Introduction to Optimal degeneration problems (Eiji Inoue)

Introduction to Optimal degeneration problems
Eiji INOUE （RIKEN iTHEMS, Japan）

8, September, 2023, Rutgers University

Introduction to Optimal degeneration problems (Eiji Inoue)

I want to explain three analogous frameworks on optimal degeneration
1

The framework of extremal metric2

The framework of Kähler–Ricci soliton

3

The framework of µ-cscK metric

From K-stability viewpoint, all of these are covered in Lahdili’s weighted
cscK framework.
Today, we focus K-instability aspect, which cannot be discussed inweighted cscK framework (too general).

Introduction to Optimal degeneration problems (Eiji Inoue)

Canonical metric of space
Let X be a compact Kähler manifold and L be a Kähler class.
H(X , L) := {ω ∈ L | ω > 0}
¯Let s(ω) = −g i j ∂i ∂j¯ log det g be the scalar curvature. We put∫s(ω)ω nŝ(ω) = s(ω) − X∫ n .
ωXA Kähler metric ω is called a cscK metric ifŝ(ω) = 0.
Uniqueness: any two cscK metrics ω, ω ′ in the same Kähler class L isisometric by some holomorphic map f : X → X .
Conjecture (Yau–Tian–Donaldson conjecture)
Given (X , L), the existence of a cscK metric in H(X , L) is equivalent toK-stability of (X , L).

Introduction to Optimal degeneration problems (Eiji Inoue)

Degeneration of space
A pair (X , L) is called a test configuration if
X is a scheme/complex analytic space flat over A1 endowed with aGm -action compatible with the scale action on A1 .
2 Gm -equivariant isomorphism X × Gm ∼= X ∗ = X \ X0 away from the1fibre X0 over 0 ∈ A .
13L ∈ HG2 m (X , R) is a Gm -equivariant relatively ample cohomology
class whose restriction to X ∗ coincides with pX∗ L as Gm -equivariant
class via the above isomorphism.

Canonical compactification using Gm -action: (X̄ , L̄)We putDF(X , L) := (KX̄ /P1 .L̄n ) −n (KX .Ln−1 ) n+1(L̄ ).
n+1(Ln )(X , L) is called K-semistable if DF(X , L) ≥ 0 for every test configuration.

Introduction to Optimal degeneration problems (Eiji Inoue)

Question

It is known that if (X , L) admits a cscK metric, then it is K-semistable(more strongly K-polystable).
In view of GIT analogue, K-semistable varieties are believed to form an
Artin moduli stack which admits a good moduli space of K-polystablevarieties.
... Good!
Question: What we can say for K-unstable (X , L)?

Introduction to Optimal degeneration problems (Eiji Inoue)

normalized base change
We want to formulate the notion of “test configuration with the least
Donaldson–Futaki invariant”.
For a normalized base change Xd of X along z d : A1 → A1 , we haveDF(Xd , Ld ) ≤ d.DF(X , L)
andd −1 DF(Xd , Ld ) = (d ′ )−1 DF(Xd ′ , Ld ′ ) =: M(X , L)

for suﬃciently divisible d, d ′ ∈ N.
When M(X , L) < 0, M(Xd , Ld ) = d.M(X , L) gets arbitrary small bytaking d → ∞.
⇝ Want to define a “norm” ∥(X , L)∥ to normalize M(X , L).

Introduction to Optimal degeneration problems (Eiji Inoue)

normalized Donaldson–Futaki invariant
We consider the weak limit of measures on R:1 ∑DH(X ,L) := limdim H 0 (X0 , L⊗m0 )λ .δλ/m .
m→∞ mnλ∈ZWe put∫
E (X , L) :=and use∫

RtDH(X ,L) (t)/R

DH(X ,L)(∫∥(X , L)∥ :=)1/2(t − E (X , L)) DH(X ,L)
2

R

as the norm. It satisfies ∥(X , L)∥ ≥ 0 and∥(Xd , Ld )∥ = d.∥(X , L)∥.
Optimal degeneration problem
Show the existence and the uniqueness of the minimizer ofM(X , L)/∥(X , L)∥ modulo scaling.

Introduction to Optimal degeneration problems (Eiji Inoue)

Donaldson’s inequalityThe Calabi energy Ca : H(X , L) → R∫
Ca(ω) =ŝ(ω)2 ω n .
XTheorem (Donaldson ’05)
(2πM(X , L)∥(X , L)∥)2≤infω∈H(X ,L)

for any (X , L) with M(X , L) ≤ 0.
Minimizer of Ca is extremal metric:∂ # s(ω)
is holomorphic.

Ca(ω)

Introduction to Optimal degeneration problems (Eiji Inoue)

Extension
It is NOT known if there exists a tc with minimal M(X , L)/∥(X , L)∥.
2We can find a minimizer on a larger space ENA⊃ { tc } (Xia ’21).
There are two ways to extend
() 2πM(X ,L) 2ext∥(X ,L)∥µ̂ (X , L) :=0M(X , L) ≤ 0M(X , L) > 0

2to ENA:1

use geometric pluripotential theory µ̂ext
geo or2

use non-archimedean pluripotential theory µ̂ext
NA .

These are expected to coincide (= regularization of NA entropy).
Similarly, there exists an extension of Ca : H(X , ω) → R to a larger space
E 2 ⊃ H(X , ω).

Introduction to Optimal degeneration problems (Eiji Inoue)

Xia’s result
The Calabi flow√d¯ t)ωt = −1∂ ∂s(ωdtdecreases the Calabi energy.
Long time existence of smooth solution is not known, but there exists[0, ∞) → E 2 which can be regarded as Calabi flow.
Using this and compactness on E 2 , Xia proved the following.
Theorem (Xia ’21)
2There exists ϕext ∈ ENA
which maximizes µ̂ext
geo and we havemax µ̂ext
geo (ϕ) = inf 2 Ca(φ).

2
ϕ∈ENA

φ∈E

Li–Lian–Sheng: Toric ϕext is bounded ⇝ filtration on⊕mH0(X , L⊗m ).

Introduction to Optimal degeneration problems (Eiji Inoue)Framework of Kähler–Ricci solitonµ̂ext (X , L) = max µext (X , L; ρ) := max(−4πM(X , L).ρ − ∥(X , L)∥2 .ρ2 )ρ≥0ρ≥0For (X , L) = (X , −2πKX ) is a Fano manifold, there is another frameworkon optimal degeneration: Framework of Kähler–Ricci soliton.
∫H(X , L; ρ) = −L(X , L).ρ − log e ρ.t DH(X ,L) (t)Rµext
Ca: Calabi energy
Extremal metricCalabi flow

H: H-entropy
He: He entropyKähler–Ricci soliton
Kähler–Ricci flow

Introduction to Optimal degeneration problems (Eiji Inoue)

Geometric property of optimal degeneration
A good thing in the framework of Kähler–Ricci soliton is that the optimal
degeneration has a geometric realization (cf. Chen–Sun–Wang,Dervan–Székelyhidi, Han–Li, Blum–Liu–Xu–Zhuang):
X → Ak ⟲ (Gm )k with ξ ∈ [0, ∞)k .
Theorem (Han–Li ’20)
The central fibre X0 is a Q-Fano variety which is modified K-semistablewith respect to the vector field generated by ξ.
Existence of Kähler–Ricci soliton ⇒ modified K-semistable
Optimal degeneration is associated to the following filtration:
Fmλ = {s ∈ H 0 (X , mL) | lim t −1 log ∥s∥g (t) ≤ λ},t→∞where g (t) is a Kähler–Ricci flow.

Introduction to Optimal degeneration problems (Eiji Inoue)

Framework of µ-cscK metric

There is a family of similar framework parametrized by λ ∈ R:
µλ : µ-entropyµλPe : Perelman µ-entropyµλ -cscK metric“µλ -flow”
For Fano manifold (X , L) = (X , −2πλKX ), µλ -cscK metric is equivalentto Kähler–Ricci soliton. (Here λ−1 = 2π(−KX .Ln−1 )/(Ln ) > 0)
For general (X , L), a rescaling limit of the framework of µλ -cscK metricas λ → −∞ is the framework of extremal metric.

Introduction to Optimal degeneration problems (Eiji Inoue)

Perelman entropy∫∫For f ∈ C ∞ (X ) normalized as X e f ω n = X ω n , we consider(∫)∫1
λf nf n∫(s(ω) + □f )e ω − λfe ω .
W (ω, f ) := −ωnXXX∫∫√¯ log νIf we put νω = ω n / X ω n , νω,f := e f ω n / X ω n and ρ(ν) := −1∂∂for ν ∈ Ωn,n (X ) then(∫)∫νω,fνω,fW λ (ω, f ) = −trω (ρ(νω,f ))νω,f − λlogνω .
νωXX νω(cf. free energy∑

E (x)p(x) + Tx∈Ω∑)p(x) log p(x)x∈Ω

We putµλPe (ω, f ) :=sup∫∫f : X e f ωn = X ωn

W λ (ω, f ).

Introduction to Optimal degeneration problems (Eiji Inoue)µ-cscK metricTheorem (arXiv:2101.11197)
The critical points of W λ : H(X , L) × C ∞ (X )/R → R are precisely thosepairs (ω, f ) satisfying the following couple of equations:¯ #f = 0∂∂
1s(ω) + ∆f − |∇f |2 = λf + const.
2We call ω a µλ -cscK metric if there exists some f satisfying this equation.
Proposition
For λ ≤ 0, the functional µλPe : H(X , ω) → R is smooth and its criticalpoints are all minimizers and are µλ -cscK metrics.
Conjecture
For λ ≤ 0, µλ -cscK metrics are unique modulo Aut(X ).

Introduction to Optimal degeneration problems (Eiji Inoue)µ-entropy
Theorem (arXiv:2101.11197)
There exists an algebro-geometric quantityµ̂λ (X , L) = sup µλ (X , L; ρ)ρ≥0which satisfies Donaldson type inequality
sup µ̂λ (X , L) ≤ inf µλPe (ω).
(X ,L)

ωConjectureFor λ ≤ 0,sup µ̂λ (X , L) = inf µλPe (ω).
(X ,L)ωThis is true when there exists a µλ -cscK metric, i.e. minimizer of RHS.

Introduction to Optimal degeneration problems (Eiji Inoue)

Toric description
Let (X , L) be a toric variety and (X , L) be a toric tc associated to aconvex function
on the moment polytope P ⊂ t∨ . Put∫ qρ.q
ρ.quρ.q := e / P e dµ, then∫∫µλ (X , L; ρ) = −(2πuρ.q dσ − λ uρ.q log uρ.q dµ) =: µλ (ρ.q)∂P

PTheorem (arXiv:2303.09090)
For every λ ∈ R, there exists a lsc convex function q : P → (−∞, ∞]which maximizes µλ .
When λ ≤ 0, the maximizer is unique up to the addition of constant.
Moreover, q is Lp for every p ∈ [1, ∞).
Conjecture
The maximizer is piecewise aﬃne.

Introduction to Optimal degeneration problems (Eiji Inoue)

Where we are now

Set out nice framework↓
Construct optimal degeneration↓
Analyze asymptotic behavior of geometric flow↓
Analyze the regularity of optimal degeneration