Skip to main content
SpringerLink
Log in

Futaki Invariant and CM Polarization

  • Conference paper
  • First Online:
Book cover Geometry and Topology of Manifolds

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 154))

  • 1817 Accesses

Abstract

This is an expository paper. We will discuss various formulations of Futaki invariant and its relation to the CM line bundle. We will discuss their connections to the K-energy. We will also include proof for certain known results which may not have been well presented or less accessible in the literature. We always assume that M is a compact Kähler manifold. By a polarization, we mean a positive line bundle L over M, then we call (M, L) a polarized manifold.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In [21], the integrand is given in a different but equivalent form.

  2. 2.

    In [6], we assume that \(M_0\) is a \({\mathbb Q}\)-Fano variety with \(L_0\) being the anti-canonical bundle \(K_{M_0}^{-1}\). However, as one can see in the subsequent discussion, the general cases can be done by following similar arguments.

  3. 3.

    We refer the readers to [12] for definition of the twisted K-energy which extends the usual K-energy to conic cases.

  4. 4.

    Our presentation here differs slightly from that in [7], but they are equivalent.

  5. 5.

    One can even have an \({\mathbf G}_0\)-equivaraint resolution, but it is not needed here.

  6. 6.

    This is a pointwise version of the adjunction formula.

  7. 7.

    Here the \({\mathbf G}_0\)-action on \(M_0'\) is given by \(v'\) which covers the m-multiple of the action generated by v on \(M_0\), so we need to add m when using \(\hat{\theta }\) et al. in the subsequent computations.

References

  1. Arezzo, C., Lanave, G., Vedova. A.: Singularities and K-semistability. Preprint, arXiv:0906.2475

  2. Bando, S., Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions. Algebraic Geometry, Adv. Stud. Pure Math. 10 (1987)

    Google Scholar 

  3. Berman, R.: K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics. Preprint, arXiv:1205.6214

  4. Bott, R., Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114, 71–112 (1965)

    Google Scholar 

  5. Calabi, E.: Extremal Kähler metrics, II. Differential Geometry and Complex Analysis, pp. 95–114. Springer, Berlin (1985)

    Google Scholar 

  6. Ding, W., Tian, G.: Kähler-Einstein metrics and the generalized Futaki invariants. Invent. Math. 110, 315–335 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Donaldson, S.: Scalar curvature and stability of toric varieties. J. Diff. Geom. 62, 289–349 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Fujiki, A., Schumacher, G.: The moduli space of extremal compact Kähler manifolds and generalized Well-Petersson metrics. Publ. RIMS 26, 101–183 (1990)

    Google Scholar 

  9. Futaki, A.: An obstruction to the existence of Einstein-Kähler metrics. Inv. Math. 73, 437–443 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Futaki, A.: On compact Kähler manifolds of constant scalar curvatures. Proc. Jpn. Acad. Ser. A Math. Sci. 59(8), 401–402 (1983)

    Google Scholar 

  11. Futaki, A: Kähler-Einstein Metrics and Integral Invariants. Lecture Notes in Mathematics, vol. 1314. Springer

    Google Scholar 

  12. Jeffres, T., Mazzeo, R., Rubinstein, Y.: Kähler-Einstein metrics with edge singularities. Preprint, arXiv:1105.5216

  13. Li, C.: Remarks on logarithmic K-stability. Preprint, arXiv:1104.0428

  14. Li, C., Xu, C.Y.: Special test configuration and K-stability of Fano varieties. Ann. Math. 180(1), 197–232 (2014)

    Google Scholar 

  15. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34, 292 pp. Springer, Berlin (1994)

    Google Scholar 

  16. Paul, S.: Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics. Ann. Math. (2) 175, (1) 255–296 (2012)

    Google Scholar 

  17. Paul, S., Tian, G.: CM stability and the generalized Futaki invariant I. math.DG/0605.278

    Google Scholar 

  18. Paul, S., Tian, G.: CM stability and the generalized Futaki invariant II. Asterisque No. 328, 339–354 (2009)

    Google Scholar 

  19. Tian, G: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys. 1, 629–646, WSP, Singapore (1987)

    Google Scholar 

  20. Tian, G.: The K-energy on hypersurfaces and stability. Comm. Anal. Geom. 2(2), 239–265 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tian, G: Kähler-Einstein metrics on algebraic manifolds. Transcendental methods in algebraic geometry (Cetraro, 1994), 143–185, Lecture Notes in Mathematics, vol. 1646. Springer, Berlin (1996)

    Google Scholar 

  22. Tian, G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–39 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tian, G.: Canonical Metrics on Kähler Manifolds. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag (2000)

    Google Scholar 

  24. Tian, G: K-stability implies CM-stability. Preprint, arXiv:1409.7836

Download references

Author information

Authors and Affiliations

  1. Beijing University, Beijing, China

    Gang Tian

  2. Princeton University, Princeton, USA

    Gang Tian

Authors
  1. Gang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gang Tian .

Editor information

Editors and Affiliations

  1. The University of Tokyo, Meguro-ku, Tokyo, Japan

    Akito Futaki

  2. Mathematical Institute, Tohoku University, Sendai,Aoba-ku, Japan

    Reiko Miyaoka

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, China

    Zizhou Tang

  4. Chern Institute of Mathematics, Nankai University, Tianjin, China

    Weiping Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Japan

About this paper

Cite this paper

Tian, G. (2016). Futaki Invariant and CM Polarization. In: Futaki, A., Miyaoka, R., Tang, Z., Zhang, W. (eds) Geometry and Topology of Manifolds. Springer Proceedings in Mathematics & Statistics, vol 154. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56021-0_18

Download citation

Publish with us

Policies and ethics

Search

Navigation