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Moduli of K-stable Varieties pp 153–181Cite as

GIT Stability, K-Stability and the Moduli Space of Fano Varieties

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 31))

Abstract

This is a slightly extended version of the lecture notes of a mini-course in the workshop of Moduli of K-stable Varieties given by the author, in which the main construction of the proper moduli space of \(\mathbb {Q}\)-Gorenstein smoothable K-semistable Fano varieties in Li et al. (On proper moduli space of smoothable Kähler-Einstein Fano varieties, ArXiv:1411.0761 v3, 2014) is outlined.

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Notes

  1. 1.

    We assume \(\mathcal {O}_Z(1)\) being very ample only to simplify our notation.

  2. 2.

    From now on, we will abuse our notation by regarding the moment map μ K to be \(\mathfrak {k}\) valued.

  3. 3.

    Note, r will be chosen sufficiently divisible (whose existence is guaranteed by [33, Lemma 8.3]) so that all \(\mathbb {Q}\)-Gorentstein smoothable K-semistable Fano varieties are embedded in \(\mathbb {P}^N\) (cf. Definition of Z in (17) and Sect. 5).

  4. 4.

    To avoid lengthy context of technicality, here we include a slight modification of what is already included in the introduction section of [33].

  5. 5.

    which we know in the end that it also holds in the Zariski topology.

  6. 6.

    For a \(\mathbb {Q}\)-Fano variety X together with a \(\mathbb {Q}\)-Cartier divisor D ∈|− K X|, we define the K-semistable threshold for the log pair (X, D) as following:

    $$\displaystyle \begin{aligned} \mathrm{kst}(X, D):=\sup \left\{\beta \in (0,1] \left | \ \ (X, D) \mbox{ is}\ \beta\mbox{-K-semistable} \right. \right\}. \end{aligned} $$
    (21)

    In particular, it is positive by [33, Theorem 5.2].

  7. 7.

    Where [⋅]Aut(X) and [⋅]SL(N+1) denote the equivalent classes of the categorical quotients of U W and Z ∗ by Aut(X) and SL(N + 1) respectively.

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Acknowledgements

This note is an extended version of the lectures given by the author at the INDAM workshop on Moduli of K-stable Varieties at Rome on 10-14 July 2017. It is based on my joint work with Chi Li and Chenyang Xu in [33, 34]. I want to express my gratitude to my coauthors for their collaboration and sincere comments on this note, and I am responsible for all the mistakes and inaccuracies if there is still any. The author also wants to express his deep gratitude to the anonymous referee, whose meticulous proofreading of the draft tremendously improve the exposition. The author is partially supported by a Collaboration Grants for Mathematicians from the Simons Foundation:281299 and NSF:DMS-1609335. Last but not least, the author wants to thank the organizers of the workshop in Rome for creating such an exciting event.

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  1. Department of Mathematics and Computer Sciences, Rutgers University, Newark, NJ, USA

    Xiaowei Wang

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Editors and Affiliations

  1. Dept. of Mathematics and Physics, Roma Tre University, Rome, Italy

    Giulio Codogni

  2. DPMMS, University of Cambridge, Cambridge, UK

    Ruadhaí Dervan

  3. Dept. of Mathematics and Physics, Roma Tre University, Rome, Italy

    Filippo Viviani

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Wang, X. (2019). GIT Stability, K-Stability and the Moduli Space of Fano Varieties. In: Codogni, G., Dervan, R., Viviani, F. (eds) Moduli of K-stable Varieties. Springer INdAM Series, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-030-13158-6_8

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