Syzygies of compact complex hyperbolic manifolds
Authors:
Jun-Muk Hwang and Wing-Keung To
Journal:
J. Algebraic Geom. 22 (2013), 175-200
DOI:
https://doi.org/10.1090/S1056-3911-2012-00578-5
Published electronically:
April 17, 2012
MathSciNet review:
2993051
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We give a sufficient condition for the pluri-canonical bundles of a compact complex hyperbolic manifold to satisfy the property $(N_p)$ on linear syzygies in terms of the hyperbolic injectivity radius. In the process, we obtain sharp lower bounds for the volumes of one-dimensional complex analytic subvarieties in geodesic tubular neighborhoods of the Cartesian self-product of a compact complex hyperbolic manifold.
References
- Aaron Bertram, Lawrence Ein, and Robert Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), no. 3, 587β602. MR 1092845, DOI https://doi.org/10.1090/S0894-0347-1991-1092845-5
- Jean-Pierre Demailly, A numerical criterion for very ample line bundles, J. Differential Geom. 37 (1993), no. 2, 323β374. MR 1205448
- Jean-Pierre Demailly, $L^2$ vanishing theorems for positive line bundles and adjunction theory, Transcendental methods in algebraic geometry (Cetraro, 1994) Lecture Notes in Math., vol. 1646, Springer, Berlin, 1996, pp. 1β97. MR 1603616, DOI https://doi.org/10.1007/BFb0094302
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
- Mark L. Green, Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20 (1984), no. 1, 279β289. MR 772134
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Jun-Muk Hwang and Wing-Keung To, On Seshadri constants of canonical bundles of compact complex hyperbolic spaces, Compositio Math. 118 (1999), no. 2, 203β215. MR 1713311, DOI https://doi.org/10.1023/A%3A1001074801110
- Jun-Muk Hwang and Wing-Keung To, On Seshadri constants of canonical bundles of compact quotients of bounded symmetric domains, J. Reine Angew. Math. 523 (2000), 173β197. MR 1762959, DOI https://doi.org/10.1515/crll.2000.046
- Jun-Muk Hwang and Wing-Keung To, Volumes of complex analytic subvarieties of Hermitian symmetric spaces, Amer. J. Math. 124 (2002), no. 6, 1221β1246. MR 1939785
- J.-M. Hwang and W.-K. To, Buser-Sarnak invariant and projective normality of abelian varieties, in Complex and differential geometry, Springer Proceedings in Mathematics, 8, Springer-Verlag, Berlin-Heidelberg, 2011, 157β170.
- J.-M. Hwang and W.-K. To, Injectivity radius and gonality of a compact Riemann surface, Amer. J. Math. 134 (2012), 259β283.
- S. P. Inamdar, On syzygies of projective varieties, Pacific J. Math. 177 (1997), no. 1, 71β76. MR 1444773, DOI https://doi.org/10.2140/pjm.1997.177.71
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
- Li Li, Wonderful compactification of an arrangement of subvarieties, Michigan Math. J. 58 (2009), no. 2, 535β563. MR 2595553, DOI https://doi.org/10.1307/mmj/1250169076
- Robert Lazarsfeld, Giuseppe Pareschi, and Mihnea Popa, Local positivity, multiplier ideals, and syzygies of abelian varieties, Algebra Number Theory 5 (2011), no. 2, 185β196. MR 2833789, DOI https://doi.org/10.2140/ant.2011.5.185
- Walter Rudin, Function theory in the unit ball of ${\bf C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
References
- Bertram, A., Ein, L. and Lazarsfeld, R.: Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. Amer. Math. Soc. 4 no. 3 (1991), 587-602. MR 1092845 (92g:14014)
- J.-P. Demailly, A numerical criterion for very ample line bundles, J. Differential Geom. 37 (1993), 323-374. MR 1205448 (94d:14007)
- J.-P. Demailly, $L^2$ vanishing theorems for positive line bundles and adjunction theory. Lecture Notes in Math. 1646 (1996), 1-97. MR 1603616 (99k:32051)
- P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience, New York, 1978. MR 507725 (80b:14001)
- M. Green, Koszul cohomology and the geometry of projective varieties II. J. Diff. Geom. 20 (1984), 279-289. MR 772134 (86j:14011)
- S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978. MR 514561 (80k:53081)
- J.-M. Hwang and W.-K. To, On Seshadri constants of canonical bundles of compact complex hyperbolic spaces. Compositio Math. 118 (1999), 203-215. MR 1713311 (2000i:32034)
- J.-M. Hwang and W.-K. To, On Seshadri constants of canonical bundles of compact quotients of bounded symmetric domains. J. Reine Angew. Math. 523 (2000), 173-197. MR 1762959 (2001f:32039)
- J.-M. Hwang and W.-K. To, Volumes of complex analytic subvarieties of Hermitian symmetric spaces. Amer. J. Math. 124 (2002), 1221-1246. MR 1939785 (2003i:32041)
- J.-M. Hwang and W.-K. To, Buser-Sarnak invariant and projective normality of abelian varieties, in Complex and differential geometry, Springer Proceedings in Mathematics, 8, Springer-Verlag, Berlin-Heidelberg, 2011, 157β170.
- J.-M. Hwang and W.-K. To, Injectivity radius and gonality of a compact Riemann surface, Amer. J. Math. 134 (2012), 259β283.
- S. P. Inamdar, On syzygies of projective varieties. Pacific. J. Math. 177 (1997), 71-76. MR 1444773 (98a:14010)
- R. Lazarsfeld, Positivity in algebraic geometry I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge. Springer-Verlag, Berlin, 2004. MR 2095471 (2005k:14001a)
- L. Li, Wonderful compactifications of an arrangement of subvarieties. Michigan Math. J. 58 (2009), 415-443. MR 2595553
- R. Lazarsfeld, G. Pareschi and M. Popa, Local positivity, multiplier ideals, and syzygies of abelian varieties, Algebra Number Theory 5 (2011), 185β196. MR 2833789
- W. Rudin, Function theory in the unit ball of $C^{n}$. Grundlehren der Mathematischen Wissenschaften, 241, Springer-Verlag, New York-Berlin, 1980. MR 601594 (82i:32002)
Additional Information
Jun-Muk Hwang
Affiliation:
Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea
MR Author ID:
362260
Email:
jmhwang@kias.re.kr
Wing-Keung To
Affiliation:
Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
MR Author ID:
267228
Email:
mattowk@nus.edu.sg
Received by editor(s):
April 28, 2010
Received by editor(s) in revised form:
November 23, 2010
Published electronically:
April 17, 2012
Additional Notes:
The first author was supported by National Researcher Program 2010-0020413 of NRF and MEST. The second author was partially supported by the research grant R-146-000-106-112 from the National University of Singapore and the Ministry of Education.
