Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Optimal degeneration of algebraic variety andPerelman entropy
Eiji INOUE (RIKEN iTHEMS, Japan)

23, May, 2024, CRM, Montreal

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Menu

1

From Calabi energy, He entropy ...

2

To Perelman µ-entropy

3

New formula on Non-archimedean µ-entropy

4

Distortion and Moment measureOptimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

1. From Calabi energy, He entropy ...

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

cscK metric(X , ω): a compact Kähler manifold with a background metric ω.
H(ω) := {ϕ ∈ C ∞ (X ) | ω + dd c ϕ > 0}.
The Mabuchi functional M : H(ω) → R is defined as the anti-derivationof δM : T H(ω) → R given byZ
δM(ϕ, ϕ̇) = −ŝ(ωϕ )ϕ̇ωϕn .
X

The critical points of M is cscK metric: s(ωϕ ) = s̄([ω]).

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Donaldson–Futaki invariant / Mabuchi invariant
For a normal test configuration (X , L), we consider
M NA (X , L) := (KXlog/P1 · L·n ) −n(KX · L·n−1 ) (L·n+1 ).
(L·n )
n+1

CompareZ
δM(ϕ, ϕ̇) =X(Ric(ωϕ ) − □ϕ̇)(ωϕ − ϕ̇)nZ
1s̄([ω]) (ωϕ − ϕ̇)n+1 .
−n+1XFor a test configuration (X , L), we can assign a “finite energy” geodesic
ray Φ(X ,L) (t) ∈ C 1,1 H(ω). “δM” makes sense for C 1,1 -geodesic ray.
Theorem (Tian ’97, Boucksom–Hisamoto–Jonsson ’19, C. Li ’22)lim δM(Φ(X ,L) (t), Φ̇(X ,L) (t)) = M NA (X , L).

t→∞

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

YTD conjecture and K-moduli
1K-polystability (reduced uniform K-stability) DF > 0 is conjectured
to be equivalent to the existence of cscK metrics. (YTD conjecture)2

K-semistability DF ≥ 0 is conjectured to be a Zariski open
condition, and the moduli stack of K-semistable polarized variety
admits a good moduli space of K-polystable polarized variety.

For L = −λKX (cscK = KE), we have good tools on “regularity”:
1✓: Ric > 0 / Ding functional (..., Chen–Donaldson–Sun, Tian, ... /Berman–Boucksom–Jonsson)2✓: MMP / boundedness / finite generation (..., Blum–Liu–Xu, ...)For general L,
1

Reduced to NA entropy regularization conjecture (...,
Darvas–Rubinstein, Berman–Darvas–Lu, Chen–Cheng, Chi Li/
Boucksom–Jonsson, Chi Li).

2

Little known (Chen–Sun and Dervan–Naumann)Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Today’s main interest
Q. What can we expect for K-unstable (X , L)?
A naive expectation is:
There exists a unique test configuration which is optimal in some sense.
How can we formulate ‘optimal’ ?
What kind of geometric property can we expect?
We hope to
1 introduce a “well-behaved” quantity for test configuration which
would be maximized by optimal degeneration.
2 show the central fibre of optimal degeneration is K-stable in a
generalized sense: we introduce K-stability for polarized dynamics(X0 , L0 ) ⟲ Gm = “well-behaved”.
1
2This question is related to algebraic proof of properness of K-moduli space:construction of Θ-stratification of moduli stack (—).

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Quick overview of two model frameworks
We have two model frameworks related to extremal metric (Calabi flow)
and Kähler–Ricci soliton (Kähler–Ricci flow).
For ϕ ∈ H(ω), we considerZ
1
C (ϕ) =ŝ(ωϕ )2 ωϕn Calabi functional,
2 X
Z
η(ϕ) =ĥϕ e ĥϕ ωϕn He functional,
X

R√
where ĥϕ is Ric(ωϕ ) − ωϕ = −1∂ ∂¯ĥϕ with X e ĥϕ ωϕn = [ω n ]. We
consider η for [ω] = c1 (X ) (Fano).
¯ ♯ s(ωϕ ) = 0
Critical points = Extremal metric ∂∂¯ ♯ hϕ = 0/ Kähler–Ricci soliton ∂∂Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Key observation
We introduceZ1ŝ(ωϕ )ϕ̇ωϕn −WCal (ϕ, ϕ̇) =X

2

ϕ̇ ωϕn −1 [ω n ]

Z

ϕ̇ωϕn

2 

X
X
Z
Z
n


(e − 1)ϕ̇ωϕ
X R
n
−ϕ̇ n
ϕ̇ω
.
−
log
e
ω
+
ϕ
ϕ
e hϕ ωϕnX
X
X

R
WHe (ϕ, ϕ̇) = −

2

Z

hϕ

We can observe
C (ϕ) =

supWCal (ϕ, ϕ̇)( compeleting square ϕ̇ = ŝ(ωϕ ))WHe (ϕ, ϕ̇)( Legendre duality on rel. entropy ϕ̇ = −hϕ )ϕ̇∈C ∞ (X )η(ϕ) =

supϕ̇∈C ∞ (X )Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Limit along geodesic ray
Recall a test configuration φ = (X , L), we can assign a geodesic ray
Φ(X ,L) (t) ∈ C 1,1 H(ω).
Theorem (Hisamoto ’16)
Φ̇(X ,L) (t)∗ MA(Φ(X ,L) (t)) = DHφ .
Thus for (ϕ(t), ϕ̇(t)) = (Φ(X ,L) (t), Φ̇(X ,L) (t)) we have
Z
Z
2 
1
1 
NA
2
lim WCal (ϕ(t), ϕ̇(t)) = −M (φ) −σ DHφ − ·n
σDHφ
,
t→∞
2
(L ) RRZ
 Z

lim WHe (ϕ(t), ϕ̇(t)) = −D NA (φ) − log e −σ DHφ +σDHφ .
t→∞

R

R

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Algebro-geometric quantity
For φ = (X , L), we consider
Z
Z
2 
1
1 
σ 2 DHφ − ·nσDHφ,2(L ) RRZ Z
ηNA (φ) = −D NA (φ) − log e −σ DHφ +σDHφ ,C NA (φ) = −M NA (φ) −RRrespectively, where we consider ηNA for L = −KX (Fano). These are
essentially introduced by Donaldson (’05) and Dervan–Székelyhidi (’20),
respectively.
By convexity of M and D, we obtain
C (ϕ) ≥ WCal (ϕ(0), ϕ̇(0)) ↘ lim WCal (ϕ(t), ϕ̇(t)) = C NA (φ)t→∞η(ϕ) ≥ WHe (ϕ(0), ϕ̇(0)) ↘ lim WHe (ϕ(t), ϕ̇(t)) = ηNA (φ)
t→∞

⇝

sup
φ∈HNA (L)C NA (φ) ≤inf
ϕ∈H(ω)C (ϕ) : Donaldson inequality

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Well-behavedness
Theorem (Dervan, Han–C. Li)
1

C NA (resp. ηNA ) is maximized by the trivial test configuration if
and only if (X , L) is K-semistable.

2

If C NA (resp. ηNA ) is maximized by a “product test configuration”
induced by “(X , L) ⟲ Gm ”, then “(X , L) ⟲ Gm ” is relatively
K-semistable (resp. modified K-semistable). (⇐ is known for ηNA )3

If a “normal test configuration” (X , L) maximizes C NA (resp.
ηNA ), then the central fibre “(X0 , L0 ) ⟲ Gm ” is relatively
K-semistable (resp. modified K-semistable).

Relative K-stability is related to the existence of extremal metric.
Modified K-stability is equivalent to the existence of Kähler–Ricci soliton.
These stability notions are defined for polarized dynamics“(X , L) ⟲ Gm ”= (X , L; ξ), not only for pol. variety (X , L) = (X , L; 0).

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Existence
For each test configuration (X , L), we can assign a (maximal, finite
energy) geodesic ray Φ = {Φt }t∈[0,∞) ∈ R1I (ω) of Kähler potentials. We
can extend C NA to the space of (maximal finite energy) geodesic rays
1
R1I (ω) ∼(L) by putting
= ENA
Z
Z
2 
1
C ray (Φ) = −M ′ (Φ) −σ 2 DHφ −
σDHφ
.
2
R
R
Theorem (Xia + C. Li (cf. Székelyhidi, A-M. Li–Lian–Sheng))
There exists a maximal finite energy geodesic ray Φ ∈ R2I which
maximizes C ray .
On the other hand, for L = −KX (Q-Fano) and ηNA ,Theorem (Chen–Sun–Wang, Dervan–Székelyhidi, Blum–Liu–Xu–Zhuang)
There exists a finitely generated filtration F which maximizes ηNA . Its
central fibre is modified K-semistable Q-Fano variety.

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Optimal degeneration | canonical metric

Theorem (Donaldson, Xia / Dervan–Székelyhidi)supC NA (φ) =

2 (L)
φ∈ENA

sup
φ∈HR
NA (L)

ηNA (φ) =

inf

C (ϕ)

inf

η(ϕ).

ϕ∈E 2 (ω)ϕ∈H(ω)

Furthermore, optimal destabilizers are assymptotic to Calabi flow andKähler–Ricci flow, respectively.

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Beautiful results!

Sakasa-Fuji

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

2. To Perelman µ-entropy

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Perelman W -entropy (in Ricci flow)
Perelman’s original convention: for a manifold M with dimR M = m,
Z
Z
e −f
e −f
W(g , f ; τ ) = τ(R(g )+|∇f |2 )vol−(m−f )volgg
m/2(4πτ )(4πτ )m/2
M
M
R
e −f
volg = 1. Hefor τ > 0, a metric g and f ∈ C ∞ (M) with M (4πτ)m/2
proved monotonicity along the following evolving equationġt = −2Ric(gt ),ft = −∆gt ft + |∇ft |2gt − R(gt ) +

n
,
2τt

τ̇t = −1.

On Fano manifold c1 (X ) > 0, we consider normalized Kähler–Ricci flowω̇t = ωt − Ric(ωt ). For this, W(ωt , ft ; τt ) is monotonic alongω̇t = ωt − Ric(ωt ),ft = ...,

τt ≡ 1/2.

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Perelman W -entropy in Kähler geometry
In Kähler geometry, we use the following convention
ZZ
1 ˆ 2 −ϕ̇ˆ nˆ −ϕ̇ˆ ω n
λWPer(ϕ, ϕ̇) = − (s(ωϕ ) + |∇ϕ̇|)e ωϕ + λ (n − ϕ̇)eϕ2XXfor (ϕ, ϕ̇) ∈ T H(ω) and λ ∈ R, whereˆRXe −ϕ̇ ωϕn = 1. NoteλWPer(ϕ, ϕ̇) = −λ(2πλ−1 )n · (W(gϕ , ϕ̇;1) + n).
2λ0
Today, we focus on λ = 0 (for simplicity): WPer (ϕ, ϕ̇) := WPer
(ϕ, ϕ̇).

Theorem (arXiv:2101.11197)
For a smooth test configuration (X , ) and its geodesic rayλΦ(t) = Φ(X ,L) (t), we can define “WPer(Φ, Φ̇)(t)” and we haveλWPer(Φ, Φ̇)(t) ↘ µNA (X , L)

as t → ∞.

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Non-archimedean µ-entropy | Perelman µ-entropy
For a ntc (X , L), we defineµNA (X , L) :=

1
m
·n−1
LGm
) − (KXlog,G) n(KX .L·n−1 )/P1 · e(n−1)! (KX .L
−
.
1
LGm )
·n
(L·n )n! (L ) − (eCompareZ
WPer (ϕ, ϕ̇) = −n!

(Ric(ωϕ ) − □ϕ̇)e ωϕ −ϕ̇ .

XµλPer (ωϕ ) := sup W λ (ωϕ , f ).
fCritical points = µ-cscK metric
Corollary (arXiv:2101.11197)
sup µλNA (X , L) ≤ inf µλPer (ωϕ ).
(X ,L)

ωϕOptimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Conjecture

Conjecture (For λ ≤ 0)
max

exp,1
φ∈ENA
(L)

µλNA (φ) =infωϕ ∈H(ω)µλPer (ωϕ )Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Non-archimedean µ-entropy is well-behaved

Theorem (Well-behavedness, arXiv:2202.12168)
If a normal test configuration (X , L) maximizes µNA , then its central
fibre “(X0 , L0 ) ⟲ Gm ” is µK-semistable.
µK-stability is related to the existence of µ-cscK metric: a generalizationof cscK metric to polarized dynamics “(X , L) ⟲ Gm ” which unifies theframework of extremal metric and Kähler–Ricci soliton.

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

3. New formula on Non-archimedean µ-entropy

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Main applications I: New formula for µNA
Theorem A (To appear)
1
There exists a map Dist : HNA (L) → ENA(L) such that NA
D (Dist(X , L))+1ηNA (X , L) = log
E NA (Dist(X , L))µNA (X , L) =M NA (Dist(X , L))
E NA (Dist(X , L))

with
E

NA

Z
(Dist(X , L)) = −e −σ DH(X ,L) (σ) < 0.

R

By this formula, we can extend
exp,1µNA : ENA
(L) → [−∞, ∞) : φ 7→

M NA (Dist(φ)).
E NA (Dist(φ))

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Main applications II: Existence of optimal destabilizing ray
Let R1 denote the space of finite energy geodesic rays and R1I denotethe space of (I-)maximal finite energy geodesic rays (cf.
Berman–Boucksom–Jonsson, Darvas–Xia). We considerµray : E exp,1 (L) → [−∞, ∞) : φ 7→ Dist(φ)ray 7→

M ′ (Dist(φ)ray )E ′ (Dist(φ)ray )By C. Li’s inequality M NA ≤ M ′ , we haveµray ≤ µNA .
Theorem B (To appear)
exp,1
Assume X is smooth. There exists φ ∈ ENA(L) which maximizes µray .

If NA entropy regularization conjecture is true, the above theorem can be
exp,1
read as “there exists φ ∈ ENA(L) which maximizes µNA ”.

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Main applications III: Miscellaneous
Conjecture (For λ ≤ 0)
max

exp,1
φ∈ENA
(L)

µray (φ) =

sup
exp,1
φ∈ENA
(L)

µNA (φ) =inf
ϕ∈H(ω)µPer (ωϕ )Theorem (For λ ≤ 0, to appear)
If maxφ∈E exp,1 (L) µray (φ) ≤ inf ϕ∈H(ω) µPer (ωϕ ), µ-cscK metrics areNAunique modulo Aut(X , L).
Theorem (to appear)
The following are equivalent:
1

1(X , L) is K-semistable over HNA (L) (resp. ENA
(L))

2

1
Trivial configuration maximizes µNA over HNA (L) (resp. ENA(L)).

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

4. Distortion and Moment measureOptimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Distortion of Non-archimedean L-psh function
Recall for a tc (X , L), we have a continuous function φ(X ,L) : X ℶ → R.
For vE ∈ X div ⊂ X ℶ associated to E ⊂ Y → X , we haveφ(X ,L) (vE ) =ordE (L − p ∗ L).
ordE (X0 )A function φ : X ℶ → [−∞, ∞) is called L-psh function if there exists a
net of tc’s (Xi , Li ) such that φ(Xi ,Li ) is a decreasing net converging to φpointwisely. Let PSHNA (L) denote the set of L-psh functions.
For an increasing concave χ : R → R, we introduce χ-distortion of φ:Distχ (φ) := inf (t ▷ φ + χ▲ (t)),
t>0

where
(t ▷ φ)(v ) := tφ(t −1 v ),

χ▲ (t) := sup (χ(σ) − tσ).
σ∈R

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Let’s get a feeling of Distortion
Proposition (To appear)
For φ ∈ PSHNA (L), we have Distχ (φ) ∈ PSHNA (L).
Suppose (X , L) is a toric variety. Let P ⊂ MR = t∨ be the moment
polytope.
{T -invariant φ} ←→ {f : NR → R}
φ 7−→ fφ (ξ) := φ(vξ ) + sup ⟨µ, ξ⟩µ∈PFor the Legendre transform gφ (µ) := inf ξ∈NR (⟨µ, ξ⟩ + fφ (ξ)), we havegDistχ (φ) = χ(gφ )thanks to the duality:
inf (tσ + χ▲ (t)).

t>0Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Profile of Distortion
For a continuous L-psh function φ ∈ CPSHNA (L), we can assign afiltration Fφ by putting
λFφ,m:= {s ∈ H 0 (X , L⊗m ) | inf (φ(v ) + v (s/sv )/m) ≥ λ/m}.
v ∈X ℶ

For a normal test configuration (X , L), we have Fφ = Fb(X ,L) .
We can easily observe
−1

λ
mχ
FDist
= Fφ,mχ (φ),m(λ/m).

(cf. Reboulet–Witt Nyström)e.g. For χ(σ) = −e −σ ,
(
λ
FDist
:=
χ (φ),m−m log(−λ/m)

Fφ,m
0

λ<0
λ≥0

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Distortion and Duistermaat–Heckman measure
Proposition (arXiv:2202.12168, see also M. Xia)
For each σ ∈ R, the map
Zφ(X ,L) 7→

DH(X ,L)[σ,∞)

is monotonic. For generalR φ ∈ PSHNA (L), we can define a unique Radon
measure DHφ satisfying R DHφ ≤ 1 andZnZoDHφ = inf
DH(X ,L) φ ≤ φ(X ,L) .
[σ,∞)

[σ,∞)

Proposition (To appear)
For φ ∈ PSHNA (L), we have DHDistχ (φ) = χ∗ DHφ .

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Distortion and Ding invariant
Let us recallZ
DNA (X , L) := inf (AX (v ) + φ(X ,L) (v )) −v ∈X ℶ

σDHφ(X ,L)R= lct(X ,X0 −(L+KX /A1 )) (X0 ) −n! (L·n+1 ).
(L·n ) (n + 1)!

For LNA (φ) := inf v ∈X ℶ (AX (v ) + φ(v )), we compute
LNA (Distχ (φ)) = inf inf (AX (v ) + tφ(t −1 v ) + χ▲ (t))
v ∈X ℶ t>0= inf inf (tAX (w ) + tφ(w ) + χ▲ (t))
t>0 w ∈X ℶ

= inf (t inf (AX (w ) + φ(w )) + χ▲ (t))
t>0w ∈X ℶ= χ(LNA (φ)),
using the duality
inf(tσ + χ▲ (t)) = χ(σ).

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Let’s observe µNA
RecallµNA (X , L) :=

1
m
·n−1
LGm
) − (KXlog,G)n(KX .L·n−1 )/P1 · e(n−1)! (KX .L
−
.
1
·n
L
·n
Gm )
(L )
n! (L ) − (e

Using
∗
m
KXlog,G/P1 − p KX =X

ordE (X0 )AX (vE )E Gm ,E ⊂X0

we can write
LGm
∗
LGm
m
m
) = ((KXlog,G) + (p ∗ KX · e LGm )(KXlog,G/P1 − p KX ) · e/P1 · eX
ordE (X0 )AX (vE )(E Gm · e LGm )
=
E ⊂X0
∗
∗
d
(e p (sKX +L)+(LGm −p L) )ds s=0Z
X(L·n )=
ordE (X0 )AX (vE )e −σ DH(E ,L|E ) + · · ·
n! R

+

E ⊂X0

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Moment measure
For a normal test configuration (X , L), we introduce the momentmeasure D(X ,L) on X ℶ × R by
D(X ,L) :=ℶn! X
ordE (X0 )δvXE ⊗ DH(E ,L|E )⟲Gm .
(L·n )E ⊂X0

Theorem (arXiv:2202.12168 + to appear)ℶWe
R can define a measure Dφ on X × R for φ ∈ PSHNA (L) with
DHφ = 1 so thatR
Dφi → Dφ weakly

for any φi ↘ φ.

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

Distortion and Moment measure
For an increasing concave function χ, consider the map∆χ : X ℶ × R → X ℶ × R : (v , σ) 7→ (χ̇(σ).v , χ(σ)).
Theorem (To appear)
For φ ∈ ENA (L), we have
DDistχ (φ) = ∆χ∗ Dφ .
Using AX (t.v ) = tAX (v ), we computeZ
ZZ
AX (v ) χ̇(σ)Dφ (v , σ) =AX (χ̇(σ).v )Dφ (v , σ)XℶR
X ℶ ×RZ
Z=(∆χ )∗ AX Dφ =AX ∆χ∗ Dφ
X ℶ ×RX ℶ ×RZZ=
AX DDistχ (φ) =AX MA(Distχ (φ)).
X ℶ ×RX ℶ ×R

Optimal degeneration of algebraic variety and Perelman entropy (Eiji Inoue)

The name “Distortion” is inspired by effect pedal for electric guitar

Thank you!