Arakelov type inequalities for Hodge bundles over algebraic varieties, Part I: Hodge bundles over algebraic curves
Authors:
Jürgen Jost and Kang Zuo
Journal:
J. Algebraic Geom. 11 (2002), 535-546
DOI:
https://doi.org/10.1090/S1056-3911-02-00299-0
Published electronically:
February 13, 2002
MathSciNet review:
1894937
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Abstract |
References |
Additional Information
Abstract: We prove Arakelov inequalities for systems of Hodge bundles over algebraic varieties, generalizing the classical ones for families of semi-stable curves and abelian varieties. These inequalities are derived from the semi-stability of an associated Higgs bundle, a consequence of the existence of a Hermitian Yang-Mills metric.
[A]A A. Arakelov Families of algebraic curves with fixed degeneracies, Izv. Ak. Nauk. S.S.S.R, Ser. Math. 35 (1971), 1277-1302.
[BV]BV E. Bedulev and E. Viehweg On the Shafarevich conjecture for surfaces of general type over function fields Invent. Math. 139 (2000) 603-615.
[CKS]CKS E. Cattani, A. Kaplan and W. Schmid Degeneration of Hodge structures, Ann. Math. 123 (1986), 457-536.
[E1]E1 P. Eyssidieux Ph.D. Thesis, Orsay, 1994.
[E2]E2 P. Eyssidieux La characteristique d’Euler du complexe de Gauss-Manin, J. reine angew. Math. 490 (1997), 155-212.
[F]F G. Faltings Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983), 337-348.
[G]G P. Griffiths Topic in transendental algebraic geometry, Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, N.J. (1984).
[GS]GS P. Griffiths and W. Schmid Locally homogeneous complex manifolds, Acta Math. 123 (1969) 253-302.
[H]H N.J. Hitchin Lie groups and Teichmüller space, Preprint, Warwick University, 5/1990.
[JY]JY J. Jost and S. T. Yau Harmonic mappings and algebraic varieties over function fields, Amer. J. Math., 115 (1993) 1197-1227.
[K]K J. Kollár Subadditivity of Kodaira dimension: Fibers of general type, Adv. Studies in Pure Math. 10, 1987, pp.361-398.
[L]L K-F. Liu Geometric height inequalities Math. Research Letters 3, (1996), 637-702.
[P1]P1 C. Peters On Arakelov’s finiteness theorem for higher dimensional varieties Rend. Sem. Mat. Univ. Politec. Torino (1986) 43-50.
[P2]P2 C. Peters Arakelov-type inequalities for Hodge bundles Preprint, 26.10.1999
[Sch]Sch W. Schmid Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211-319.
[S1]S1 C.T. Simpson Constructing variations of Hodge structure using Yang-Mills theory and applications to unformization, JAMS 1 (1988), 867-918.
[S2]S2 C.T. Simpson Harmonic bundles on non-compact curves, JAMS 3(1990),713-770.
[UY]UY K.K. Uhlenbeck and S.T. Yau On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39-S (1986), 257-293.
[V1]V1 E. Viehweg Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Adv. Studies in Pure Math. 1, 1983 pp. 329-353.
[V2]V2 E. Viehweg Weak positivity and the additivity of the Kodaira dimension, II. The local Torelli Map, Classification of algebraic and analytic manifolds, Birkhäuser, Boston-Basel-Stuttgart, 1983, pp. 567-589.
[Z]Z K. Zuo On the Negativity of kernels of Kodaira-Spencer Maps on Hodge bundles and Applications, Preprint, 2000.
[A]A A. Arakelov Families of algebraic curves with fixed degeneracies, Izv. Ak. Nauk. S.S.S.R, Ser. Math. 35 (1971), 1277-1302.
[BV]BV E. Bedulev and E. Viehweg On the Shafarevich conjecture for surfaces of general type over function fields Invent. Math. 139 (2000) 603-615.
[CKS]CKS E. Cattani, A. Kaplan and W. Schmid Degeneration of Hodge structures, Ann. Math. 123 (1986), 457-536.
[E1]E1 P. Eyssidieux Ph.D. Thesis, Orsay, 1994.
[E2]E2 P. Eyssidieux La characteristique d’Euler du complexe de Gauss-Manin, J. reine angew. Math. 490 (1997), 155-212.
[F]F G. Faltings Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983), 337-348.
[G]G P. Griffiths Topic in transendental algebraic geometry, Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, N.J. (1984).
[GS]GS P. Griffiths and W. Schmid Locally homogeneous complex manifolds, Acta Math. 123 (1969) 253-302.
[H]H N.J. Hitchin Lie groups and Teichmüller space, Preprint, Warwick University, 5/1990.
[JY]JY J. Jost and S. T. Yau Harmonic mappings and algebraic varieties over function fields, Amer. J. Math., 115 (1993) 1197-1227.
[K]K J. Kollár Subadditivity of Kodaira dimension: Fibers of general type, Adv. Studies in Pure Math. 10, 1987, pp.361-398.
[L]L K-F. Liu Geometric height inequalities Math. Research Letters 3, (1996), 637-702.
[P1]P1 C. Peters On Arakelov’s finiteness theorem for higher dimensional varieties Rend. Sem. Mat. Univ. Politec. Torino (1986) 43-50.
[P2]P2 C. Peters Arakelov-type inequalities for Hodge bundles Preprint, 26.10.1999
[Sch]Sch W. Schmid Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211-319.
[S1]S1 C.T. Simpson Constructing variations of Hodge structure using Yang-Mills theory and applications to unformization, JAMS 1 (1988), 867-918.
[S2]S2 C.T. Simpson Harmonic bundles on non-compact curves, JAMS 3(1990),713-770.
[UY]UY K.K. Uhlenbeck and S.T. Yau On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39-S (1986), 257-293.
[V1]V1 E. Viehweg Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Adv. Studies in Pure Math. 1, 1983 pp. 329-353.
[V2]V2 E. Viehweg Weak positivity and the additivity of the Kodaira dimension, II. The local Torelli Map, Classification of algebraic and analytic manifolds, Birkhäuser, Boston-Basel-Stuttgart, 1983, pp. 567-589.
[Z]Z K. Zuo On the Negativity of kernels of Kodaira-Spencer Maps on Hodge bundles and Applications, Preprint, 2000.
Additional Information
Jürgen Jost
Affiliation:
Max Planck Institute for Mathematics, Inselstrasse 22-26, D-04103 Leipzig, Germany
Email:
jost@mis.mpg.de
Kang Zuo
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T. Hong Kong
MR Author ID:
269893
Email:
kzuo@math.cuhk.edu.hk
Received by editor(s):
December 2, 1999
Received by editor(s) in revised form:
October 17, 2000
Published electronically:
February 13, 2002
Additional Notes:
The second author was supported by a Heisenberg fellowship of the DFG
