talk: Beyond YTD conjectureµ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Beyond YTD conjecture: Perelman’s µ-entropy
Define µ̌λ : H(X , [ω]) → R byµ̌λ (ω) :=supW̌ λ (ω, f )f ∈C ∞ (X )

where

∫
¯ −λ(n+f ))e f ω n(s(ω)+□fW̌ λ (ω, f ) :=/∫X

e f ω n −λ log

(1 ∫

X

n!

)
e f ωn .

X

Theorem (to appear)
The functional µ̌λ is smooth when λ ≤ 0 (mountain pass theorem).
Its critical points of µ̌λ are precisely µλ -cscK metrics (for some ξ).
The critical points are global minimizer.
A Kähler metric ω is µλξ -cscK metric iﬀ µ̌λ (ω) = µλ (ξ).

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Beyond YTD conjecture: Non-archimedean µ-entropyµλNA : HNA (X , L) → R
Theorem (Essentially proved in I. ’20)
If a ‘test configuration’ ϕξ ∈ HNA (X , L) associated to a vector field
λmaximizes µNA, then (X , L) is µλξ K-semistable.
Conjecture
sup µλNA ≤ inf µ̌λX admits a µλ -cscK metric ω⇐⇒ µ̌λ (ω) = inf µ̌λ ⇐⇒ µ̌λ (ω) = sup µλNA .
X is µλ K-semistable with respect to ξ⇐⇒ µλNA (ϕξ ) = sup µλNA ⇐⇒ µλNA (ϕξ ) = inf µ̌λ .

µ-cscK metric and µK-stability (Eiji Inoue)µK-stability

Remarks: Han-Li’s result when L = −KX
Berman–Witt-Nyström proved that (X , −KX ) is µ2π K-polystablewith respect to special degenerations if X admits a KRs (=µ2π -cscK metric).
Recently, J. Han and C. Li introduced G -uniform g-Ding stabilityNA‘D NAg (ϕ) ≥ γ · J g (ϕ)’ and proved the equivalence of G -uniform
g-Ding stability of (X , −KX ) for ‘maximal’ G is equivalent to theexistence of KR g-soliton.
They also show that the (G -uniform) g -Ding stability of (X , −KX )is equivalent to that with respect to special degenerations, using
MMP with scaling. The proof works also for M NA
g . (I guess it works
λalso for Futξ . )g -Mabuchi stability for g = e ⟨ξ,−⟩ must be equivalent toµλξ K-stability. (λ is determined from ξ. )Thus, ∃ KRs on X ⇐⇒ (X , −KX ) is µ2π K-polystable.