Growth of balls of holomorphic sections and energy at equilibrium.(English)Zbl 1208.32020
Let \(L\) be a big line bundle on a compact complex manifold \(X\). Given a non-pluripolar compact set \(K\subset X\) and a continuous Hermitian metric \(e^{-\phi}\) on \(L\), the authors define \({\mathcal E}_{eq}(K,\phi)\), the energy at equilibrium of \((K,\phi)\), as the Monge-Ampère energy \(\mathcal E\) of the extremal plurisubharmonic weight \(P_K\phi\) (upper envelope of plurisubharmonic weights not exceeding \(\phi\) on \(K\)), given by \[ {d\over dt}{\mathcal E}((1-t)\phi_1+t\phi_2)|_{t=0}=\int_x(\phi_2-\phi_1)(dd^c\phi_1)^n.\] The functional \(\phi\mapsto {\mathcal E}_{eq}(K,\phi)\) is shown to be concave, continuous and Gâteau differentiable. Furthermore, this energy describes the asymptotic behavior, as \(k\to\infty\), of the volume of the unit ball \({\mathcal B}^\infty(K,k\phi)\) in the space \(H^0(kL)\) of global holomorphic sections in the \(L^\infty\)-norm as follows. Let \[ {\mathcal L}_k(K,\phi)=(2k h^0(kL))^{-1} \log\text{vol}_k{\mathcal B}^\infty(K,k\phi), \] then \({\mathcal L}_k(K_1,\phi_1) -{\mathcal L}_k(K_2,\phi_2)\to {\mathcal E}_{eq}(K_1,\phi_1)-{\mathcal E}_{eq}(K_2,\phi_2)\); here \(\text{vol}_k\) is the Lebesgue measure on the vector space \(H^0(kL)\). A similar formula is obtained for the volumes of the unit balls \({\mathcal B}^2(\mu,k\phi)\) in \(H^0(kL)\) with respect to \(L^2(\mu)\) for a probability measure \(\mu\) on K with the Bernstein-Markov property. As a corollary, Zaharjuta’s and Rumelely’s results on transfinite diameter are extended to the case of big line bundles. Applications to Arakelov geometry (an asymptotic description of the analytic torsion and an equidistribution theorem for algebraic points of small height) are given as well.
Reviewer: Alexandr Yu. Rashkovsky (Stavanger)
MSC:
| 32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |
| 32A36 | Bergman spaces of functions in several complex variables |
| 32C05 | Real-analytic manifolds, real-analytic spaces |
| 32J27 | Compact Kähler manifolds: generalizations, classification |
| 32L10 | Sheaves and cohomology of sections of holomorphic vector bundles, general results |
| 32U15 | General pluripotential theory |
| 32W20 | Complex Monge-Ampère operators |
Cite
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