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Notes on Kähler-Ricci Flow

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Ricci Flow and Geometric Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2166))

Abstract

In these notes some aspects of the Analytic Minimal Model Program through Kähler-Ricci flow which was initiated by J. Song and the author are discussed. Some open problems will be also presented.

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Notes

  1. 1.

    This means that [ω0 ] − tc 1 (X) > 0 represents a Kähler class, where c 1 (X) denotes the 2π multiple of the first Chern class of X in the usual terminology.

  2. 2.

    The flow equation in [27] is not the same as, but equivalent to, (4.1).

  3. 3.

    We always use C ε , c ε ′ etc. to denote some uniform constants which may depend on ε and T′.

  4. 4.

    In [27], K X is assumed to be big. It is clear from the arguments in the proof that this assumption was not used.

  5. 5.

    In [25], it was misnamed as the stable base locus.

  6. 6.

    The singularities are caused by those singular fibers, but the developing map should be meromorphic.

  7. 7.

    It will be interesting to construct an explicit example of such a singular X T , even though no one doubts its existence.

  8. 8.

    A projective variety is \(\mathbb{Q}\)-factorial if it is normal and any \(\mathbb{Q}\)-Weil divisor is \(\mathbb{Q}\)-Cartier.

  9. 9.

    Without loss of generality, we may assume that N is independent of i.

  10. 10.

    It is likely that such a Fano-like manifold is actually Fano. This is indeed the case if the dimension is not greater than 3.

  11. 11.

    Reference [15] is mainly for complex surfaces, but the part on limiting metrics works for any dimensions.

  12. 12.

    In fact, one can prove that \(\left (\pi ^{{\ast}}\omega \right )^{\kappa }\wedge \varTheta\) extends to a continuous function on X, where Θ is the (n-κ, n-κ)-form which restricts to polarized flat volume form on each smooth fiber (see [16, p. 15]).

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Acknowledgements

This work was partially supported by NSF grants.

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Authors and Affiliations

  1. Beijing Normal University, Beijing, China

    Gang Tian

  2. Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Gang Tian

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  1. Gang Tian
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Correspondence to Gang Tian .

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Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Riccardo Benedetti

  2. Department of Mathematics, University of Naples, Naples, Italy

    Carlo Mantegazza

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Tian, G. (2016). Notes on Kähler-Ricci Flow. In: Benedetti, R., Mantegazza, C. (eds) Ricci Flow and Geometric Applications. Lecture Notes in Mathematics(), vol 2166. Springer, Cham. https://doi.org/10.1007/978-3-319-42351-7_4

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