Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-25T01:59:47.262Z Has data issue: false hasContentIssue false
  • English
  • Français

Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds

Part of: Differential topology Complex manifolds Homology and cohomology theories

Published online by Cambridge University Press:  26 February 2021

D. KOTSCHICK
D. KOTSCHICK*
Affiliation:
Mathematisches Institut, LMU München, Theresienstr. 39, 80333München, Germany. e-mail: dieter@math.lmu.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

MSC classification

Primary: 32Q15: Kähler manifolds
Secondary: 32Q55: Topological aspects of complex manifolds 55N22: Bordism and cobordism theories, formal group laws 57R20: Characteristic classes and numbers
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 95 - 103
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

References

Amorós, J., Burger, M., Corlette, K., Kotschick, D. and Toledo, D.. Fundamental Groups of Compact Kähler Manifolds, Math. Surv. and Monogr., Vol. 44 (Amer. Math. Soc., Providence, R.I. 1996).CrossRefGoogle Scholar
Barth, W., Peters, C. and Van de Ven, A.. Compact Complex Surfaces (Springer-Verlag, Berlin, 1984).CrossRefGoogle Scholar
Beauville, A.. Complex Algebraic Surfaces, Second Edition. LMS Student Texts 23 (Cambridge University Press, 1996).CrossRefGoogle Scholar
Bott, R.. Vector fields and characteristic numbers. Michigan Math. J. 14 (1967), 231–244.CrossRefGoogle Scholar
Carrell, J. B.. Holomorphic one forms and characteristic numbers. Topology 13 (1974), 225–228.CrossRefGoogle Scholar
Carrell, J. B. and Lieberman, D. I.. Holomorphic vector fields and Kaehler manifolds. Invent. Math. 21 (1973), 303–309.CrossRefGoogle Scholar
Crew, R. and Fried, D.. Nonsingular holomorphic flows. Topology 25 (1986), 471–473.CrossRefGoogle Scholar
Friedman, R. and Morgan, J. W.. Smooth four-manifolds and complex surfaces. Ergeb. Math. Grenzgeb. (3), 27., (Springer-Verlag, Berlin, 1994).CrossRefGoogle Scholar
Green, M. and Lazarsfeld, R.. Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville. Invent. math. 90 (1987), 389–407.CrossRefGoogle Scholar
Griffiths, H. B.. The fundamental group of a surface, and a theorem of Schreier. Acta Math. 110 (1963), 1–17.CrossRefGoogle Scholar
Hacon, C. D. and Kovács, S.. Holomorphic one-forms on varieties of general type. Ann. Sci. Norm. Sup. 38 (2005), 599–607.CrossRefGoogle Scholar
Kotschick, D.. Characteristic numbers of algebraic varieties. Proc. Natl. Acad. USA. 106, no. 25 (2009), 10114–10115.CrossRefGoogle ScholarPubMed
Kotschick, D.. Topologically invariant Chern numbers of projective varieties. Adv. Math. 229 (2012), 1300–1312.CrossRefGoogle Scholar
Kotschick, D. and Schreieder, S.. The Hodge ring of Kähler manifolds. Composit. Math. 149 (2013), 637–657.CrossRefGoogle Scholar
Neumann, W. D.. Fibering over the circle within a cobordism class. Math. Ann. 192 (1971), 191–192.CrossRefGoogle Scholar
Popa, M. and Schnell, C.. Kodaira dimension and zeros of holomorphic one-forms. Ann. of Math. 179 (2014), 1109–1120.CrossRefGoogle Scholar
Scárdua, B.. Holomorphic Anosov systems, foliations and fibering complex manifolds. Dyn. Syst. 18 (2003), 365–389.CrossRefGoogle Scholar
Tischler, D.. On fibering certain foliated manifolds over S 1. Topology 9 (1970), 153–154.CrossRefGoogle Scholar
Zhang, Q.. Global holomorphic one-forms on projective manifolds with ample canonical bundle. J. Alg. Geometry 6 (1997), 777–787.Google Scholar