Growth of balls of holomorphic sections and energy at equilibrium

By Robert J. Berman, S. Boucksom. 2008

226 citations recorded by the source corpus.

Let L be a big line bundle on a compact complex manifold X. Given a non-pluripolar compact subset K of X and a continuous Hermitian metric e−φ on L, we define the energy at equilibrium of (K,φ) as the Monge-Ampère energy of the extremal psh weight associated to (K,φ). We prove the differentiability of the energy at equilibrium with respect to φ, and we show that this energy describes the asymptotic behaviour as k→∞ of the volume of the sup-norm unit ball induced by (K,kφ) on the space of global holomorphic sections H0(X,kL). As a consequence of these results, we recover and extend Rumely’s Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan’s equidistribution theorem for algebraic points of small height to the case of a big line bundle.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b3ded9235f99bd1b36c9664d9ac0f8e1aa73d8ee

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, methodology): “This follows from general maximality properties of envelopes (compare [11]), but in the present setting it also follows”
Smooth and singular Kahler-Einstein metrics (2014, methodology): “In addition, the result of Berman–Boucksom is essentially needed if one replaces ωnϕ by a more general measure or even volume form ν; the result can be seen as the starting point of the variational approach to constructing a weak solution of the equation ωnϕ = ν.”
Monge–Ampère equations in big cohomology classes (2008, background): “[2], [33], and [6] for the extension to the singular setting).”
Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry (2018, methodology): “ vol(L,φ,ψ) = E(P(φ),P(ψ)). In the Archimedean case, classicial results in pluripotential imply that continuity of envelopes holds whenever Xis normal, and Corollary B is indeed also a consequence of [BB10] in that case. In the non-Archimedean case, we similarly expect continuity of envelopes to hol”
A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics (2010, background): “l-known formula: (2.4) Eω(u) := 1 (n+1)!V Xn j=0 ˆ X uωj u∧(ω)n−j. Conversely, one can simply take this latter formula as the deﬁnition of Eω and observe that the following proposition holds (compare [7] for a more general singular setting): Proposition 2.3. The following holds (i) The diﬀerential o”
Polynomial Interpolation and Approximation in C^d (2011, background): “The main results in [3], which are stated and proved in a much more general setting than weighted pluripotential theory in C and lead to the results in [4], require the weighted theory.”
Degenerate complex Hessian equations on compact K\"ahler manifolds (2014, background): “One could prove the lemma by following [5].”
On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations (2020, unknown): “By [12], d(t) is differentiable at t = 0, d′(0) = ∫ X v(−MAg(φ) + e−φdμ) = 0.”