Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-23T22:18:41.523Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimising CM degree and slope stability of projective varieties

Part of: Complex manifolds Families, fibrations

Published online by Cambridge University Press:  24 February 2021

KENTARO OHNO
KENTARO OHNO*
Affiliation:
Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan e-mail: ohken322@gmail.com
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 discuss a minimisation problem of the degree of the Chow–Mumford (CM) line bundle among all possible fillings of a polarised family with fixed general fibers, motivated by the study of the moduli space of K-stable Fano varieties. We show that such minimisation implies the slope semistability of the fiber if the central fiber is smooth.

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 32Q26: Notions of stability
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 125 - 137
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

Alexeev, V.. Moduli spaces M g,n(W) for surfaces. In: Higher Dimensional Complex Varieties: Proceedings of the International Conference Held in Trento, Italy, June 15-24, 1994, page 1 (Walter de Gruyter, 1996).Google Scholar
Blum, H., Liu, Y. and Xu, C.. Openness of K-semistability for Fano varieties. arXiv preprint arXiv:1907.02408 (2019).Google Scholar
Blum, H. and Xu, C.. Uniqueness of K-polystable degenerations of Fano varieties. Ann. of Math., 190(2), (2019), 609–656.CrossRefGoogle Scholar
Boucksom, S., Hisamoto, T. and Jonsson, M.. Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs. In Ann. l’Inst. Fourier 67, (2017), pages 743–841.CrossRefGoogle Scholar
Donaldson, S. K.. Scalar curvature and stability of toric varieties. J. Differential Geom., 62(2), (2002), 289–349.CrossRefGoogle Scholar
Eisenbud, D.. Commutative Algebra: With a View Toward Algebraic Geometry (Springer, 1995).CrossRefGoogle Scholar
Fine, J. and Ross, J.. A note on positivity of the CM line bundle. Int. Math. Res. Notices, 2006(2006), id. 95875.Google Scholar
Fujiki, A. and Schumacher, G.. The moduli space of extremal compact Kähler manifolds and generalised Weil–Petersson metrics. Publ. Res. Inst. Math. Sc., 26(1), (1990), 101–183.CrossRefGoogle Scholar
Fulton, W.. Intersection Theory. Ergeb. Math. Grenzgeb. (3). (Springer Berlin Heidelberg, 1997).Google Scholar
Kollár, J. and Shepherd-Barron, N. I.. Threefolds and deformations of surface singularities. Invent. Math., 91(2), (1988), 299–338.CrossRefGoogle Scholar
Li, C. and Xu, C.. Special test configuration and K-stability of Fano varieties. Ann. Math., 180, (2014), 197–232.CrossRefGoogle Scholar
Li, C. and Xu, C.. Stability of valuations and Kollár components. J. Eur. Math. Soc., 22(8)(2020), 2573–2627.CrossRefGoogle Scholar
Liu, Y. and Zhuang, Z.. Birational superrigidity and K-stability of singular Fano complete intersections. Internat. Math. Res. Notices, 202(1)(2018), 382–401.CrossRefGoogle Scholar
Matsumura, H.. Commutative ring theory . Cambridge Stud. Adv. Math. (Cambridge University Press, Cambridge, 1986).Google Scholar
Odaka, Y.. A generalisation of the Ross–Thomas slope theory. Osaka J. Math., 50(1), (2013), 171–185.Google Scholar
Odaka, Y.. On the moduli of Kähler–Einstein Fano manifolds. In: Proceedings of Kinosaki Algebraic Geometry Symposium (2013), arXiv preprint arXiv:1211.4833.Google Scholar
Panov, D. and Ross, J.. Slope stability and exceptional divisors of high genus. Mathe. Ann., 343(1), (2009), 79–101.CrossRefGoogle Scholar
Paul, S. T. and Tian, G.. CM stability and the generalised Futaki invariant I. arXiv preprint arXiv:math/0605278 (2006).Google Scholar
Ross, J. and Thomas, R.. A study of the Hilbert–Mumford criterion for the stability of projective varieties. J. Algebraic Geom., 16(2), (2007), 201–255.CrossRefGoogle Scholar
Stoppa, J.. A note on the definition of K-stability. arXiv preprint arXiv:1111.5826 (2011).Google Scholar
Wang, X., Xu, C., et al. Nonexistence of asymptotic GIT compactification. Duke Math. J., 163(12), (2014), 2217–2241.CrossRefGoogle Scholar