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

DOI:
https://doi.org/10.1090/S0002-9947-10-04801-4

Published electronically:
April 27, 2010

MathSciNet review:
2645040

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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 of critical points of random holomorphic sections of the th tensor power of a positive line bundle. The first non-topological term in the asymptotic expansion of is the Calabi functional multiplied by the constant which depends only on the dimension of the manifold. We prove that is strictly positive in all dimensions, showing that the expansion is non-topological for all , and that the Calabi extremal metric, when it exists, asymptotically minimizes .

**[As]**R. Askey,*Some basic hypergeometric extensions of integrals of Selberg and Andrews*, SIAM J. Math. Anal. 11 (1980), 938-951. MR**595822 (82e:33002)****[Ba]**B. Baugher,*Asymptotics and Dimensional Dependence of the Number of Critical Points of Random Holomorphic Sections*, Comm. Math. Phys. 282 (2008), 419-433. MR**2421483 (2009h:32028)****[Bo]**R. Bott,*On a theorem of Lefschetz*, Michigan Math. J. 6 (1959), 211-216. MR**0215323 (35:6164)****[Ca1]**E. Calabi,*Extremal metrics*, in `Seminar on Differential Geometry. Papers presented at seminars held during the academic year 1979-1980', edited by S.-T. Yau. Annals of Mathematics Studies, 102, Princeton University Press, Princeton, N.J, 1982, 259-290. MR**645728 (83a:53002)****[Ca2]**E. Calabi,*Extremal metrics II*, in `Differential geometry and complex analysis', edited by I. Chavel and H. M. Farkas, Springer, Berlin, 1985, 95-114. MR**780039 (86h:53067)****[Don]**S. K. Donaldson,*Scalar curvature and projective embeddings*, I. J. Differential Geom. 59 (2001), 479-522. MR**1916953 (2003j:32030)****[DSZ1]**M. R. Douglas, B. Shiffman and S. Zelditch,*Critical points and supersymmetric vacua*, Comm. Math. Phys. 252 (2004), no. 1-3, 325-358. MR**2104882 (2005h:32047)****[DSZ2]**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. MR**2219939 (2007j:32012)****[DSZ3]**M. R. Douglas, B. Shiffman and S. Zelditch,*Critical points and supersymmetric vacua, III: String/M models*, Comm. Math. Phys. 265 (2006), no. 3, 617-671. MR**2231684 (2007j:32013)****[Hw]**A. D. Hwang,*On the Calabi energy of extremal Kähler metrics*, Internat. J. Math. 6 (1995), no. 6, 825-830. MR**1353997 (96i:58032)**

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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

DOI:
https://doi.org/10.1090/S0002-9947-10-04801-4

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.