Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)

Canonical metrics and Perelman’s entropy inKähler geometry
Eiji INOUE （RIKEN iTHEMS, Japan）

16–20, August, 2021, Conference on Complex Analysis and Geometry

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)

Contents

1

Canonical metrics and K-stability in Kähler geometry2

Brief introduction to µ-cscK metrics

3µ-cscK metric and Perelman’s entropy

4µK-stability and non-archimedean µ-entropy

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
1. Canonical metrics and K-stability in Kähler geometry

1. Canonical metrics and K-stability in Kähler geometry

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
1. Canonical metrics and K-stability in Kähler geometry

Extremal metric and Calabi functional
The Calabi functional C : H(X , L) → R is given by∫
1C (ω) :=ŝ(ω)2 ω n , ŝ(ω) = s(ω) − s̄
2 X
The critical points are extremal metrics: ∂ ♯ s(ω) = g i ȷ̄ sȷ̄ ∂i is holomorphic.
The notion generalizes cscK metric s(ω) = const., hence alsoKähler–Einstein metric Ric(ω) = λω.
For a compact complex manifold X and a Kähler class L = [ω],ω ∈ L can be KE metric only when c1 (X ) ∝ L or = 0. Conversely,
cscK metric in such Kähler class is automatically KE.
∫Extremal metric in L is cscK metric iﬀ Fut(X ,L) = − X ŝµ• ω n = 0.
Berman–Berndtsson, et al. Given (X , L), extremal metrics are uniquemodulo Aut0 (X ) if it exists.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
1. Canonical metrics and K-stability in Kähler geometry(Relative) K-stability
A text configuration (X , L) is a C× -equivariant flat family of polarized
ϖschemes (X , L) −→ C endowed with an isomorphism X × C∗ ∼= ϖ−1 (C∗ ).
Donaldson–Futaki invariant ⇐ moment map picture:
DF(X , L) := (KX̄ /P1 .L̄·n ) −(KX .L·n−1 ) ·n+1(L̄).
(n + 1)(L·n )Relative Donaldson–Futaki invariant: when (X , L) ⟲ T , for ξ ∈ tDFrelξ (X , L) := DF(X , L) +()−1 ∫1 n+2
c1,T (L̄; ξ)⌣n+2 .
2π2X̄

Székelyhidi, et al. If (X , L) has an extremal metric with ξ = Im(∂ ♯ s),then we haveDFrelξ (X , L) ≥ 0 for ∀(X , L),i.e. (X , L) is relatively K-semistable.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
1. Canonical metrics and K-stability in Kähler geometry

Donaldson’s inequality
Donaldson’s lower bound:sup −(X ,L)

2πDF(X , L)
≤ inf (2C (ω))1/2 ,ω∈L∥(X , L)∥

where we put(∫n+2∥(X , L)∥ = (t − b) DH(X , L) = −2R∫with the barycenter b := R tDH(X , L).
22)−1(L̄·n+2C× ; η) −(L̄·n+1 )2(n + 1)2

optimal destabilization conjecture
We have the equality
sup −(X ,L)

2πDF(X , L)
= inf (2C (ω))1/2∥(X , L)∥
ω∈H(X ,L)and the maximum of LHS is achieved by some test configuration.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
1. Canonical metrics and K-stability in Kähler geometryKähler–Ricci soliton
Kähler–Ricci soliton is a self-similar solution of Kähler–Ricci flow:
Ric(ω) − LJξ ω = λωfor some ξ ∈ iso(X , ω) with holomorphic ξ J = Jξ +

√

−1ξ.

The notion generalizes Kähler–Einstein metric Ric(ω) = λω.
ω ∈ L can be KRs only when c1 (X ) ∝ L or = 0 .
KRs is KE metric iﬀ Fut(X ,L) = 0, which holds for λ ≤ 0.
Usually, L = 2πc1 (X ), hence X is Fano.
Tian–Zhu, et al. Uniqueness modulo Aut0 (X ) (volume minimization)
Berman–Witt-Nyström, et al. If X admits a KRs with ξ, thenFutmod(X , L) ≥ 0 for ∀(X , L),ξi.e. X is modified K-semistable.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
1. Canonical metrics and K-stability in Kähler geometry

H-entropy
The H-entropy for ω ∈ L is defined by∫∫∫ωnh nh nH(ω) =he ω /e ω − logeh ,n!
XXX√¯ = Ric(ω) − ω. KRs are thewhere h is the Ricci potential −1∂ ∂hcritical points of H.
The following is a generalization of Tian–Zhu’s volume functional:∫ordE (L − L)
HNA (X , L) := − min (AX (vE ) +) − log e −t DH(X ,L)
E ⊂X0
ordE X0
RTheorem (Chen–Sun–Wang, Dervan–Székelyhidi, Han–Li, et al.)
We have the equality
sup HNA (X , L) = inf H(ω)(X ,L)

ω∈Land the maximum of LHS is achieved by a unique f.g. filtration.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
Brief introduction to µ-cscK metrics

2. Brief introduction to µ-cscK metrics

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
Brief introduction to µ-cscK metricsµ-cscK metric
CscK metric is a natural generalization of Kähler–Einstein metric.
What about Kähler–Ricci soliton?
In view of K-stability, we want moment map picture.
In view of optimal degeneration, we want volume minimization.
⇝ µ-cscK metricDefinition
A Kähler metric ω is called µ̌λξ -cscK metric if¯ ξ ) + (□µ¯ ξ + |∂ ♯ µξ |2 ) = λµξ + const.
sµξ := (s(ω) + □µλ−1µ2πλ= µ̌2πλξ−2ξ -cscK metric in λ c1 (X ) ⇔ KRs Ric(ω) − Lξ ω = 2π ω.

Extremal metric = the limit of µλ -cscK metrics as λ → −∞.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
Brief introduction to µ-cscK metrics

Moment map picture for µ-cscK metric(M, ω) ↶ T : Hamiltonian torus action. For ξ ∈ t and λ ∈ R, the mapµξ nSξλ : JT → ham∨
T : J 7→ (sµξ (gJ ) − λµξ )e ωgives a moment map for a symplectic manifold (JT , Ωξ ) ↶ HamT .
µλξ -Futaki invariant∫Futλξ (ζ) := −µζ (sµξ (ω) − λµξ )e µξ ω n .
X⇝ generalized to test configuration via equivariant intersection formula⇒ µλξ K-stability ⊃ modified K-stability
Theorem (Lahdili ’19, I. ’20, Apostolov–Jubert–Lahdili ’21)
If (X , L) admits a µ̌λξ -cscK metric, then it is µ̌λξ K-semistable.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
Brief introduction to µ-cscK metrics

Volume minimization: µ-Futaki invariant vanishing
Proposition (A generlaization of Tian–Zhu’s result. I. ’19)
(X , ω) ↶ K : Hamiltonian action. There is a functionalµ̌λ : k −→ R
ξ 7→ W̌ λ (ω, µωξ)
satisfying the following.
1

Its derivative at ξ ∈ k is Futλξ .

2

It is proper and bounded from above, hence attains a maximum,
which implies the existence of ξ with Futλξ = 0.

3The value′λice (X , L) := sup{λ ∈ R | µ̌λ has a unique crit. pt. for ∀λ′ < λ}is finite (never ±∞). ⇝ phase transition

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
3. µ-cscK metric and Perelman’s entropy

3. µ-cscK metric and Perelman’s entropy

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
3. µ-cscK metric and Perelman’s entropy

Perelman’s W -entropy
Perelman’s W -entropy
For a Kähler metricω ∈ H(X , L) and a smooth function f ∈ C ∞ (X , R)
∫normalized as X e f ω n /n! = 1, we put∫W̌ (ω, f ) := − (s(ω) + |∂ ♯ f |2 )e f ω n /n!,∫ X
Š(ω, f ) := (n + f )e f ω n /n!,XλW̌ (ω, f ) := W̌ (ω, f ) + λŠ(ω, f ).

W̌ λ : T H(X , L) = H(X , L) × C ∞ (X )/R → R.
1Original:(4πτ )n∫X()
τ (R(ω) + |∇f |2 ) − (2n − f ) e −f volgCanonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
3. µ-cscK metric and Perelman’s entropyµ-cscK metric is critical points of W -functional
Theorem (I. ’21)
A state (ω, f ) ∈ T H(X , L) is a critical point of W̌ λ if and only if ξ = ∇fis real holomorphic and ω is a µ̌λξ -cscK metric.
We putµ̌λ (ω) := ∫supW̌ λ (ω, f ).

e f ω n /n!=1X

Theorem (Rothaus ’81, I. ’21)
For each λ and ω, there exists f attaining the maximum of W̌ λ (ω, ·).
When λ ≤ 0, such f is unique. Consequently, µ̌λ is smooth on H(X , L)for λ ≤ 0. The critical points are precisely µλ -cscK metrics in this case.
∫The Perelman µ-entropy is analogous to Calabi ∫functional X ŝ 2 ω n /n! forextremal metric in a Kähler class or H-entropy X he h ω n /n! for KRs.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
3. µ-cscK metric and Perelman’s entropy

Algebraic bound on Perelman µ-entropy
For (X , L) = (X , −KX ), we have µ̌2π ≥ 2πH.
µ̌2π : Hanalogy∼Mabuchi : DingTheorem (I. ’21)
sup µ̌λNA (X , L; τ ) ≤(X ,L;τ )µ̌λ (ω)infω∈H(X ,L)

CororallyFor λ ≤ 0, if there is a µ̌λξ -cscK metric ω, we havesup µλNA (X , L; τ ) = µλNA (ξ) = µλ (ω) =(X ,L;τ )inf

ωφ ∈H(X ,L)µλ (ωφ ).

In particular, all the critical points of µλ are global minimizers, which are
precisely µλ -cscK metrics.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
3. µ-cscK metric and Perelman’s entropyCharacteristic µ-entropy
For a test configuration (X , L), we have the expansion∫∫×1−τ.t
LC× |X0eνm (t) =e −τ.t DH(X ,L) − (κC; τ ).m−1 + O(m−2 )X0 .e2RRforνm = −1 ∑dim H 0 (X0 , L⊗m0 )λ
mnm→∞−→DH(X ,L) .

λ∈ZFor τ ≥ 0,××
(KX .e L ) − τ.(KX̄log,C.e LC× ; τ )(κCX .e LC× |X0 ; τ )/P1∫µ̌NA (X , L; τ ) := 2π ∫ 0−τ.t= 2πeDH(X ,L)e −τ.t DH(X ,L)R∫ R∫−τ.t(n − τ.t)e
DH(X ,L)σ̌(X , L; τ ) := R ∫ −τ.t− log e −τ.t DH(X ,L)e
DH(X ,L)RR

λ(X , L; τ ) := µ̌NA (X , L; τ ) + λσ̌(X , L; τ )µ̌NACanonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
3. µ-cscK metric and Perelman’s entropy

Sketch of proofW̌ λ (ωϕτ.t , −ϕ̇τ.t ) is monotonically decreasing.
∫tConsider the action functional A(t) :=“− 0 W̌ λ (ωϕτ.s , −ϕ̇τ.s )ds”
and show its convexity by a tensor calculus on equivariant
diﬀerential forms and Berman–Berndtsson’s subharmonicity
argument. (cf. the convexity of Mabuchi functional)
limt→∞ W̌ λ (ωϕτ.t , −ϕ̇τ.t ) = µλNA (X , L; τ )The equivariant Stokes theorem and the regularity of geodesic ray
across the central fibre (Chu–Tosatti–Weinkove).
µ̌λ (ω) ≥ W̌ λ (ωϕτ.t , −ϕ̇τ.t ) → µλNA (X , L; τ )Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
3. µ-cscK metric and Perelman’s entropy

Universal aspect∫
H-entropy
Introduce

he h ω n− loge h ωnXH(ω) = ∫X∫e h ω n /n!
X∫∫
fe h ω ne f ω n /n!,Ľ(ω, f ) = ∫X h n − loge ωXXthen H(ω) = supf Ľ(ω, f ) and HNA (X , L; τ ) = limt→∞ Ľ(ωϕτ.t , ϕ̇τ.t )Calabi functional

C (ω) =∫

Introduce
Wext (ω, f ) = −X

∫ 2 n
1 X ŝ ω∫
2 X ωn∫ 2 n(ŝ − fˆ)2 ω nŝ ω∫+ X∫ n ,2 X ωn2 Xωthen C (ω) = supf Wext (ω, f ) and CNA (X , L; τ ) := limt→∞ Wext (ωϕτ.t , ϕ̇τ.t ),2(X ,L)2
when DF (X , L) ≤ 0.
for which we have maxτ ≥0 CNA (X , L; τ ) = (L2π·n ) DF∥(X ,L)∥2

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
4. µK-stability and non-archimedean µ-entropy

4. µK-stability and non-archimedean µ-entropy

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
4. µK-stability and non-archimedean µ-entropy

Characteristic µ-entropy and µK-semistability
We can extend µ̌λ to f.g. filtration of R =⊕mH0(X , L⊗m ).

Proposition (I. ’21 (to appear), essentially proved in I. ’20)
There is a family of f.g. filtrations {Fξ+τ (X ,L) }τ ∈[0,∞) satisfyingµ̌λ (Fξ ) = µ̌λ (ξ) anddµ̌λ (Fξ+τ (X ,L) ) = −Futλξ (X , L).
dτ τ =0

Cororally (I. ’21)
If µ̌λ is maximized by ξ ∈ t, then (X , L) is µ̌λξ K-semistable.
This is the case when (X , L) admits a µ̌λξ -cscK metric for λ ≤ 0.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
4. µK-stability and non-archimedean µ-entropy

Optimal degeneration and µK-semistability
Theorem (I. ’21, to appear)
If a f.g. filtration F⊕maximizesµ̌λ , then the central fibre⊕−λ(Xo , L|Xo ) = Proj m λ∈R t .F λ Rm /F λ+ Rm is µ̌λξ K-semistable.
Question: Show the existence and uniqueness of maximizers of µ̌λ .
Proposal :Consider a ‘completion’ of the space of f.g. filtrations and extendµ̌λNA to the completion as an usc function.
Show compactness maximizing sequences ⇝ existence.
Prove the ‘regularity’ of the maximizer.
cf. Similar approach by M. Xia in the context of extremal metric.

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
4. µK-stability and non-archimedean µ-entropy

Completion approach for optimal degeneration
1
Boucksom–Jonsson: a completion ENA(X , L) of the space of TCs.

Theorem (I. ’21, to appear)
exp1
There is a complete metric space ENA(X , L) ⊂ ENA(X , L) and aexp
λfunctional µNA : ENA (X , L) → [−∞, ∞) which extends µλNA for test
configurations.

We can also extend the H-entropy as HNA : E exp (X , L) → R. We haveµ̌2π
NA ≤ 2πHNA for (X , L) = (X , −KX ).

Canonical metrics and Perelman’s entropy in Kähler geometry (Eiji Inoue)
4. µK-stability and non-archimedean µ-entropy

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