A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

A Thermodynamical formalism for Optimal
degeneration problem in Kähler geometry
Eiji INOUE （RIKEN iTHEMS, Japan）

29, March, 2022, RIKEN iTHEMS Now and Next

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Entropy and Equilibrium in Stochastic thermodynamicsΩ: a finite set (for simiplicity) called systemE : Ω → R: a function called Hamiltonian
For a probability distribution p on Ω (non-equilibrium state), we consider∑p(i) log p(i) . . . EntropyS(p) := −i∈ΩU(p) :=∑E (i)p(i)...

Internal energyi∈ΩA state p is called equilibrium if it maximizes S among all p ′ with thesame internal energy U(p ′ ) = U(p).
Lagrange multiplier β = 1/T⇝ Equilibrium is a minimizer of U − TS for a suitable∑ choice of T .
⇝ Equilibrium = Canonical distribution p = e −βE / e −βE .

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Setup in Kähler geometry
Today, I explain optimal degeneration problem in Kähler geometry can beformalized in an analogous way.
Mathematical setup:
X : a compact (= closed) complex manifold
L: a polarization or a Kähler class ∈ H 2 (X , R)
Complex = Cn = (R ⊕ Ri)n ⇝ X is real 2n-dimensional• Locally, X = {f1 (z1 , . . . , zN ) = · · · = fk (z1 , . . . , zN ) = 0}, using
multivariable holomorphic/polynomial functions.
• Global structure (compactness) is important in today’s talke.g. X can be a Riemann surface, K3 surface, CP n , toric manifolds ....
Comment: Closely related to Symplectic manifold (phase space).

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Holomorphic structure
X ∼= X ′ “⇔” ∃ a holomorphic change of variables: zi′ = hi (z1 , . . . , zN ).
e.g. the upper half space H = {z ∈ C | Imz > 0} and a disc∆ = {z ∈ C | |z| < 1} are biholomorphic, while shapes are diﬀerent.

“Geometric shape” of X is not yet determined.
⇝ Consider “geometric shapes” of X and study the “best shape”.

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Canonical metric
Consider a Riemannian metric g on X .
Kähler condition: we assume the following compatibility with (X , L)
1. On a holomorphic coordinate, g can be written as√
g = gi j¯dz i d z̄ j + gi j¯d z̄ i dz j , dz i = dx i + −1dy i .
2. The 2-form√−1gi j¯dz i ∧ d z̄ j is closed and its deRham class is L.

Curvatures of g :Ricci curvature: Rici j¯ := −∂i ∂¯j log det gk l¯¯Scalar curvature: s := g i j Rici j¯
g is calledKähler–Einstein metric: Rici j¯ = λgi j¯Kähler–Ricci soliton: Rici j¯ − ∂i ∂¯j f = λgi j¯
cscK metric: s = const⇝ 2πc1 (X ) = λL

⇝ const = 2πn(c1 (X ).L·n−1 )/(L·n )A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Two situations

Given a space (X , L), there are two possibilities:
1. There exists a canonical metric, in which case solution is unique.
2. There is no canonical metric.
Intuitively,
1. Space (X , L) admits a static equilibrium.
2. Space (X , L) admits no static equilibrium.
The case 1 ... Yau–Tian–Donaldson conjecture.
The case 2 ... more important, interesting and challenging.

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Dynamics of shape of space

Conjecture (Optimal degeneration)
When there is no canonical metric, there would be dynamics g (t) ofspace (X , L) which develops to a static equilibrium g∞ of another space(X0 , L0 ) at t = ∞.
e.g. Ricci flow (limit: Ricci soliton), Calabi flow, ... “µ-flow”?
How one can find “canonical flow” satisfying the above property?
Idea: Regard dynamics g (t) as a state of spacetime X × [0, ∞) andcharacterize dynamics with the above property as “equilibrium ofspacetime”.
Especially interested in the limit behavior t = ∞.
⇝ Want to determine boundary condition for equilibrium.

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Boundary condition for spacetime: schematic degeneration
Let’s put the vision into a mathematical form.
We will introduce the notion of degeneration of (X , L), assign quantities
to it and characterize optimal degeneration by these quantities.
Scheme theoretic definition: a degeneration is a C× -equivariant flatfamily ϖ : X → C of schemes over C together with a relative
polarization L and the base change diagram
X × (C \ {0}) ⊔ X0(C \ {0}) ⊔ {0}C× -equiv fb-wise isom.

XC“Equilibrium of spacetime” g (t) must converge towards a canonicalmetric on X0 = ϖ−1 (0). (Not precise, for simplicity)⇝ X represents a boundary condition.

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Illustration of degeneration

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Toric manifold: for brevity
Before introducing quantities for degeneration...
Let’s focus on a convenient class: { toric manifolds } ⊂ { Kähler mfds }.
It helps to describe the quantities in a simpler way.
(I need 100 pages to explain general case. )Figure: This represents Figure: This represents Figure: This represents
CP 2 #CP 2
L on CP 2 #CP 2L′ on CP 2 #CP 2

A convex polytope P ⊂ Rn spanned by vertices v1 , . . . , vN ∈ Zn .
⇝ (X , L) associated to P, which is a real 2n-dimensional manifold with atorus symmetry X ⟲ T = U(1)×n and a moment map µ : X → Rn .

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)“Non-archimedean” Equilibrium
A “nice” convex
function q on P assigns a degeneration of (X , L).
∫dµ qP∫We put u := e q dµ e andP∫
1∫
SP (u) := −u log udµ · · · Entropydµ PP∫
1UP (u) := ∫udσ · · · Internal µ-energydµ ∂PP
Put UP := [min U, U(1P )]. For U ∈ UP , we call u equilibrium of internalenergy U if
UP (u) = U,SP (u) = sup{SP (u ′ ) | UP (u ′ ) = U}.

Theorem (’23)
For each U ∈ UP , there exists a unique equilibrium of internal energy U.
If UP ̸= {U(1P )}, space (X , L) admits no cscK metric (no staticequilibrium).

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Free µ-energy
Now we consider
FP (T , u) = UP (u) − TSP (u)···Free µ-energy.

Theorem (’23)
For T ≥ 0, there exists a unique minimizer u of FP (T , •), which isequilibrium of some internal energy U ∈ U∗P ∪ {U(1P )}.
Moreover, if the minimizer is a “nice” convex function, thenX0 = ϖ−1 (0) of the associated degeneration X is µ−2πT K-semistable.
To show the existence, we use compactness: for a minimizing sequence
ui , we prove that we can take a subsequence which converges to some
u∞ in a suitable topology. Diﬃcult to describe the minimizer explicitly.
µK-semistability is related to the existence of µ-cscK metric: a kind of
canonical metric which generalizes cscK metric and Kähler–Ricci soliton.

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Perelman entropy and minimax principle
These are related to Perelman’s entropy.
1volgX∫

1(s(g ) + |∇f |2 + 2πTf )e f volg ,2X∫∫µPer (T , g ) := inf {W (T , g , f ) |e f volg =volg }.

W (T , g , f ) := ∫fX

X

Theorem (’21)
We havemin 2πFP (T , q) ≥ sup µPer (T , g ).
qg−2πT

For T ≥ 0, g is a µ-cscK metric if and only if g maximizesµPer (T , g ). If its maximizer exists, we havemin 2πFP (T , q) = max µPer (T , g ).
qg

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Mathematical questions

1

Prove the equality minq 2πFP (T , q) = supg µPer (T , g ) for general
case.

2

Show the existence of optimal degeneration for general Kähler
manifold by establishing the following two steps.
• Establish a compactness on non-archimedean psh metrics.
• Establish a regularity theory on non-archimedean psh metrics.

3

Construct moduli theory on µ−2πT K-semistable polarized schemes

4and observe “wall-crossing” of moduli spaces as T varies.

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)

Subtle questions
1Temperature is a new parameter in Kähler geometry. Can we use it
to understand “equilibrium of spacetime” g (t), using Wick rotation?

2

Can we find other physical structure in this framework?
Statistical mechanics, quantum mechanics, information theory...
(Since the framework only shares an abstract feature withthermodynamics, we might need to consider some abstraction ofthese theories. )3Is there any relation to quantum entanglement and “emergence ofspacetime”?

4

As usual in thermodynamics, free µ-energy can be derived as thediﬀerence of entropy of the composition with a heat bath (infinitedimensional space). This implicates the geometry of optimal
degeneration is of infinite dimensional nature (“infinite dimensionalYamabe problem”). Perelman finds a heuristic argument on
monotonicity of W -entropy which utilizes this picture. Can we findsimilar heuristics on minq 2πFP (T , q) = supg µPer (T , g )?

A Thermodynamical formalism for Optimal degeneration problem in Kähler geometry (Eiji Inoue)Thank you for listening!