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 -metrics, projective flatness and families of polarized abelian varieties
Author(s):
Wing-Keung
To;
Lin
Weng
Journal:
Trans. Amer. Math. Soc.
356
(2004),
2685-2707.
MSC (2000):
Primary 14K99, 32G08, 32G13, 32Q20
Posted:
December 9, 2003
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Abstract:
We compute the curvature of the  -metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the  -metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space.
References:
-
- [A]
- M.F. Atiyah, Complex analytic connections in fiber bundles, Trans. Amer. Math. Soc. 85(1957), 181-207. MR 19:172c
- [APW]
- S. Axelrod, S.D. Pietra and E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Diff. Geom. 33(1991), 787-902. MR 92i:58064
- [Bo]
- A. Borel, Introduction aux groupes arithmétiques, Hermann, Paris, 1969. MR 39:5577
- [D1]
- S.K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Diff. Geom. 18 (1993), 269-277. MR 85a:32036
- [D2]
- S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50(1985), 1-26. MR 86h:58038
- [FS]
- A. Fujiki and G. Schumacher, The moduli space of compact extremal Kähler manifolds and generalized Petersson-Weil metrics, Publ. RIMS Kyoto Univ. 26(1990), 101-183. MR 92e:32012
- [G]
- H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. Math. IHES 5(1960), 233-292. MR 22:12544
- [GH]
- P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley and Sons, 1978. MR 80b:14001
- [H]
- N.J. Hitchin, Flat connections and geometric quantization, Commun. Math. Phys, 131(1990), 347-380. MR 91g:32022
- [Ko1]
- S. Kobayashi, Curvature and stability of vector bundles, Proc. Japan Acad. Ser. A. Math. Sci. 58(1982), 158-162. MR 83i:53090
- [Ko2]
- S. Kobayashi, Differential geometry of complex vector bundles, Princeton, NJ: Iwanami Shoten and Princeton University Press, 1987.
- [KN]
- S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. 1, Interscience Publishers, New York, 1963. MR 27:2945
- [Ku]
- M. Kuga, Fiber varieties over a symmetric space whose fibers are abelian varieties, I, II, Lect. Notes, Univ. of Chicago, 1963-64.
- [Lü]
- M. Lübke, Stability of Einstein-Hermitian vector bundles, Manuscripta Math. 42(1983), 245-247. MR 85e:53087
- [MFK]
- D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, 3rd Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1994. MR 95m:14012
- [Mok]
- N. Mok, Aspects of Kähler geometry on arithmetic varieties, Proc. Symp. Pure Math. 52, part 2(1991), 335-396. MR 92j:32027
- [M1]
- D. Mumford, On the equations defining abelian varieties I, Invent. Math. 1(1966) 287-354; II, Invent. Math. 3(1967) 75-135; III, Invent. Math. 3(1967) 215-244.
- [M2]
- D. Mumford, Towards an enumerative geometry of the moduli space of curves, M. Artin and J.Tate (eds.), Arithmetic and geometry, Birkhäuser, Basel-Boston, 1983, pp. 271-328. MR 85j:14046
- [MT]
- N. Mok and W.-K. To, Eigensections on Kuga families of abelian varieties and finiteness of their Mordell-Weil groups, J. Reine Angew. Math. 444(1993), 29-78. MR 94m:32045
- [NS]
- M.S. Narasimhan and C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math. 82(1965), 540-564. MR 32:1725
- [Sa]
- I. Satake, Algebraic structures on symmetric domains, Iwanami Shoten and Princeton Univ. Press, 1980. MR 82i:32003
- [Sch]
- G. Schumacher, The curvature of the Petersson-Weil metric on the moduli space of Kähler-Einstein manifolds, in V. Ancona and A. Silva (eds.), Complex Analysis and Geometry, Plenum Press, New York, 1993, pp. 339-354. MR 94g:32038
- [SD]
- H.P.F. Swinnerton-Dyer, Analytic theory of abelian varieties, Cambridge Univ. Press, Cambridge, 1974. MR 51:3180
- [Siu1]
- Y.-T. Siu, Complex analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Diff. Geom. 17(1982), 55-138. MR 83j:58039
- [Siu2]
- Y.-T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, Birkhäuser, Basel-Boston, 1987. MR 89d:32020
- [ST]
- G. Schumacher and M. Toma, On the Petersson-Weil metric for the moduli space of Hermitian-Einstein bundles and its curvature, Math. Ann. 293(1992), 101-107. MR 93h:32040
- [TUY]
- A. Tsuchiya, K. Ueno and Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Advanced Studies in Pure Math. 19(1989), 459-566. MR 92a:81191
- [TW]
- W.-K. To and L. Weng, curvature of the
 -metric on the direct image of a family of Hermitian-Einstein vector bundles, Amer. J. Math. 120 (1998), 649-661. MR 99d:32038 - [UY]
- K. Uhlenbeck and S.T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39(1986), 257-293. MR 88i:58154
- [W]
- G. Welters, Polarized abelian varieties and the heat equation, Compositio Math. 49(1983), 173-194. MR 85f:14045
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Additional Information:
Wing-Keung
To
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email:
mattowk@nus.edu.sg
Lin
Weng
Affiliation:
Graduate School of Mathematics, Kyushu University, Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan
Email:
weng@math.kyushu-u.ac.jp
DOI:
10.1090/S0002-9947-03-03488-3
PII:
S 0002-9947(03)03488-3
Received by editor(s):
November 14, 2002
Posted:
December 9, 2003
Copyright of article:
Copyright
2003,
American Mathematical Society
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