Balanced metrics and Chow stability of projective bundles over Riemann surfaces
Author:
Reza Seyyedali
Journal:
Proc. Amer. Math. Soc. 141 (2013), 28412853
MSC (2010):
Primary 32Q26; Secondary 53C07
Published electronically:
May 1, 2013
MathSciNet review:
3056574
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Abstract: In 1980, I. Morrison proved that slope stability of a vector bundle of rank over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. In a previous work, we generalized Morrison's result to higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit a constant scalar curvature metric and have a discrete automorphism group. In this article, we give a simple proof for polarizations , where is a positive integer, and the base manifold is a compact Riemann surface of genus .
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 [D2]
 S. K. Donaldson, Anti selfdual YangMills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), no. 1, 126. MR 765366 (86h:58038)
 [D3]
 S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479522. MR 1916953 (2003j:32030)
 [D4]
 S. K. Donaldson, Scalar curvature and projective embeddings. II, Q. J. Math. 56 (2005), no. 3, 345356. MR 2161248 (2006f:32033)
 [DZ]
 A. Della Vedova and F. Zuddas, Scalar curvature and asymptotic Chow stability of projective bundles and blowups. Trans. Amer. Math. Soc. 364 (2012), no. 12, 64956511. MR 2958945
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 D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2) 106 (1977), no. 1, 4560. MR 466475 (81h:14014)
 [H1]
 Y.J. Hong, Ruled manifolds with constant Hermitian scalar curvature, Math. Res. Lett. 5 (1998), no. 5, 657673. MR 1666868 (2000j:32039)
 [H2]
 Y.J. Hong, Constant Hermitian scalar curvature equations on ruled manifolds, J. Differential Geom. 53 (1999), no. 3, 465516. MR 1806068 (2001k:32041)
 [H3]
 Y.J. Hong, Gaugefixing constant scalar curvature equations on ruled manifolds and the Futaki invariants, J. Differential Geom. 60 (2002), no. 3, 389453. MR 1950172 (2004a:53040)
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 H. Luo, Geometric criterion for GiesekerMumford stability of polarized manifolds, J. Differential Geom. 49 (1998), no. 3, 577599. MR 1669716 (2001b:32035)
 [M]
 I. Morrison, Projective stability of ruled surfaces, Invent. Math. 56 (1980), no. 3, 269304. MR 561975 (81c:14007)
 [PS]
 D. H. Phong and J. Sturm, Stability, energy functionals, and KählerEinstein metrics, Comm. Anal. Geom. 11 (2003), no. 3, 565597. MR 2015757 (2004k:32041)
 [PS2]
 D. H. Phong and Jacob Sturm, Scalar curvature, moment maps, and the Deligne pairing. Amer. J. Math. 126 (2004), no. 3, 693712. MR 2058389 (2005b:53137)
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 J. Ross and R. Thomas, An obstruction to the existence of constant scalar curvature Kähler metrics, J. Differential Geom. 72 (2006), no. 3, 429466. MR 2219940 (2007c:32028)
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 [S2]
 R. Seyyedali, Balanced metrics and Chow stability of projective bundles over Kähler manifolds. II, preprint.
 [UY]
 K. Uhlenbeck and S.T. Yau, On the existence of HermitianYangMills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257S293. MR 861491 (88i:58154)
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 X. Wang, Balance point and stability of vector bundles over a projective manifold, Math. Res. Lett. 9 (2002), no. 23, 393411. MR 1909652 (2004f:32034)
 [W2]
 Xiaowei Wang, Canonical metrics on stable vector bundles. Comm. Anal. Geom. 13 (2005), no. 2, 253285. MR 2154820 (2006b:32031)
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 S. Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 1998, no. 6, 317331. MR 1616718 (99g:32055)
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Additional Information
Reza Seyyedali
Affiliation:
Department of Mathematics, University of California, Irvine, California 92697
Address at time of publication:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
Email:
rseyyeda@math.uci.edu, rseyyedali@uwaterloo.ca
DOI:
http://dx.doi.org/10.1090/S000299392013115480
Received by editor(s):
November 28, 2010
Received by editor(s) in revised form:
November 11, 2011
Published electronically:
May 1, 2013
Communicated by:
Michael Wolf
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
