Perelman’s entropy and optimal degeneration (Eiji Inoue)

Perelman’s entropy and optimal degeneration
Eiji INOUE （RIKEN iTHEMS, Japan）

20, July, 2021, ZAG seminar

Perelman’s entropy and optimal degeneration (Eiji Inoue)

Contents

1

Model story: optimal degeneration along Kähler–Ricci flow

2

Brief introduction to µ-cscK metrics

3µK-stability and non-archimedean µ-entropy

4µ-cscK metric and Perelman’s entropy

Perelman’s entropy and optimal degeneration (Eiji Inoue)
1. Model story: optimal degeneration along Kähler–Ricci flow

1. Model story: optimal degeneration along Kähler–Ricci flow

Perelman’s entropy and optimal degeneration (Eiji Inoue)
1. Model story: optimal degeneration along Kähler–Ricci flow

Optimal degeneration along Kähler–Ricci flow
Theorem (Chen–Sun–Wang ’15)
For any Fano manifold X and any initial Kähler metric ω, theGromov–Hausdorﬀ limit along Kähler–Ricci flow exists as a Fano varietyX̂ (ω) metrized by a Kähler–Ricci soliton ω̂: Ric(ω̂) − Lξ ω̂ = ω̂.
Moreover, there exist the following two step degenerations:
12X ⇝ X̄ (ω) via filtration Fv , which generates X̄ (ω) ⟲ RX̄ (ω) ⇝ X̂ (ω) via R-equivariant special degeneration, on whichX̂ (ω) ⟲ R = exp(Rξ).

X̄ (ω) = Proj(⊕⊕)
t −λ F λ H 0 (X , −mKX )/F λ+ H 0 (X , −mKX )m≥0 λ∈RX̄ (ω) is a modified K-semistable Fano variety.
cf. I.’18: moduli of mK-ss Fano manifolds (+ thesis ⇝ algebraic moduli)

Perelman’s entropy and optimal degeneration (Eiji Inoue)
1. Model story: optimal degeneration along Kähler–Ricci flow

Non-archimedean H-entropy
Dervan–Székelyhidi ’16: The H-entropy
HNA (F) := − inf (AX (w ) + φF (w )) − logw∫Re −t DHF

is maximized by CSW’s degeneration Fv : X ⇝ X̄ (ω) (for any ω).
Here φF (w ) = inf{σ ∈ R | F λ ⊂ Fwλ [σ] = Fwλ+•σ }.
∫Moreover, for H(ωφ ) = X hφ e hφ ωφn /n!, we have
sup HNA (F) = inf H(ωφ ).

F :f.g.

φHan–Li ’20: The maximizers of HNA are unique, hence X̄ = X̄ (ω) isindependent of the initial metric ω: the degeneration X ⇝ X̄ iscanonically attached to each Fano manifold X .
Even when X is singular,
Han–Li ’20: The maximizer of HNA is realized by a q.m. valuation v .
Blum–Liu–Xu–Zhuang ’21: grv R is f.g. ⇝ X̄ is mK-ss Fano variety.

Perelman’s entropy and optimal degeneration (Eiji Inoue)
Brief introduction to µ-cscK metrics

2. Brief introduction to µ-cscK metrics

Perelman’s entropy and optimal degeneration (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) + □µ−1µ̌2πλ−2ξ -cscK metric in λ c1 (X ) ⇔ KRs Ric(ω) − Lξ ω = λω.

Extremal metric ⇒ µλ -cscK metric for λ ≪ 0.

Perelman’s entropy and optimal degeneration (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.

Perelman’s entropy and optimal degeneration (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 functionalµ̌λNA : k −→ Rξ 7→ W̌ λ (ω, µωξ)
satisfying the following.
1Its 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 | µ̌λNA has a unique crit. pt. for ∀λ′ < λ}is finite (never ±∞). ⇝ phase transition

Perelman’s entropy and optimal degeneration (Eiji Inoue)
3. µK-stability and non-archimedean µ-entropy

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

Perelman’s entropy and optimal degeneration (Eiji Inoue)
3. µK-stability and non-archimedean µ-entropy

Characteristic µ-entropy
For a test configuration (X , L), we have the expansion∫∫×1−τ.t
LC× |X0eνm (t) =; τ ).m−1 + O(m−2 )e −τ.t DH(X ,L) − (κC
X0 .e2RRforνm = −1 dt
dim F(X,L) Rm
mn dt

m→∞−→DH(X ,L) .

Definition (Characteristic µ-entropy)
For τ ≥ 0,××

(KX .e L ) − τ.(KX̄C /P1 .e LC× ; τ )(κCX .e LC× |X0 ; τ )∫µ̌(X , L; τ ) := 2π ∫ 0−τ.t= 2πeDH(X ,L)e −τ.t DH(X ,L)RR∫∫(n − τ.t)e −τ.t DH(X ,L)σ̌(X , L; τ ) := R ∫ −τ.t− log e −τ.t DH(X ,L)eDH(X,L)RRµ̌λ (X , L; τ ) := µ̌(X , L; τ ) + λσ̌(X , L; τ )Perelman’s entropy and optimal degeneration (Eiji Inoue)
3. µK-stability and non-archimedean µ-entropy

Non-archimedean µ-entropy
Definition (Non-archimedean µ-entropy)
×(KX .e L ) − τ.(KX̄log,C.e LC× ; τ )/P1∫µ̌NA (X , L; τ ) := 2π≥ µ̌(X , L; τ )e −τ.t DH(X ,L)Rµ̌λNA (X , L; τ ) := µ̌NA (X , L; τ ) + λσ̌(X , L; τ )We have µ̌NA (Xd , Ld ; τ ) = µ̌NA (X , L; dτ ).

Non-archimedean µ-entropy for toric test configuration
Let P be the moment polytopeof a toric variety (X , L). For a convex∫function q normalized as P e q dµ = 1, we put∫∫λqµNA (q) :=e dσ − λ (n + q)e q dµ.
∂PP

Then we have µλNA (X , L; τ ) = µλNA (τ q(X ,L) ).

Perelman’s entropy and optimal degeneration (Eiji Inoue)
3. µK-stability and non-archimedean µ-entropy

Perturbation of test configurationsσ ⊂ N ⊗ R: a strictly convex full-dim Q-polyhedral cone
⇝ Bσ = SpecC[σ ∨ ∩ M] ↶ T = N ⊗ C× .
Definition (Polyhedral configuration)
A polyhedral configuration consists of a triple (X /Bσ , L; ξ) ofa T -equivariant projective flat family X over Bσ ,a relatively ample Q-line bundle L on X /Bσ ,a vector ξ ∈ σPolyhedral configuration defines a f.g. filtration∑F t Rm :={s ∈ H 0 (X , L⊗m ) | s.χ−µ ∈ H 0 (X , L⊗m )}.
⟨µ,ξ⟩≥λWe can define µ̌λ (F) similarly as µ̌λ (X , L; τ ).

Perelman’s entropy and optimal degeneration (Eiji Inoue)
3. µK-stability and non-archimedean µ-entropy

Characteristic µ-entropy and µK-semistability
Suppose (X , L) ↶ T . Take ξ ∈ t and σ ⊂ t so that ξ ∈ σ.
⇝ (Bσ × X /Bσ , pX∗ L; ξ) endowed with the diagonal Bσ × X ↶ T .
∗For a T -equiv. tc (X , L), (Bσ × X /Bσ×[0,∞) , pXL; ξ + τ.η) endowed with×′
the diagonal Bσ × X ↶ T × C is a pc for σ = σ × [0, ∞).

Let Fξ+τ (X ,L) denote the associated filtration.
Proposition (I. ’21 (to appear), essentially proved in I. ’20)dµ̌λ (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.

Perelman’s entropy and optimal degeneration (Eiji Inoue)
3. µK-stability and non-archimedean µ-entropy

Optimal degeneration and µK-semistability
Theorem (I. ’21, to appear)
If a polyhedral configuration (X /Bσ , L; ξ) maximizes µ̌λ , then thecentral fibre (Xo , L|Xo ) is µ̌λξ K-semistable.
Question: Show the existence and uniqueness of maximizers of µ̌λ .
Remark: Uniqueness seems diﬃcult since µ̌λ is not concave even on t.
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.

Perelman’s entropy and optimal degeneration (Eiji Inoue)
3. 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)exp
1
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 haveµ̌2πNA ≤ 2πHNA for (X , L) = (X , −KX ).

Perelman’s entropy and optimal degeneration (Eiji Inoue)
4. µ-cscK metric and Perelman’s entropy

4. µ-cscK metric and Perelman’s entropy

Perelman’s entropy and optimal degeneration (Eiji Inoue)
4. µ-cscK metric and Perelman’s entropy

Perelman’s W -entropyPerelman’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!,X
λW̌ (ω, f ) := W̌ (ω, f ) + λŠ(ω, f ).

W̌ λ : T H(X , L) = H(X , L) × C ∞ (X )/R → R.

Perelman’s entropy and optimal degeneration (Eiji Inoue)
4. 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 s 2 ω n /n! forextremal metric in a Kähler class or H-entropy X he h ω n /n! for KRs.

Perelman’s entropy and optimal degeneration (Eiji Inoue)
4. µ-cscK metric and Perelman’s entropy

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 )infµ̌λ (ω)ω∈H(X ,L)

cf. Dervan–Székelyhidi (and Donaldson, Hisamoto, Xia):
sup HNA (F) = inf H(ωφ ).

F :f.g.

φPerelman’s entropy and optimal degeneration (Eiji Inoue)
4. 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 )CororallyFor λ ≤ 0, all the critical points of µλ are global minimizers, which are
precisely µλ -cscK metrics. If there is a µ̌λξ -cscK metric ω, we havesup µλNA (X , L; τ ) = µλNA (ϕξ ) = µλ (ω) =(X ,L;τ )inf

ωφ ∈H(X ,L)µλ (ωφ ).

Perelman’s entropy and optimal degeneration (Eiji Inoue)
4. µ-cscK metric and Perelman’s entropy

Universal aspect∫

he h ω n− loge h ωnXH(ω) = ∫X

H-entropy∫Introduce

fe h ω n− loge h ωnX

Ľ(ω, f ) = ∫X∫e h ω n /n!
X∫e f ω n /n!,Xthen H(ω) = supf Ľ(ω, f ) and HNA (X , L; τ ) = limt→∞ Ľ(ωϕτ.t , ϕ̇τ.t )∫Calabi functionalIntroduce∫
Wext (ω, f ) = − XC (ω) =X(s − s̄)2 ω n∫
2 X ωn((s − f ) − (s̄ − f¯))2 ω n∫+2 X ωn∫X(s − s̄)2 ω n∫,2 X ωnthen C (ω) = supf Wext (ω, f ) and CNA (X , L; τ ) := limt→∞ Wext (ωϕτ.t , ϕ̇τ.t ),2(X ,L)2
for which we have maxτ ≥0 CNA (X , L; τ ) = (L2π·n ) DF
when DF (X , L) ≤ 0.
∥(X ,L)∥2

Perelman’s entropy and optimal degeneration (Eiji Inoue)
4. µ-cscK metric and Perelman’s entropy

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