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.
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Notes
- 1.
In [21], the integrand is given in a different but equivalent form.
- 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.
We refer the readers to [12] for definition of the twisted K-energy which extends the usual K-energy to conic cases.
- 4.
Our presentation here differs slightly from that in [7], but they are equivalent.
- 5.
One can even have an \({\mathbf G}_0\)-equivaraint resolution, but it is not needed here.
- 6.
This is a pointwise version of the adjunction formula.
- 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
Arezzo, C., Lanave, G., Vedova. A.: Singularities and K-semistability. Preprint, arXiv:0906.2475
Bando, S., Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions. Algebraic Geometry, Adv. Stud. Pure Math. 10 (1987)
Berman, R.: K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics. Preprint, arXiv:1205.6214
Bott, R., Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114, 71–112 (1965)
Calabi, E.: Extremal Kähler metrics, II. Differential Geometry and Complex Analysis, pp. 95–114. Springer, Berlin (1985)
Ding, W., Tian, G.: Kähler-Einstein metrics and the generalized Futaki invariants. Invent. Math. 110, 315–335 (1992)
Donaldson, S.: Scalar curvature and stability of toric varieties. J. Diff. Geom. 62, 289–349 (2002)
Fujiki, A., Schumacher, G.: The moduli space of extremal compact Kähler manifolds and generalized Well-Petersson metrics. Publ. RIMS 26, 101–183 (1990)
Futaki, A.: An obstruction to the existence of Einstein-Kähler metrics. Inv. Math. 73, 437–443 (1983)
Futaki, A.: On compact Kähler manifolds of constant scalar curvatures. Proc. Jpn. Acad. Ser. A Math. Sci. 59(8), 401–402 (1983)
Futaki, A: Kähler-Einstein Metrics and Integral Invariants. Lecture Notes in Mathematics, vol. 1314. Springer
Jeffres, T., Mazzeo, R., Rubinstein, Y.: Kähler-Einstein metrics with edge singularities. Preprint, arXiv:1105.5216
Li, C.: Remarks on logarithmic K-stability. Preprint, arXiv:1104.0428
Li, C., Xu, C.Y.: Special test configuration and K-stability of Fano varieties. Ann. Math. 180(1), 197–232 (2014)
Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34, 292 pp. Springer, Berlin (1994)
Paul, S.: Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics. Ann. Math. (2) 175, (1) 255–296 (2012)
Paul, S., Tian, G.: CM stability and the generalized Futaki invariant I. math.DG/0605.278
Paul, S., Tian, G.: CM stability and the generalized Futaki invariant II. Asterisque No. 328, 339–354 (2009)
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)
Tian, G.: The K-energy on hypersurfaces and stability. Comm. Anal. Geom. 2(2), 239–265 (1994)
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)
Tian, G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–39 (1997)
Tian, G.: Canonical Metrics on Kähler Manifolds. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag (2000)
Tian, G: K-stability implies CM-stability. Preprint, arXiv:1409.7836
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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
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DOI: https://doi.org/10.1007/978-4-431-56021-0_18
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