Pluripotential Theory and Convex Bodies Turgay Bayraktar Sabanci - - PowerPoint PPT Presentation

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Pluripotential Theory and Convex Bodies

Turgay Bayraktar

Sabanci University (Istanbul)

December 19, 2019

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Outline

1

Introduction

2

Review of Literature

3

Pluripotential theory associated to a convex body

4

Main Results

5

Ingredients of proofs The results are based on joint works with T. Bloom (Toronto) & N. Levenberg (Indiana) & C. H. Lu (Orsay)

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Weighted transfinite diameter of a compact set

Let K ⊂ Cd be a non-pluripolar compact set and Q : K → R be a continuous weight function. Let also {ej(z) := zα(j)}j=1,...,dN be the standard monomial basis (the ordering is unimportant) for the space of polynomials Pn where dN := dim Pn. For points ζ1, ..., ζdN ∈ Cd, let VDMN(ζ1, ..., ζdN) : = det[ei(ζj)]i,j=1,...,dN = det    e1(ζ1) e1(ζ2) . . . e1(ζdN) . . . . . . ... . . . edN(ζ1) edN(ζ2) . . . edN(ζdN)    . We denote the weighted N-th order diameter by δQ,N(K) :=

• max

(x1,...,xdN )∈K dN |VDM(x1, . . . , xdN)|e−N dN

j=1 Q(xj) 1 ℓN

where ℓN := dN

j=1 deg(ej) = d d+1NdN. When Q ≡ 0 we simply write

δN(K).

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Weighted transfinite diameter of a compact set

Theorem (Zaharyuta ’75, Bloom-Levenberg ’10, Berman-Boucksom ’10) The limit δ

Q(K) := lim N→∞(δQ,N)

d d+1

exists and it is called the weighted transfinite diameter of K. Remark: In complex dimension one (i.e. d = 1) the δ(K) is equal to the Chebyshev constant T(K) := lim

N→∞[inf{pNK : pN(z) = zN + N−1

• j=1

ajzj}]

1 N

which is also equal to e−ρ(K) where ρ(K) := lim

|z|→∞[gK(z) − log |z|]

is the Robin constant of K and gK is the Green’s function of K with pole at infinity.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Weighted transfinite diameter of a compact set

Remarks: The case Q ≡ 0 is due to Zaharyuta. Berman & Boucksom obtained a far reaching generalization in the line bundle setting: Let L be a holomorphic line bundle on a compact complex manifold X of dimension d. Let H0(X, L) denote the space of global holomorphic sections. Let s1, . . . , sk be a basis for H0(X, L) and (x1, . . . xk) be k-tuple of points on X then the Vandermonde type determinant det[si(xj)]1≤i,j≤k is a section of the pull-back line bundle L⊠k over X k. For a given continuous Hermitian metric h on a big line bundle L → X and closed subset K ⊂ X; the role of δQ,N is played by the maximum of the point-wise norm | det[si(xj)]|h⊗N on K.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Global Weighted Extremal Function

Let K ⊂ Cd be a non-pluripolar compact set and Q : K → R be a continuous weight function. We define the weighted extremal function by V ∗

K,Q(z) := lim sup ζ→z

VK,Q(ζ) where VK,Q(z) := sup{u(z) : u ∈ L(Cd) and u ≤ Q on K}. Basic Facts: V ∗

K,Q ∈ L+(Cd)

When K is sufficiently regular, VK,Q(z) = sup{

1 deg(p) log |p(z)| :

p is a polynomial s.t. pe−deg(p)QK ≤ 1} By Bedford-Taylor theory µK,Q := (ddcV ∗

K,Q)d is a probability

measure called the weighted equilibrium measure, here ddc = i

π∂ ¯

∂.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Equidistribution of Fekete Points

Definition An array (f1, . . . , fdN) ∈ K dN is called Fekete array of order N if δQ,N(K) =

• |VDM(f1, . . . , fdN)|e−N dN

j=1 Q(fj) 1 ℓN .

Theorem (Berman-Boucksom-Nystr¨

• m ’11)

Let (x(N)

1

, . . . , x(N)

dN ) ∈ K dN denote sequence of array of points satisfying

lim

N→∞ |VDM(x(N) 1

, . . . , x(N)

dN )|e−N dN

j=1 Q(x(N) j

)

1 NdN = δ

Q(K)

(i.e. (x(N)

1

, . . . , x(N)

dN ) are asymptotically Fekete) then

1 dN

dN

• j=1

δx(N)

j

→ µK,Q weak-*

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

P-pluripotential Theory

Fix a convex body P ⊂ (R+)d; i.e., P is compact, convex and Po = ∅ (e.g., P is a non-degenerate convex polytope, i.e., the convex hull of a finite subset of (Z+)d in (R+)d with nonempty interior). We consider the finite-dimensional polynomial spaces Poly(NP) := {p(z) =

• J∈NP∩(Z+)d

cJzJ : cJ ∈ C} for N = 1, 2, ... where zJ = zj1

1 · · · zjd d for J = (j1, ..., jd). We let dN be

the dimension of Poly(NP). For P = Σ where Σ := {(x1, ..., xd) ∈ Rd : 0 ≤ xi ≤ 1,

d

• j=1

xi ≤ 1}, we have Poly(NΣ) = PN, the usual space of holomorphic polynomials of degree at most N in Cd. We assume that Σ ⊂ kP for some k ∈ Z+: note 0 ∈ P so Poly(NP) are linear spaces.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

A general notion of “degree”

Convexity of P implies that pN ∈ Poly(NP), pM ∈ Poly(MP) ⇒ pN · pM ∈ Poly((N + M)P). We may define an associated “norm” for x = (x1, ..., xd) ∈ Rd

+ via

xP := inf

λ>0{x ∈ λP}.

We remark that this defines a true norm on all of Rd if P is the positive “octant” of a centrally symmetric convex body B, i.e., P = B ∩ (R+)d. We may thus define a general degree of a polynomial p(z) =

α∈Zd

+ aαzα associated to the convex body P as

degP(p) := max

aα=0 αP.

Then Poly(NP) = {p : degP(p) ≤ N}.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

P−extremal functions in Cd

We briefly describe some basics of the “P-pluripotential theory” associated to a convex body P ⊂ Rd

+. Recall we have the logarithmic

indicator function on Cd HP(z) := sup

J∈P

log |zJ| := sup

J∈P

log[|z1|j1 · · · |zd|jd]. Define LP = LP(Cd) := {u ∈ PSH(Cd) : u(z) − HP(z) = 0(1), |z| → ∞}.

• Eg. For p ∈ Poly(NP), N ≥ 1 we have 1

N log |p| ∈ LP.

Given K ⊂ Cd and Q : K → R the weighted P−extremal function of K is V ∗

P,K,Q(z) where

VP,K,Q(z) := sup{u(z) : u ∈ LP(Cd), u ≤ Q on K}. Example: K = T d, the unit d−torus in Cd. Then VP,T d(z) = HP(z) = max

J∈P log |zJ|.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Siciak-Zaharyuta Theorem in P-setting

Using H¨

• rmander’s L2 estimates for ¯

∂−operator we obtain the following: Theorem (B. ’17) The function V ∗

P,K,Q ∈ L+ P . Moreover,

VP,K,Q(z) = lim

N→∞

• sup{ 1

N log |p(z)| : p ∈ Poly(NP), ||pe−NQ||K ≤ 1}

• .

This is our starting point to develop a P−pluripotential theory. In the special case P = Σ = {(x1, ..., xd) ∈ Rd : 0 ≤ xi ≤ 1,

d

• j=1

xi ≤ 1}, Poly(NΣ) = Pn, and we recover “classical” pluripotential theory: HΣ(z) = max[0, log |z1|, ..., log |zd|] = max[log+ |z1|, ..., log+ |zd|] and LΣ = L; VΣ,K = VK.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

P-Pluripotential Theory

Theorem (B. ’17) Let K ⊂ Cd be a non-pluripolar compact set, Q : K → R a continuous weight function and Pj ⊂ Rd

+ be a convex body for j = 1, . . . , d. Then

for uj ∈ L+

Pj the mixed complex Monge-Amp´

ere i π ∂ ¯ ∂u1 ∧ · · · ∧ i π ∂ ¯ ∂um is well defined and of total mass MVd(P1, . . . , Pd). In particular, the mass of µP,K,Q := (ddcV ∗

P,K,Q)d depends only on the dimension d and

volume of P. (Warning: MVd(Σ, . . . , Σ) = 1) Sketch of proof. The idea of the proof is to obtain a Bedford-Taylor type domination principle for the class LP and then using results of Passare & Rullg˚ ard in convex analysis on Rd

+ we obtain the assertion.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Applications

Random Polynomial Mappings and Value Distribution Theory (B. ’17) Approximation by polynomials of various degree (N. Trefethen ’17;

• L. Bos & N. Levenberg ’18)

Tropical Algebraic Geometry: Random Amoebas (on going) Determinantal Point Processes (joint works with T. Bloom & N. Levenberg ’18 & C. H. Lu ’19)

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Discretization

Recall dN is the dimension of Poly(NP). We can write Poly(NP) = span{e1, ..., edN} where {ej(z) := zα(j)}j=1,...,dN are the standard basis monomials. For points ζ1, ..., ζdN ∈ Cd, let VDMP

N(ζ1, ..., ζdN) :

= det[ei(ζj)]i,j=1,...,dN = det    e1(ζ1) e1(ζ2) . . . e1(ζdN) . . . . . . ... . . . edN(ζ1) edN(ζ2) . . . edN(ζdN)    . We denote the weighted N-th order diameter by δP,Q,N(K) :=

• max

(x1,...,xdN )∈K dN |VDM(x1, . . . , xdN)|e−N dN

j=1 Q(xj) 1 ℓN

where ℓN := dN

j=1 deg(ej). When Q ≡ 0 we simply write δP,N(K).

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Asymptotic P−Fekete arrays and (ddcVP,K,Q)d

Theorem (B. & Bloom & Levenberg ’18) Let P, K, Q as above. Then

1 The limit δ

P,Q(K) := limN→∞ δP,Q,N(K) exists.

2 For each N ∈ N and dN-tuple of points z(N)

1

, · · · , z(N)

dN ∈ K

satisfying lim

N→∞

• |VDMP

N(z(N) 1

, · · · , z(N)

dN )|e−N dN

j=1 Q(z(N) j

) 1

ℓN = δ

P,Q(K)

we have 1 dN

dN

• j=1

δz(N)

j

→ µP,K,Q := 1 d!Vol(P)(ddcVP,K,Q)d weak- ∗ .

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

A Determinantal Point Process

Let P, K, Q as before. A Bernstein-Markov measure ν is a finite measure

• n K satisfying for each p ∈ Poly(NP)

pe−NQK ≤ MNpe−NQL2(ν) where lim sup

N→∞

M

1 N

N = 1.

We define a probability measure on K dN via ProbN := 1 ZN |VDMP

N(z(N) 1

, · · · , z(N)

dN )|2e−2N dN

j=1 Q(xj)dν(z1) . . . dν(zdN)

where ZN is the normalizing constant. Proposition (BBL ’18) Let P, K, Q as before then lim

N→∞ Z

1 2ℓN

N

= δ

P,Q(K).

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

A Determinantal Point Process

In this context, we consider the probability space (χ, P) :=

∞

• N=1

(K dN, ProbN) Proposition (BBLL ’19) (SLLN) Let P, K, Q as before and ν be a Bernstein-Markov measure. Then for P- a.e. array {z(N)

j

} ∈ χ we have 1 dN

dN

• j=1

δz(N)

j

→ 1 γd µP,K,Q where γd := d!Vol(P).

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Large Deviation Principle

Definition Let X be a Polish space (i.e. separable, complete metrizable space) and PN be a sequence of probability measures on X, αN be a sequence of positive real numbers such that limN αN = ∞. Let also I : X → [0, ∞] be a lower semicontinous function on X. We say that the sequence {Pn} satisfy large deviation principle with speed αN and rate function I if for each Borel measurable set E ⊂ X − inf

x∈E ◦ I(x) ≤ lim inf N

α−1

N log PN(E) ≤ lim sup N

α−1

N log PN(E) ≤ − inf x∈ ¯ E I(x).

We say that I is a good rate function if the level sets are compact subsets of X. Here, E ◦ (resp. ¯ E) denotes the interior (resp. closure) of E.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Large Deviation Principle

Next, we consider the map jN : K dN → MP(K) jN(z1, . . . , zdN) := γd dN

dN

• j=1

δzj, where γd := d!Vol(P). Theorem (BBLL ’19) (LDP) The sequence σN := (jN)∗ProbN of probability measures on MP(K) satisfies a large deviation principle with speed 2ℓN and good rate function I(µ) := log JQ(µP,K,Q) − log JQ(µ) where JQ(µ) := inf

u∈C(K)[log δ P,u(K) + bd

• K

udµ] − bd

• K

Qdµ

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Sketch of Proof of LDP

Theorem (G¨ artner-Ellis ) Let C(K)∗ be the topological dual of C(K), and let {σǫ} be a family of probability measures on MP(K) ⊂ C(K)∗ (equipped with the weak-* topology). Suppose for each λ ∈ C(K), the limit Λ(λ) := lim

ǫ→0 ǫ log

• C(K)∗ e

1 ǫ <λ,x>dσǫ(x)

exists as a finite real number and assume Λ is Gˆ ateaux differentiable; i.e., for each λ, θ ∈ C(K), the function f (t) := Λ(λ + tθ) is differentiable at t = 0. Then {σǫ} satisfies an LDP in C(K)∗ with the convex, good rate function Λ∗. Here Λ∗(x) := sup

λ∈C(K)

• < λ, x > −Λ(λ)
• ,

is the Legendre transform of Λ.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Sketch of Proof of LDP

Now, we observe that for every u ∈ C(K) Λ(u) : = lim

N→∞

1 2ℓN log

• C(K)∗ e2ℓN<u,µ>dσN(µ)

= log ¯ δP,Q− u

bd

¯ δP,Q Theorem (BBL ’18 Rumely type formula) log ¯ δP,Q(K) = − 1 bd E(V ∗

P,K,Q)

where E(u) := EHP(u) = 1 d + 1

d

• j=0
• Cd(u − HP)ddcuj ∧ (ddcHP)d−j.

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Sketch of Proof of LDP

Thus it is enough to show that for u1, u2 ∈ C(K), f (t) := E(V ∗

P,K,Q−(u1+tu2))

is Gˆ ateaux differentiable. Indeed, this is the case and f ′(t) =

• Cd u2(ddcV ∗

P,K,Q+(u1+tu2))d.

Hence, the sequence σN satisfies LDP with a good rate function Λ∗. Finally, we computing Λ∗(µ) = I(µ): I(µ) = bd

• sup

u∈C(K)

E(V ∗

P,K,u) −

• K

udµ

• − bd
• E(V ∗

P,K,Q) −

• K

Qdµ

• =

log JQ(µK,Q) − log JQ(µ)

Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs

Thank you!

Turgay Bayraktar Pluripotential Theory and Convex Bodies