Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

    [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