A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)

A sketch of the future of µ-cscK metrics andµK-stability
Eiji INOUE （RIKEN iTHEMS, Japan）

9, October, 2022, The 28th Symposium on Complex Geometry

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)

Contents

1

Motivation

2

Theory on “stability”: µλξ -cscK metric and µλξ K-stability

3

Theory on “instability”: (toric) NA µ-entropy and Perelman entropyA sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

1. Motivation

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

Dream

Want to construct a moduli theory of polarized varieties, generalizingthat of Fano varieties, which encloses “unstable” objects.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

What is the moduli theory of Fano varieties?, unstable case
What can we say about “unstable (= general)” Fano manifolds?
Let X be a Q-Fano variety. In some way, we want to assign another“semistable” Q-Fano variety X0 for which we have a nice moduli theory.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

What is the moduli theory of Fano varieties?, unstable case
What can we say about “unstable (= general)” Fano manifolds?
Let X be a Q-Fano variety. In some way, we want to assign another“semistable” Q-Fano variety X0 for which we have a nice moduli theory.
F

In this talk, a “degeneration” (X , L) ⇝ (X0 , L0 ) of (X , ⊕L) means a f.g.
filtration F = {F λ }λ∈R on the graded ring R(X , L) = m H 0 (X , L⊗m ),which is an “irrational version” of test configuration (see [BHJ] and[CSW]). A prime divisor E ⊂ Y → X (over X ) gives a (not necessarilyf.g.) filtration FE .
F

For a “special degeneration” X ⇝E X0 of a Q-Fano variety (X , −KX ), we
can define the H-entropy ȞNA (F) ∈ R following [DS] and [HL]:∫
ȞNA (F) := −2π((1 + ordE (KY /KX )) + log e −t DHvE ).
R

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

What is the moduli theory of Fano varieties?, unstable case
Theorem (CSW, DS, HL, BLXZ)Let X be a Q-Fano variety. Then there exists a unique “specialF
degeneration” X ⇝E X0 which maximizes ȞNA . Moreover,the central fibre X0 is a Q-Fano variety with a vector field ξF .
X0 admits a unique ξF -equivariant test configuration whose central
fibre is a Q-Fano variety Z with Kähler–Ricci soliton w.r.t ξF .
When X is smooth, Z is the Gromov–Hausdorﬀ limit of thenormalized Kähler–Ricci flow.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

What is the moduli theory of Fano varieties?, unstable case
Theorem (CSW, DS, HL, BLXZ)Let X be a Q-Fano variety. Then there exists a unique “specialF
degeneration” X ⇝E X0 which maximizes ȞNA . Moreover,the central fibre X0 is a Q-Fano variety with a vector field ξF .
X0 admits a unique ξF -equivariant test configuration whose central
fibre is a Q-Fano variety Z with Kähler–Ricci soliton w.r.t ξF .
When X is smooth, Z is the Gromov–Hausdorﬀ limit of thenormalized Kähler–Ricci flow.
Remark: later we introduce another invariant µ̌λNA
ȞNA ... Ding type invariant ... makes sense for L = −KXµ̌λNA ... DF/Mabuchi type invariant ... makes sense in general
The same result is true also for µ̌2π
NA instead of ȞNA .

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

What is the moduli theory of Fano varieties?, stable caseKähler–Ricci soliton:
Ric(ω) − LJξ ω = 2πω.
Theorem (i + thesis)
There exists an algebraic moduli space of Fano manifolds withKähler–Ricci solitons.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

What is the moduli theory of Fano varieties?, stable caseKähler–Ricci soliton:
Ric(ω) − LJξ ω = 2πω.
Theorem (i + thesis)
There exists an algebraic moduli space of Fano manifolds withKähler–Ricci solitons.
Conjecture (Compact moduli space of KRs Q-Fano varieties)
There exists a projective moduli space of Q-Fano varieties withKähler–Ricci solitons.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

What is “stable”, then?

“Stability” in this line ismodified Futaki invariant [BW] ... is defined for special test
configuratione ⟨−,ξ⟩ -Ding invariant [HL] ... is a Ding type invariant defined forgeneral test configurationµ2πξ -Futaki invariant [iii] ... is a DF/Mabuchi type invariant definedfor general test configuration
These are a posteriori equivalent.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

Dream, again

Want to construct a moduli theory of polarized varieties, generalizingthat of Fano varieties.
||
Want to extend the theory of Kähler–Ricci soliton / modified K-stability
to general polarized variety.

How?
..
.
K-stability is introduced based on moment map picture of cscK metric.
Observe the moment map picture of Kähler–Ricci soliton.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

Dream, again

Want to construct a moduli theory of polarized varieties, generalizingthat of Fano varieties.
||
Want to extend the theory of Kähler–Ricci soliton / modified K-stability
to general polarized variety.

How?
..
.
K-stability is introduced based on moment map picture of cscK metric.
Observe the moment map picture of Kähler–Ricci soliton.
..
.
There appears a new degree of freedom λ ∈ R, which enriches the theory.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

Tasks

Let’s establish:Diﬀerential geometric theory on “stability”: µλξ -cscK metricDiﬀerential geometric theory on “instability”: Perelman entropy
Algebro-geometric theory on “stability”: µλξ K-stability
Algebro-geometric theory on “instability”: µλ K-optimal degeneration

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Motivation

Tasks

Let’s establish:Diﬀerential geometric theory on “stability”: µλξ -cscK metricDiﬀerential geometric theory on “instability”: Perelman entropy
Algebro-geometric theory on “stability”: µλξ K-stability
Algebro-geometric theory on “instability”: µλ K-optimal degeneration
We find two “phases” of the theory:
Good behavior when λ ≤ 0Phase transition when λ ≫ 0

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stability

2. Theory on “stability”: µλξ -cscK metric and µλξ K-stability

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stabilityµλξ -cscK metric
The moment map picture on Kähler–Ricci soliton observed in [i] yields
the following notion.
Definition
For λ ∈ R, a Kähler metric ω ∈ L is called a µλ -cscK metric w.r.t ξf if
there exists f ∈ C ∞ (X ) such that Lξf J = 0 for the Hamiltonian vector
field ξf (i.e. iξf ω + df = 0) andsfλ (ω) := (s(ω) + □f ) + (□f + |∂ ♯ f |2 ) − λfis constant.
Y. Nakagawa (2011) independently introduced the equation for
λ = −2π(KX · L·n−1 )/(L·n )(in view of K-energy on KRs) and studied Calabi ansatz example.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stability

Basic property
A Kähler–Ricci soliton Ric(ω) − LJξf ω = 2πβω is a µ2πβ-cscKfmetric, whose Kähler class L must satisfy βL = −KX .
Conversely, a µ2πβ -cscK metric in the Kähler class L satisfying
βL = −KX is a Kähler–Ricci soliton.
A cscK metric s(ω) = const. in any Kähler class L is a µλ -cscK
metric with f = 0 and every λ ∈ R.
Trivial construction:ω ∈ L is a µλ -cscK metric iﬀ cω ∈ cL is a µλ/c -cscK metric (c > 0).
If (X , ωX ) and (Y , ωY ) are µλ -cscK manifolds, then(X × Y , ωX × ωY ) is a µλ -cscK manifold.
Perturbation: the invertibility of ∆ − ∇f − λ : Cξ∞(X ) → Cξ∞(X )ff
If ω ∈ L is a µλ -cscK metric for λ ≤ 0, then there exists a µλ -cscK
metric ω ′ ∈ L′ in a small perturbation L′ of L.
If ω ∈ L is a µλ -cscK metric for λ ≤ 0, then there exists a µλ̃ -cscK
metric ω̃ in L for a small perturbation λ̃ of λ.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stability

DG theory on “stability”
For a given Kähler class L, we are interested in the existence and theuniqueness modulo Aut(X , L) of µλ -cscK metrics.
Proposition (Reductiveness, ii)
If a Kähler class L = c1 (L) admits a µλ -cscK metric ω w.r.t. ξ, thenAut0ξ (X , L) is reductive with a maximal compact subgroup Isom0ξ (X , ω).
If λ ≤ 0, Aut0ξ (X , L) is a maximal reductive subgroup in Aut0 (X , L).
Theorem (Uniqueness of µλξ -cscK metric, Lahdili2)
Given a Kähler class L, λ ∈ R and ξ, µλ -cscK metrics w.r.t. ξ in L areunique modulo Autξ (X , L).

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stability

DG theory on “stability”
For a given Kähler class L, we are interested in the existence and theuniqueness modulo Aut(X , L) of µλ -cscK metrics.
Proposition (Reductiveness, ii)
If a Kähler class L = c1 (L) admits a µλ -cscK metric ω w.r.t. ξ, thenAut0ξ (X , L) is reductive with a maximal compact subgroup Isom0ξ (X , ω).
If λ ≤ 0, Aut0ξ (X , L) is a maximal reductive subgroup in Aut0 (X , L).
Theorem (Uniqueness of µλξ -cscK metric, Lahdili2)
Given a Kähler class L, λ ∈ R and ξ, µλ -cscK metrics w.r.t. ξ in L areunique modulo Autξ (X , L).
Conjecture (Uniqueness of phase)
Given a Kähler class L and λ ≤ 0, µλ -cscK metrics in L are uniquemodulo Aut(X , L).

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stability

Moving λ ∈ R: extremal limit and phase transition
Proposition (Extremal limit, ii )Suppose there exists an extremal metric ωext ∈ L, then there exists afamily of µλ -cscK metrics {ωλ } in L for λ ≪ 0 such that ωλ converges
smoothly to ωext as λ → −∞.
Proposition (Phase transition, ii )Consider L = c1 (O(1)) on CP 1 .
For λ ≤ 2π, every µλ -cscK metric has trivial ξ = 0.
For λ > 2π, there exists a µλ -cscK metric w.r.t. non-trivial ξ ̸= 0.
The latter metric is not the Fubini–Study metric and the isometry groupreduces to U(1) ⊂ SU(1).
Corollary (Non-uniqueness result)
Given a Kähler class L and λ ≫ 0, µλ -cscK metrics in L are NOT uniquemodulo Aut(X , L), in general.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stability

Examples by Calabi ansatz
Let Σ be a proper curve and L be a line bundle of degree ℓ ≥ 1. Consider
the ruled surface X := PΣ (L ⊕ O). In
H 2 (X , R) = [H ∗ (Σ)[B]/(B 2 − ℓB.F )](2) = RB ⊕ RF , the Kähler cone isgiven byℓ
CX = {xB + yF | x > 0, y > − x}2
Proposition (Nakagawa, ii)
Every Kähler class in the subcone CX′ = {xB + yF | x, y > 0} admits aµλ -cscK metric w.r.t. non-trivial ξ ̸= 0 for every λ ≥ 0.
Remark: There exists L ∈ CX′ with no extremal metric.
Question
What can we say about L ∈ CX \ CX′ ?
′Can we compute inf{λ ∈ R | ∃µλ -cscK in L for every λ′ > λ}?

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stability

Connecting KRs and extremal metric

Proposition (Connecting KRs and extremal metric, ii )In the Kähler class −KX of X = CP 2 #CP 2 , there exists a smooth family{ωλ }λ∈[−∞,2π] of µλ -cscK metrics, which gives an extremal metric at
λ = −∞ and a KRs at λ = 2π.
Question
What about the case λ > 2π? Observe phase transition in this example.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stabilityµλξ K-stability (in toric case, for brevity)
Since Lξf g = 0, Tcpt := exp Rξf ⊂ Isom(X , g ) is a closed torus.
Keep this in mind, we consider an algebraic torus action T on (X , L). For
λ ∈ R, ξ ∈ t := Hom(T , C× ) ⊗ R and a T -equivariant test configuration(X , L), we can define the µλξ -Futaki invariant Futλξ (X , L).
When (X , L) ⟲ T is toric, a convex function q on the moment polytope
P := −µ(X ) ⊂ t∨ encodes a T -equivariant test configuration (X , L).
µ0ξ -Futaki invariant for toric test configuration∫∫∫qe −⟨x,ξ⟩ dσe −⟨x,ξ⟩ dσ P qe −⟨x,ξ⟩ dµ
1
0
∂P
∂P
∫
Futξ (q) = ∫ −⟨x,ξ⟩− ∫ −⟨x,ξ⟩
2π
e
dµ
e
dµ P e −⟨x,ξ⟩ dµPP∫e ⟨x,−ξ⟩+tq dσd∫∂P=dt t=0 P e ⟨x,−ξ⟩+tq dµA sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)λTheory on “stability”: µλξ -cscK metric and µξ K-stability

Conjectures based on moment map picture
Conjecture (Existence of µλξ -cscK metric)
A Kähler class L admits a µλ -cscK metric w.r.t. ξ if and only if
Futλξ (X , L) ≥ 0 (µλξ K-semistable) for every ξ-equivariant (X , L) andFutλξ (X , L) = 0 only for product configurations (µλξ K-polystable).
Theorem (Lahdili1, iii, AJL)
⇒ is true (for smooth test configuration).
Conjecture (Moduli space of µλξ K-semistable polarized varieties)
For each λ, ξ ∈ t, there exists an algebraic moduli space of polarized
varieties with µλξ -cscK metrics, whose moduli stack consists of
T -equivariant family of polarized varieties which are µλξ K-semistable.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

3. Theory on “instability”: NA µ-entropy and Perelman entropyA sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

AG theory on ‘instability’: (toric) NA µ-entropy∫e ⟨x,−ξ⟩+tq dσ
1
d
0
∫∂P
Futξ (q) =.
2πdt t=0 P e ⟨x,−ξ⟩+tq dµ

RecallDefinition (Non-archimedean µ-entropy)∫(e q dσ+λ n+µ̌λNA (q) := −2π ∫∂P q
e dµ
P

∫
∫
)
qe q dµ
q
P
∫
−
log
e
dµ
e q dµPP

We can define µ̌λNA (F) also for (not necessarily T -equiv.) degeneration
F(X , L) ⇝ (X0 , L0 ) of general (X , L), but I don’t explain it here.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

NA µ-entropy and µK-stability
Theorem (Entropy maximization is optimal, v)
FSuppose a “degeneration” (X , L) ⇝ (X0 , L0 ) maximizes µ̌λNA , then(X0 , L0 ) is µλ K-semistable w.r.t. ξF .

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

NA µ-entropy and µK-stability
Theorem (Entropy maximization is optimal, v)
FSuppose a “degeneration” (X , L) ⇝ (X0 , L0 ) maximizes µ̌λNA , then(X0 , L0 ) is µλ K-semistable w.r.t. ξF .
Conjecture (Optimal destabilization)
If X has only klt singularities, there exists a unique “degeneration”
F(X , L) ⇝ (X0 , L0 ) which maximizes µ̌λNA .
Recall:Theorem (CSW, DS, HL, BLXZ, v)
Let X be a Q-Fano variety. Then there exists a unique “specialF
degeneration” X ⇝E X0 which maximizes µ̌2π
NA , which is related to theGromov–Hausdorﬀ limit of the normalized Kähler–Ricci flow.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Restriction to aﬃne functions (product tcs / vectors)
We can also write µ̌λNA (⟨·, ζ⟩) as follows:∫ λ∫s(ω)e −µζ ω n
X −µζ
∫ −µ
−
+ λ(n − loge −µζ ω n /n!).
e ζ ωnX
X
If L = −KX (Fano) and λ = 2π, the first term is n: Tian–Zhu functional.
Proposition (Volume minimization, ii )If the exists a µλ -cscK metric w.r.t. ξ ∈ t, then ξ must be a critical point(actually, a maximizer) of the functional µ̌λNA (⟨·, −ζ⟩) on t.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Restriction to aﬃne functions (product tcs / vectors)
We can also write µ̌λNA (⟨·, ζ⟩) as follows:∫ λ∫s(ω)e −µζ ω n
X −µζ
∫ −µ
−
+ λ(n − loge −µζ ω n /n!).
e ζ ωnX
X
If L = −KX (Fano) and λ = 2π, the first term is n: Tian–Zhu functional.
Proposition (Volume minimization, ii )If the exists a µλ -cscK metric w.r.t. ξ ∈ t, then ξ must be a critical point(actually, a maximizer) of the functional µ̌λNA (⟨·, −ζ⟩) on t.
Recall:Conjecture (Uniqueness of phase)
Given a Kähler class L and λ ≤ 0, µλ -cscK metrics in L are uniquemodulo Aut(X , L).
Since Isomξ (X , ω) is maximally compact for λ ≤ 0, it suﬃces to show
the uniqueness of the critical points of µ̌λNA (⟨·, −ζ⟩).

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Phase transition, revival
In other words, the conjecture is equivalent to the following:
Conjecture (Uniqueness of phase, reduced)
Given a Kähler class L, λ ≤ 0 and a maximal torus T ⊂ Aut(X , L), thecritical points of µ̌λNA (⟨·, −ζ⟩) are unique.
A sophisticated version of Laplace principle shows the following:
Proposition (Phase transition, ii )The functional µ̌λNA (⟨·, −ζ⟩) is proper and bounded from above, hence
admits a critical point. Moreover,′sup{λ ∈ R | µ̌λNA (⟨·, −ζ⟩) admits unique critical point for every λ′ < λ}is never ±∞. (The result ̸= +∞ was unexpected. )Now we have shown the conjecture is true for non-positive
λ′ < sup{λ | ...}.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

DG theory on ‘instability’: Perelman’s entropy
For ω ∈ H(X , L) and f ∈ C ∞ (X ), we put∫ λ∫s (ω)e f ω nλX ∫fW (ω, f ) := −+ λ(n − loge f ω n /n!),e f ωnX
XµλPer (ω) := sup W λ (ω, f ).
fTheorem (Donaldson type inequality, iv )sup µ̌λNA (F) ≤ inf µλPer (ω).
Fω

Conjecture (Minimax conjecture)
sup µ̌λNA (F) = inf µλPer (ω).
Fω

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Perelman’s µλ -entropy and µλ -cscK metrics

Theorem (Perelman’s µλ -entropy and µλ -cscK metrics, iv )Critical points of W λ : H(X , L) × C ∞ (X ) → R are precisely thepairs (ω, f ) of µλ -cscK metric ω w.r.t. ξf and the potential f .
For λ ≤ 0, µλPer : H(X , L) → R is smooth and its critical points are
precisely µλ -cscK metrics ω w.r.t. some ξ.
For λ ≤ 0, any µλ -cscK metric attains the minimum of µλPer .
For λ ≤ 0, the vector ξ associated to a µλ -cscK metric gives afiltration Fξ which attains the maximum of µ̌λNA and we have theequality:
sup µ̌λNA (F) = inf µλPer (ω).
Fω

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Summary
Theorem (µλ -cscK metric is characterized by Perelman µ-entropy)
For λ ≤ 0, TFAE:ω ∈ H(X , L) is a µλ -cscK metric.
ω minimizes µλPer : H(X , L) → R.
R
We put HNA(X , L) := { f.g. filtrations F}/ ∼.

Theorem (Entropy maximization is optimal)
B implies A:
R
A: The central fibre of F ∈ HNA(X , L) is µλ K-semistable w.r.t. ξF .
R
B: F maximizes µ̌λNA : HNA(X , L) → R.

Conjecture
For λ ≤ 0, A implies B.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Conjectures, I: YTD conjecture and moduli space
Conjecture (Compact moduli space of KRs Q-Fano varieties)
There exists a projective moduli space of Q-Fano varieties withKähler–Ricci solitons.
Conjecture (Existence of µλξ -cscK metric)
A Kähler class L admits a µλ -cscK metric w.r.t. ξ if and only if
Futλξ (X , L) ≥ 0 (µλξ K-semistable) for every ξ-equivariant (X , L) andFutλξ (X , L) = 0 only for product configurations (µλξ K-polystable).
Conjecture (Moduli space of µλξ K-semistable polarized varieties)
For each λ, ξ ∈ t, there exists an algebraic moduli space of polarized
varieties with µλξ -cscK metrics, whose moduli stack consists of
T -equivariant family of polarized varieties which are µλξ K-semistable.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Conjectures, II: Existence and Uniqueness of optimal
degeneration
Conjecture (Uniqueness of phase)
Given a Kähler class L and λ ≤ 0, µλ -cscK metrics in L are uniquemodulo Aut(X , L).
Conjecture (Optimal destabilization)
F∃! “degeneration” (X , L) ⇝ (X0 , L0 ) which maximizes µ̌λNA .
Conjecture (Minimax conjecture)
sup µ̌λNA (F) = inf µλPer (ω).
Fω

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Approach: non-archimedean pluripotential theory
Conjecture (Optimal destabilization)
F∃! “degeneration” (X , L) ⇝ (X0 , L0 ) which maximizes µ̌λNA .
R
Want to consider a completion of the space of f.g. filtrations HNA(X , L).
↓Use Boucksom–Jonsson’s non-archimedean pluripotential theory

Theorem (v)There exists
exp
a complete metric space ENA(X , L) consisting of functions on theNA
Berkovich space X ,expRa canonical embedding HNA(X , L) ,→ ENA(X , L) andexpRλ(X , L).
an usc extension of µ̌NA : HNA (X , L) → [−∞, ∞) to ENA

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Toric case is verified

Theorem (New!)
λThere exists∫ a lsc convex function q on P which maximizes toric µ̌NAfornρq
λ ≤ 0 and P e dµ < ∞ for ρ ∈ [1, n−1 ).

Conjecture
The maximizer q is piecewise aﬃne.

A sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

References
i: The moduli space of Fano manifolds with KRs ...
ii: Constant µ-scalar curvature ...
iii: Equivariant calculus ...
iv: Entropies in µ-framework ... , I
v: Entropies in µ-framework ... , II
AJL: Apostolov, Jubert, Lahdili, Weighted K-st. and coercivity ...
BHJ: Boucksom, Hisamoto, Jonsson, Uniform K-stability, DH ...
BLXZ: Blum, Liu, Xu, Zhuang, The existence of KRs degeneration
CSW: Chen, Sun, Wang, Kähler–Ricci flow, KE metric, and ...
DS: Dervan, Székelyhidi, The Kähler–Ricci flow and optimal ...
HL1: Han, Li, On the YTD conjecture for generalized KRs ...
HL2: Han, Li, Algebraic uniqueness of KRF limits ...
Lahdili1: Kähler metrics with constant weighted scalar curvature ...
Lahdili2: Convexity of weighted Mabuchi functional ...
Nakagawa: On generalized Kähler–Ricci solitonsA sketch of the future of µ-cscK metrics and µK-stability (Eiji Inoue)
Theory on “instability”: NA µ-entropy and Perelman entropy

Thank you!