µ-cscK metric and µK-stability (Eiji Inoue)µ-cscK metric and µK-stability of polarized
manifolds
Eiji Inoue （瑛二, 井上）

12, June, 2020 for Géométrie et Topologie seminar via Zoomµ-cscK metric and µK-stability (Eiji Inoue)

Contents:
1

Introduction to µ-cscK metrics – special features

2

How to formulate µK-stability? – equivariant calculus

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

1. 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) + □θDefinition
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 ω ∈ 2πλ−1 c1 (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 mapon J (M, ω). Namely, the map S : J (M, ω) → Lie(Ham(M, ω))∨ givenby∫⟨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. ’18, Lahdili ’18)
(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 nM

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 features

Characterization of vector fields: µλ -entropyVolλ (−2ξ)(n!e n )λ∫∫∫n¯ ξ )e θξ ω n(s + □θ(n + θξ )e θξ ω nθξ ωX ∫X∫ θ+−=−λλlogeθnnn!
e ξωe ξωXXXµλ (−2ξ) := − log∫µ =−λX∫∫ω+µ¯(Ric + □µ)e(ω + µ)e ω+µX ∫∫+λ− λ loge ω+µe ω+µe ω+µX
X
X

The functional µλ depends only on [ω].
Proposition: µλ -entropy/µλξ -Futaki invariant (I. ’19)
∃ µλξ -cscK metric in [ω] ⇒ ξ is a critical point of µλ[ω] .

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

Properties of µλ -entropy
Proposition (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
Proposition (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 )XXXX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 −xx2 sinh x) + λ(−1 +) − λ log.
tanh xtanh xxλfreeze (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 −xx2 sinh x) + λ(−1 +) − λ log.
tanh xtanh xxλfreeze (CP 1 , −KCP 1 ) = 4π.
There actually exists a µλξ -cscK metric for exactly two ξ ̸= 0 (andξ = 0) when λ > 4π.
As λ → ∞, the family of (non-cscK) µλ -cscK metrics converges onC ⊂ CP 1 , which looks like parabolic antenna.
As λ → ∞, the family of (non-cscK) µλ -cscK metrics ωλ admits afamilyof 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)
cscK × KRS (eg. toric Fano) ... ruled manifold over cscK manifold?

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

New!: 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 ).

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

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

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

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

Review on Donaldson-Futaki invariant
Recall, for a normal test configuration (X /C, L) of (X , L), theDonaldson–Futaki invariant is given byDF (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)µK-stability

Review on Donaldson-Futaki invariant
Recall, 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) ofDonaldson–Futaki invariants. (cf. Hilbert-Mumford criterion)
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 .

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

K-stability in cscK (KE) case

Theorem (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))
The Kähler class λc1 (X ) admits a cscK metric (KE metric) ⇐⇒(X , −λKX ) is K-(poly)stable.

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

K-stability and moduli problem
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 for every
C× -equivariant morphism f : C → B, the weight×−c1C (f ∗ CM(X , L)) ∈ HC2 × (C, Z) ∼= Z.η ∨ is equal to DF (f ∗ X , f ∗ L).
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)µK-stability

K-stability and moduli problem
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 for every
C× -equivariant morphism f : C → B, the weight×−c1C (f ∗ CM(X , L)) ∈ HC2 × (C, Z) ∼= Z.η ∨ is equal to DF (f ∗ X , f ∗ L).
Theorem (Odaka, Li-Wang-Xu)
Q-smoothable Fano varieties with Kähler–Einstein metrics form a properalgebraic moduli space.
Theorem (I. ’19)
Fano manifolds with Kähler-Ricci solitons (= µ2π -cscK metric in c1 (X ))form a complex analytic moduli space.
Can we compactify the moduli space (or make it algebraic)?

µ-cscK metric and µK-stability (Eiji Inoue)µK-stabilityµFutaki invariant(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:)(L̄TL̄TLTT LT.e)·(e)−(κ.e)(e)Evξ (κT1XX̄ /CPFut0ξ (X , L) := 4π∈ R.
(Evξ (e LT ))2
When X is smooth, this is equivalent to:∫∫∫ Ω+Θ∫ Ω+Θξξ¯ Θ̃ξ )e Ω+Θξ¯ ξ )e ω+θξ(Ricrel+□e− X (Ricω + □θeΩ̃X̄
X ∫X̄
Ω̃,−2ω+θ2ξ)( 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.
Proposition (I. ’20)
12

For smooth X , the µλξ K-semistability of (X , L) with respect togeneral test configurations is equivalent to the µλξ K-semistability
with respect to smooth test configurations (test configurations withsmooth total space X ).
For smooth test configuration, Futλξ is equivalent to one of Lahdili’s
weighted Futaki invariants.

Cororally (Essentially, Lahdili’s result on weighted cscK ’19)
If a smooth T -polarized manifold (X , L) admits a µλξ -cscK metric in
c1 (L), then (X , L) is µλξ K-semistable (with respect to general test
configurations).

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:
1Naturality: f ∗ Dξ µλ (X /B, L) = Dξ µλ (X ′ /B ′ , L′ ) for
X′B′

23X
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 for a polarized family is Zariski
open on the base. Then we get the following result on algebraicity.
Cororally
The moduli space of Fano manifolds with Kähler–Ricci solitons is analgebraic 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)e0µ = − X ∫ ω+µ.
eX
∫
∫
ω+µ¯
Both X (Ric + □µ)e
and X e ω+µ are the integration of equivariantforms. 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+µon t. For a G -equivariant polarized family (X /B, L), we puteXµ0X /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 diﬀerentialoperationDξ : 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. Theelement 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, wecan see Dξ µλX /C,L = Futλξ (X , L).η ∨ .

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 diﬀerentialoperationDξ : 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. Theelement 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 ∨ . Thank you for listening!

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

Latest news: 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 stability
NA‘D NAg (ϕ) ≥ γ · J g (ϕ)’ and proved the equivalence of G -uniformg-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 NAg . (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.