Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds: An addendum

Authors:
Bernard Shiffman and Steve Zelditch

Journal:
Proc. Amer. Math. Soc. **131** (2003), 291-302

MSC (2000):
Primary 53D50, 53D35, 60D05

DOI:
https://doi.org/10.1090/S0002-9939-02-06557-7

Published electronically:
May 8, 2002

MathSciNet review:
1929049

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We define a Gaussian measure on the space of almost holomorphic sections of powers of an ample line bundle over a symplectic manifold , and calculate the joint probability densities of sections taking prescribed values and covariant derivatives at a finite number of points. We prove that they have a universal scaling limit as . This result will be used in another paper to extend our previous work on universality of scaling limits of correlations between zeros to the almost-holomorphic setting.

**[Arc]**Archimedes,*On the Sphere and Cylinder*(Greek), Syracuse, ca. 257BC.**[Aur]**Denis Auroux, Estimated transversality in symplectic geometry and projective maps, to appear in Proc. International KIAS Conference (Seoul, 2000), http://xxx.lanl.gov/abs/math.SG/0010052.**[BSZ1]**P. Bleher, B. Shiffman and S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds,*Invent. Math.*142 (2000), 351-395.**[BSZ2]**P. Bleher, B. Shiffman and S. Zelditch, Universality and scaling of zeros on symplectic manifolds, in*Random Matrix Models and Their Applications*, P. Bleher and A. Its (Eds.), MSRI Publications 40, Cambridge Univ. Press, 2001, pp. 31-69.**[BSZ3]**P. Bleher, B. Shiffman and S. Zelditch, Correlations between zeros and supersymmetry,*Commun. Math. Phys.*224 (2001), 255-269.**[BoGu]**L. Boutet de Monvel and V. Guillemin,*The spectral theory of Toeplitz operators*, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR**620794****[BoSj]**L. Boutet de Monvel and J. Sjöstrand,*Sur la singularité des noyaux de Bergman et de Szegő*, Journées: Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris, 1976, pp. 123–164. Astérisque, No. 34-35 (French). MR**0590106****[Don]**S. K. Donaldson,*Symplectic submanifolds and almost-complex geometry*, J. Differential Geom.**44**(1996), no. 4, 666–705. MR**1438190****[ShZe1]**B. Shiffman and S. Zelditch, Random almost holomorphic sections of ample line bundles on symplectic manifolds, (preprint 2000), http://xxx.lanl.gov/abs/math.SG/0001102.**[ShZe2]**B. Shiffman and S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds,*J. Reine Angew. Math.*544 (2002), 181-222.**[ShZe3]**B. Shiffman and S. Zelditch, Random polynomials and Levy concentration of measure, (in preparation).**[Woo]**N. M. J. Woodhouse,*Geometric quantization*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. Oxford Science Publications. MR**1183739**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
53D50,
53D35,
60D05

Retrieve articles in all journals with MSC (2000): 53D50, 53D35, 60D05

Additional Information

**Bernard Shiffman**

Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Email:
shiffman@math.jhu.edu

**Steve Zelditch**

Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Email:
zelditch@math.jhu.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06557-7

Received by editor(s):
August 3, 2001

Published electronically:
May 8, 2002

Additional Notes:
Research partially supported by NSF grants #DMS-9800479, #DMS-0100474 (first author) and #DMS-0071358 (second author).

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 2002
American Mathematical Society