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