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 Introduction 1 Review of Literature 2 Pluripotential theory associated to a convex body 3 Main Results 4 Ingredients of proofs 5 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 ⊂ C d be a non-pluripolar compact set and Q : K → R be a continuous weight function. Let also { e j ( z ) := z α ( j ) } j =1 ,..., d N be the standard monomial basis (the ordering is unimportant) for the space of polynomials P n where d N := dim P n . For points ζ 1 , ..., ζ d N ∈ C d , let VDM N ( ζ 1 , ..., ζ d N ) : = det[ e i ( ζ j )] i , j =1 ,..., d N   e 1 ( ζ 1 ) e 1 ( ζ 2 ) . . . e 1 ( ζ d N ) . . . ... . . . = det  .   . . .  e d N ( ζ 1 ) e d N ( ζ 2 ) . . . e d N ( ζ d N ) We denote the weighted N-th order diameter by j =1 Q ( x j ) � 1 ( x 1 ,..., x dN ) ∈ K dN | VDM ( x 1 , . . . , x d N ) | e − N � dN δ Q , N ( K ) := � max ℓ N where ℓ N := � d N d d +1 Nd N . When Q ≡ 0 we simply write j =1 deg( e j ) = δ N ( 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 Q ( K ) := lim d N →∞ ( δ Q , N ) δ 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 N − 1 N →∞ [inf {� p N � K : p N ( z ) = z N + � 1 a j z j } ] T ( K ) := lim N j =1 which is also equal to e − ρ ( K ) where ρ ( K ) := | z |→∞ [ g K ( z ) − log | z | ] lim is the Robin constant of K and g K 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 H 0 ( X , L ) denote the space of global holomorphic sections. Let s 1 , . . . , s k be a basis for H 0 ( X , L ) and ( x 1 , . . . x k ) be k -tuple of points on X then the Vandermonde type determinant det [ s i ( x j )] 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 δ Q , N is played by the maximum of the point-wise norm | det[ s i ( x j )] | 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 ⊂ C d 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 V K , Q ( ζ ) ζ → z where V K , Q ( z ) := sup { u ( z ) : u ∈ L ( C d ) and u ≤ Q on K } . Basic Facts: V ∗ K , Q ∈ L + ( C d ) 1 When K is sufficiently regular, V K , Q ( z ) = sup { deg ( p ) log | p ( z ) | : p is a polynomial s.t. � pe − deg ( p ) Q � K ≤ 1 } K , Q ) d is a probability By Bedford-Taylor theory µ K , Q := ( dd c V ∗ measure called the weighted equilibrium measure, here dd c = 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 ( f 1 , . . . , f d N ) ∈ K d N is called Fekete array of order N if | VDM ( f 1 , . . . , f d N ) | e − N � dN j =1 Q ( f j ) � 1 δ Q , N ( K ) = ℓ N . � Theorem (Berman-Boucksom-Nystr¨ om ’11) Let ( x ( N ) , . . . , x ( N ) d N ) ∈ K d N denote sequence of array of points satisfying 1 d N ) | e − N � dN j =1 Q ( x ( N ) 1 Q ( K ) N →∞ | VDM ( x ( N ) , . . . , x ( N ) ) � NdN = δ lim j 1 (i.e. ( x ( N ) , . . . , x ( N ) d N ) are asymptotically Fekete) then 1 d N 1 � δ x ( N ) → µ K , Q weak-* d N j j =1 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 P o � = ∅ (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 c J z J : c J ∈ C } � Poly ( NP ) := { p ( z ) = J ∈ NP ∩ ( Z + ) d for N = 1 , 2 , ... where z J = z j 1 1 · · · z j d d for J = ( j 1 , ..., j d ). We let d N be the dimension of Poly ( NP ). For P = Σ where d Σ := { ( x 1 , ..., x d ) ∈ R d : 0 ≤ x i ≤ 1 , � x i ≤ 1 } , j =1 we have Poly ( N Σ) = P N , the usual space of holomorphic polynomials of degree at most N in C d . 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 p N ∈ Poly ( NP ) , p M ∈ Poly ( MP ) ⇒ p N · p M ∈ Poly (( N + M ) P ) . We may define an associated “norm” for x = ( x 1 , ..., x d ) ∈ R d + via � x � P := inf λ> 0 { x ∈ λ P } . We remark that this defines a true norm on all of R d 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 + a α z α associated to the convex body P as p ( z ) = � α ∈ Z d deg P ( p ) := max a α � =0 � α � P . Then Poly ( NP ) = { p : deg P ( p ) ≤ N } . Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs P − extremal functions in C d We briefly describe some basics of the “ P -pluripotential theory” associated to a convex body P ⊂ R d + . Recall we have the logarithmic indicator function on C d log[ | z 1 | j 1 · · · | z d | j d ] . log | z J | := sup H P ( z ) := sup J ∈ P J ∈ P Define L P = L P ( C d ) := { u ∈ PSH ( C d ) : u ( z ) − H P ( z ) = 0(1) , | z | → ∞} . Eg. For p ∈ Poly ( NP ) , N ≥ 1 we have 1 N log | p | ∈ L P . Given K ⊂ C d and Q : K → R the weighted P − extremal function of K is V ∗ P , K , Q ( z ) where V P , K , Q ( z ) := sup { u ( z ) : u ∈ L P ( C d ) , u ≤ Q on K } . Example: K = T d , the unit d − torus in C d . Then J ∈ P log | z J | . V P , T d ( z ) = H P ( z ) = max Turgay Bayraktar Pluripotential Theory and Convex Bodies

Introduction A Classical result Our Setting Main Results Ingredients of proofs Siciak-Zaharyuta Theorem in P-setting ormander’s L 2 estimates for ¯ Using H¨ ∂ − operator we obtain the following: Theorem (B. ’17) P , K , Q ∈ L + The function V ∗ P . Moreover, sup { 1 � N log | p ( z ) | : p ∈ Poly ( NP ) , || pe − NQ || K ≤ 1 } � V P , K , Q ( z ) = lim . N →∞ This is our starting point to develop a P − pluripotential theory. In the special case d P = Σ = { ( x 1 , ..., x d ) ∈ R d : 0 ≤ x i ≤ 1 , � x i ≤ 1 } , j =1 Poly ( N Σ) = P n , and we recover “classical” pluripotential theory: H Σ ( z ) = max[0 , log | z 1 | , ..., log | z d | ] = max[log + | z 1 | , ..., log + | z d | ] and L Σ = L ; V Σ , K = V K . 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 ⊂ C d be a non-pluripolar compact set, Q : K → R a continuous weight function and P j ⊂ R d + be a convex body for j = 1 , . . . , d. Then for u j ∈ L + P j the mixed complex Monge-Amp´ ere i ∂ u 1 ∧ · · · ∧ i π ∂ ¯ π ∂ ¯ ∂ u m is well defined and of total mass MV d ( P 1 , . . . , P d ) . In particular, the P , K , Q ) d depends only on the dimension d and mass of µ P , K , Q := ( dd c V ∗ volume of P. (Warning: MV d (Σ , . . . , Σ) = 1 ) Sketch of proof. The idea of the proof is to obtain a Bedford-Taylor type domination principle for the class L P and then using results of Passare & Rullg˚ ard in convex analysis on R d + 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