Metric Dependence and Asymptotic Minimization of the Expected Number of Critical Points of Random Holomorphic Sections

Details Metric Dependence and Asymptotic Minimization of the Expected Number of Critical Points of Random Holomorphic Sections Metric Dependence and Asymptotic Minimization of the Expected Number of Critical Points of Random Holomorphic Sections

2008-02-12

arxiv.org/abs/0802.1549v2

Physics

Mathematical Physics

19 pages, added references; also includes a Mathematica worksheet in both notebook and pdf form

Scientific paper

We prove the main conjecture from [M. R. Douglas, B. Shiffman and S. Zelditch, Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the metric dependence and asymptotic minimization of the expected number \mathcal{N}^{crit}_{N,h} of critical points of random holomorphic sections of the Nth tensor power of a positive line bundle. The first non-topological term in the asymptotic expansion of \mathcal{N}^{crit}_{N,h} is the the Calabi functional multiplied by the constant \be_2(m) which depends only on the dimension of the manifold. We prove that \be_2(m) is strictly positive in all dimensions, showing that the expansion is non-topological for all m, and that the Calabi extremal metric, when it exists, asymptotically minimizes \mathcal{N}^{crit}_{N,h}.

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