Metric dependence and asymptotic minimization of the expected number of critical points of random holomorphic sections
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Abstract:
We prove the main conjecture from Douglas, Shiffman, and Zelditch (2006) concerning the metric dependence and asymptotic minimization of the expected number $\mathcal {N}^{\operatorname {crit}}_{N,h}$ of critical points of random holomorphic sections of the $N$th tensor power of a positive line bundle. The first non-topological term in the asymptotic expansion of $\mathcal {N}^{\operatorname {crit}}_{N,h}$ is the Calabi functional multiplied by the constant $\beta _2(m)$ which depends only on the dimension of the manifold. We prove that $\beta _2(m)$ 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 $\mathcal {N}^{\operatorname {crit}}_{N,h}$.References
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Additional Information
- Benjamin Baugher
- Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
- Address at time of publication: 3353 Deep Well Ct., Abingdon, Maryland 21009
- Email: bbaugher@math.jhu.edu
- Received by editor(s): February 13, 2008
- Published electronically: April 27, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4537-4555
- MSC (2010): Primary 53C55
- DOI: https://doi.org/10.1090/S0002-9947-10-04801-4
- MathSciNet review: 2645040