Relating transfinite diameters using an Okounkov body. (English) Zbl 1435.32045

Ma‘u, Sione

Eur. J. Math. 5, No. 3, 1013-1032 (2019).

This paper studies the notion of transfinite diameter using an Okounkov body. For a given collection of \(m+1\) points \(F:=\{\zeta_0,\dots,\zeta_m\}\subset K\subset\mathbb{C}\), let \(\mathrm{VDM}(F):=\prod_{j<k}(\zeta_j-\zeta_k)\). Maximizing \(|\mathrm{VDM}(F)|^{1/m}\) over all subsets \(F\) of size \(m+1\) gives \(d_m(K)\), the \(m\)th-order diameter of \(K\). A subset for which the maximum is attained is called a set of Fekete points of order \(m\), and the transfinite diameter is \(d(K):=\lim_{m\to\infty}d_m(K)\).
An Okounkov body is a well-studied object in complex geometry that gives a simplified model of the local behaviour of a holomorphic line bundle over a complex manifold. The main result of this paper is given in Theorem 5.8, in which relations between the transfinite diameter of a locally circled subset \(K\) of the complexified sphere in \(\mathbb{C}^3\) and notions of weighted transfinite diameter of the projection of \(K\) to \(\mathbb{C}^2\) are derived. The method is based on connecting a Chebyshev transform of \(K\) on an Okounkov body of \(V\) to the classical Chebyshev transform of the projection of \(K\) to \(\mathbb{C}^2\).

Reviewer: Konstantin Malyutin (Sumy)

MSC:

32U20	Capacity theory and generalizations
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
52B20	Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Keywords:

transfinite diameter; Okounkov body; Fekete point; locally circled set; Chebyshev transform; affine variety

References:

[1]	Berman, R., Boucksom, S.: Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math. 181(2), 337-394 (2010) · Zbl 1208.32020
[2]	Bloom, T., Bos, L., Christensen, C., Levenberg, N.: Polynomial interpolation of holomorphic functions in \[\mathbf{C}C\] and \[{\mathbf{C}}^nCn\]. Rocky Mountain J. Math. 22(2), 441-470 (1992) · Zbl 0763.32009
[3]	Bloom, T., Bos, L.P., Calvi, J.-P., Levenberg, N.: Polynomial interpolation and approximation in \[{\mathbf{C}}^dCd\]. Ann. Polon. Math. 106, 53-81 (2012) · Zbl 1254.32045
[4]	Bloom, T., Levenberg, N.: Weighted pluripotential theory in \[{\mathbf{C}}^N\] CN. Amer. J. Math. 125(1), 57-103 (2003) · Zbl 1067.31007
[5]	Boucksom, S., Chen, H.: Okounkov bodies of filtered linear series. Compositio Math. 147(4), 1205-1229 (2011) · Zbl 1231.14020
[6]	Cox, D.A., Ma‘u, S.: Transfinite diameter on complex algebraic varieties. Pacific J. Math. 291(2), 279-317 (2017) · Zbl 1375.32054
[7]	DeMarco, L., Rumely, R.: Transfinite diameter and the resultant. J. Reine Angew. Math. 611, 145-161 (2007) · Zbl 1131.37054
[8]	Kaveh, K., Khovanskii, A.G.: Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176(2), 925-978 (2012) · Zbl 1270.14022
[9]	Küronya, A., Lozovanu, V., Maclean, C.: Convex bodies appearing as Okounkov bodies of divisors. Adv. Math. 229(5), 2622-2639 (2012) · Zbl 1253.14008
[10]	Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. 42(5), 783-835 (2009) · Zbl 1182.14004
[11]	Ma‘u, S.: Newton-Okounkov bodies and transfinite diameter. Dolomites Res. Notes Approx. 10(Special Issue), 138-160 (2017). arXiv:1701.00570 · Zbl 1370.14008
[12]	Witt Nyström, D.: Transforming metrics on a line bundle to the Okounkov body. Ann. Sci. Éc. Norm. Supér. 47(4), 1111-1161 (2014) · Zbl 1329.14025
[13]	Zaharjuta, V.P.: Transfinite diameter, Čebyšev constants, and capacity for compacta in \[\mathbb{C}^nCn. Sb\]. Math. 25(3), 350-364 (1975) · Zbl 0333.32004