## Osaka Journal of Mathematics

### A generalization of the Ross--Thomas slope theory

Yuji Odaka

#### Abstract

We give a formula for the Donaldson--Futaki invariants of certain type of semi test configurations, which essentially generalizes the Ross--Thomas slope theory [28]. The positivity (resp. non-negativity) of those a priori special'' Donaldson--Futaki invariants implies K-stability (resp. K-semistability). As an application, we prove the K-(semi)stability of certain polarized varieties with semi-log-canonical singularities, which generalizes some results of [28].

#### Article information

Source
Osaka J. Math., Volume 50, Number 1 (2013), 171-185.

Dates
First available in Project Euclid: 27 March 2013

https://projecteuclid.org/euclid.ojm/1364390425

Mathematical Reviews number (MathSciNet)
MR3080636

Zentralblatt MATH identifier
1328.14073

#### Citation

Odaka, Yuji. A generalization of the Ross--Thomas slope theory. Osaka J. Math. 50 (2013), no. 1, 171--185. https://projecteuclid.org/euclid.ojm/1364390425

#### References

• V. Alexeev: Log canonical singularities and complete moduli of stable pairs, arXiv:alge-geom/960813 (1996).
• X.X. Chen and G. Tian: Geometry of Kähler metrics and foliations by holomorphic discs, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 1–107.
• \begingroup S.K. Donaldson: Scalar curvature and projective embeddings, I, J. Differential Geom. 59 (2001), 479–522. \endgroup
• S.K. Donaldson: Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289–349.
• S.K. Donaldson: Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), 453–472.
• W. Fulton: Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2, Springer, Berlin, 1984.
• A. Futaki: An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983), 437–443.
• D. Gieseker: Global moduli for surfaces of general type, Invent. Math. 43 (1977), 233–282.
• D. Gieseker: Lectures on Moduli of Curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 69, Springer-Verlag, New York, 1982.
• R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
• H. Hironaka: Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann. of Math. (2) 79 (1964), 109–203;
• H. Hironaka: Resolution of singularities of an algebraic variety over a field of characteristic zero, II, Ann. of Math. (2) 79 (1964), 205–326.
• J. Kollár and S. Mori: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, Cambridge, 1998.
• J. Kollár and N.I. Shepherd-Barron: Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299–338.
• C. Li and C. Xu: Special test configurations and K-stability of $\mathbb{Q}$-Fano varieties, arXiv:1111.5398 (2011).
• \begingroup T. Mabuchi: Chow-stability and Hilbert-stability in Mumford's geometric invariant theory, Osaka J. Math. 45 (2008), 833–846. \endgroup
• T. Mabuchi: K-stability of constant scalar curvature polarization, arXiv:0812.4093 (2008).
• T. Mabuchi: A stronger concept of K-stability, arXiv:0910.4617 (2009).
• H. Matsumura: Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge Univ. Press, Cambridge, 1986.
• D. Mumford: Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge 34, Springer, Berlin, 1965.
• D. Mumford: Stability of Projective Varieties, Enseignement Math., Geneva, 1977.
• Y. Odaka: The GIT stability of polarized varieties via discrepancy, to appear in Annals of Mathematics.
• Y. Odaka: The Calabi Conjecture and K-stability, Int. Math. Res. Not. 13, 2011.
• Y. Odaka and Y. Sano: Alpha invariant and K-stability of $\mathbb{Q}$-Fano varieties, Adv. Math. 229 (2012), 2818–2834.
• Y. Odaka: On parametrization, optimization and triviality of test configurations, arXiv:1201.0692 (2012).
• D. Panov and J. Ross: Slope stability and exceptional divisors of high genus, Math. Ann. 343 (2009), 79–101.
• J. Ross and R. Thomas: An obstruction to the existence of constant scalar curvature Kähler metrics, J. Differential Geom. 72 (2006), 429–466.
• J. Ross and R. Thomas: A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), 201–255.
• J. Stoppa: K-stability of constant scalar curvature Kähler manifolds, Adv. Math. 221 (2009), 1397–1408.
• G. Tian: Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1–37.
• X. Wang: Heights and GIT weights, to appear in Math. Research Letters.
• S.T. Yau: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978), 339–411. \endthebibliography*