Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Metric dependence and asymptotic minimization of the expected number of critical points of random holomorphic sections


Author: Benjamin Baugher
Journal: Trans. Amer. Math. Soc. 362 (2010), 4537-4555
MSC (2010): Primary 53C55
Published electronically: April 27, 2010
MathSciNet review: 2645040
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C55

Retrieve articles in all journals with MSC (2010): 53C55


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

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04801-4
PII: S 0002-9947(10)04801-4
Received by editor(s): February 13, 2008
Published electronically: April 27, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.