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), 291302
MSC (2000):
Primary 53D50, 53D35, 60D05
Published electronically:
May 8, 2002
MathSciNet review:
1929049
Fulltext PDF Free Access
Abstract 
References 
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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 almostholomorphic 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), 351395.
 [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. 3169.
 [BSZ3]
P. Bleher, B. Shiffman and S. Zelditch, Correlations between zeros and supersymmetry, Commun. Math. Phys. 224 (2001), 255269.
 [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
(85j:58141)
 [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. 3435 (French).
MR
0590106 (58 #28684)
 [Don]
S.
K. Donaldson, Symplectic submanifolds and almostcomplex
geometry, J. Differential Geom. 44 (1996),
no. 4, 666–705. MR 1438190
(98h:53045)
 [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), 181222.
 [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
(94a:58082)
 [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), 351395.
 [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. 3169.
 [BSZ3]
 P. Bleher, B. Shiffman and S. Zelditch, Correlations between zeros and supersymmetry, Commun. Math. Phys. 224 (2001), 255269.
 [BoGu]
 L. Boutet de Monvel and V. Guillemin, The Spectral Theory of Toeplitz Operators, Ann. Math. Studies 99, Princeton Univ. Press, Princeton, 1981. MR 85j:58141
 [BoSj]
 L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Asterisque 3435 (1976), 123164. MR 58:28684
 [Don]
 S. K. Donaldson, Symplectic submanifolds and almost complex geometry, J. Diff. Geom. 44 (1996), 666705. MR 98h:53045
 [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), 181222.
 [ShZe3]
 B. Shiffman and S. Zelditch, Random polynomials and Levy concentration of measure, (in preparation).
 [Woo]
 N. M. J. Woodhouse, Geometric Quantization, Clarendon Press, Oxford, 1992. MR 94a:58082
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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:
http://dx.doi.org/10.1090/S0002993902065577
PII:
S 00029939(02)065577
Received by editor(s):
August 3, 2001
Published electronically:
May 8, 2002
Additional Notes:
Research partially supported by NSF grants #DMS9800479, #DMS0100474 (first author) and #DMS0071358 (second author).
Communicated by:
Christopher D. Sogge
Article copyright:
© Copyright 2002
American Mathematical Society
