K-STABILITY OF FANO VARIETIES: AN ALGEBRO-GEOMETRIC APPROACH CHENYANG XU Abstract. We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach. Contents 1. Introduction2 1.1. K-stability in algebraic geometry3 1.2. Organization of the paper6 1.3. Conventions7 Part 1. What is K-stability? 8 2. Definition of K-stability by degenerations and MMP8 2.1. One parameter group degeneration8 2.2. MMP on family of Fano varieties 11 3. Fujita-Li's valuative criterion of K-stability 14 3.1. β-invariant 14 3.2. Stability threshold 18 3.A. Appendix: General valuations 20 4. Special valuations 21 4.1. Complements 21 4.2. Degenerations and lc places 23 4.3. Bounded complements 27 4.A. Appendix: Normalized volumes and local stability 28 5. Filtrations and Ding stability 34 5.1. Ding stability 34 5.2. Filtrations and non-Archimedean invariants 35 5.3. Revisit uniform stability 40 Notes on history 43 Part 2. K-moduli space of Fano varieties 46 Kss 6. Artin stack Xn;V 47 6.1. Family of varieties 47 6.2. Boundedness 48 Date: December 10, 2020. 1 2 CHENYANG XU 6.3. Openness 49 kps 7. Good moduli space Xn;V 50 7.1. Separated quotient 51 7.2. The existence of a good moduli space 57 8. Properness and Projectivity 59 8.1. Properness 59 8.2. Projectivity 60 Notes on history 63 Part 3. Explicit K-stable Fano varieties 66 9. Estimating δ(X) via j − KX jQ 67 9.1. Fano varieties with index one 67 9.2. Index 2 Fano hypersurfaces 70 10. Explicit K-moduli spaces 75 10.1. Deformation and degeneration 75 10.2. Wall crossings and log surfaces 77 Notes on history 78 References 80 Throughout, we work over an algebraically closed field k of characteristic 0. 1. Introduction The concept of K-stability is one of the most precious gifts differential geometers brought to algebraic geometers. It was first introduced in [Tia97] as a criterion to char- acterize the existence of K¨ahler-Einsteinmetrics on Fano manifolds, which is a central topic in complex geometry. The definition involves the sign of an analytic invariant, namely the generalized Futaki invariant, of all possible normal C∗-degenerations of a Fano manifold X. Later, in [Don02], the notion of K-stability was extended to general polarized manifolds (X; L), and the generalized Futaki invariant was reformulated in completely algebraic terms, which then allows arbitrary flat degenerations. In this survey, we will discuss the recent progress on the algebraic study of K- stability of Fano varieties, using the ideas developed in higher dimensional geometry, especially the techniques centered around the minimal model program (MMP). While there is a long history for complex geometers to study K¨ahler-Einsteinmetrics on Fano varieties, algebraic geometers, especially higher dimensional geometers, only started to look at the K-stability question relatively recently. One possible reason is that only until the necessary background knowledge from the minimal model program was developed (e.g. [BCHM10]), a systematic study in general could rise above the CX is partially supported by a Chern Professorship of the MSRI (NSF No. DMS-1440140) and by the National Science Fund for Distinguished Young Scholars (NSFC 11425101) `Algebraic Geometry' (2015-2018), NSF DMS-1901849 (2019-now). K-STABILITY OF FANO VARIETIES: AN ALGEBRO-GEOMETRIC APPROACH 3 horizon. Nevertheless, there has been spectacular progress from the algebro-geometric side in the last a few years, which we aim to survey in this note. 1.1. K-stability in algebraic geometry. Unlike many other stability notions, when K-stability was first defined in [Tia97] and later formulated using purely algebraic geometric terms in [Don02], it was not immediately clear to algebraic geometers what this really means and whether it is going to be useful in algebraic geometry. The definition itself is clear: one considers all one parameter subgroup flat degenerations X =A1, and attach an invariant Fut(X ) to it, and being K-stable is amount to saying that Fut(X ) is always positive (except the suitably defined trivial ones). Following the philosophy of Donaldson-Uhlenbeck-Yau Theorem, which established the Hitchin- Kobayashi Correspondence between stable bundles and Einstein-Hermitian bundles, one naturally would like compare it to the geometric invariant theory (GIT) (see e.g. [PT04, RT07] etc.). This may make algebraic geometers feel more comfortable with the concept. However, an apparently similar nature to the asymptotic GIT stability also makes one daunted, since the latter is notoriously known to be hard to check. Later people found examples which are K-stable but not asymptotically GIT stable varieties (see [OSY12]), which made the picture even less clear. However, it is remarkable that in [Oda13a], concepts and technicals from the mini- mal model program were first noticed to be closely related to K-stability question. In particular, it was shown K-semistability of a Fano variety implies at worst it only has Kawamata log terminal (klt) singularities, which is a measure of singularities invented in the minimal model program (MMP) theory. Then by running a meticulously de- signed MMP process, in [LX14] we show to study K-stability of Fano varieties, one only needs to consider X =A1 where the special fiber over 0 2 A1 is also a klt Fano va- riety. These works make it clear that to study K-stability of Fano varieties, the MMP would play a prominent role. In fact, as a refinement of Tian's original perspective, [Oda13a] and [LX14] suggested that in the study of K-stability of Fano varieties, we should focus on Kawamata log terminal (klt) Fano varieties, since this class of varieties is equipped with the necessary compactness. The class of klt Fano varieties is a theme that has been investigated in higher dimensional geometry for decades, however such compactness was not noticed before by birational geometers. It was then foreseeable that not far from the future, there would be an intensive interplay between two origi- nally disconnected subjects, and a purely algebraic K-stability theory would be given a birth as a result. Nevertheless, such a view only fully arose after a series of intertwining works (con- tributed by K. Fujita, C. Li and others) which established a number of equivalent char- acterizations of notations in K-stability, including the ones using invariants defined on valuations (instead of the original definition using one parameter subgroup degen- erations X =A1). More specificly, in [Ber16], inspired by the analytic work [Din88], Berman realized that one can replace the generalized Futaki invariants by Ding in- variants and define the corresponding notions of Ding stability. Soon after that, it was 4 CHENYANG XU noticed in [BBJ18,Fuj19b] that one can apply the argument in [LX14] for Ding invari- ants, and conclude that Ding stability and K-stability are indeed the same for Fano varieties. A key technical advantage for Ding invariant, observed by Fujita in [Fuj18], is that we can define Ding(F) more generally for a linearly bounded multiplicative 0 filtration F on the anti-pluri-canonical ring R = ⊕m2NH (−mrKX ) which satisfies that if we approximate a filtration by its finitely generated m-truncation Fm, then limm Ding(Fm) ! Ding(F) (this is not known for the generalized Futaki invariants!). Therefore, Ding semistability implies the non-negativity of any Ding(F). Based on this, a key invariant for the later development was independently formulated by Fujita and Li (see [Fuj19b, Li18]), namely β(E) for any divisorial valuation E over K(X). They also show that there is a close relation between Ding and β, and conclude for instance X is Ding-semistable if and only if β(E) ≥ 0 for any divisorial valuation. Another remarkable conceptual progress is the definition of the normalized volumes in [Li18] which provides the right framework to study the local K-stability theory for klt singularities. As a consequence, one can investigate the local-to-global principle for K-stability as in many other higher dimensional geometry problems. In a different direction, a uniform version of the definition of K-stability was introduced in [BHJ17] and [Der16] independently, which is more natural when one consider the space of all valuations instead of only divisorial valuations (see [BJ20, BX19]). With all these progress on our foundational understanding of K-stability, we can then turn to study various questions on K-stable Fano varieties using purely algebraic geometry. One of the most important reasons for algebraic geometers' interests in K-stability of Fano varieties is the possibility of using it to construct a well behaved moduli space, called K-moduli. Constructing moduli spaces of Fano varieties once seemed to be out of reach for higher dimensional geometers, as one primary reason is that degenerations of a family of Fano varieties are often quite complicated. Nevertheless, it is not com- pletely clueless to believe that by adding the K-stability assumption, one can overcome the difficulty. In fact, explicit examples of K-moduli spaces, especially parametrizing families of surfaces, appeared in [Tia90,MM93,OSS16]. Then general compact moduli spaces of smoothable K-polystable Fano varieties were first abstractly constructed in [LWX19] (also see partial results in [SSY16,Oda15]). However, the proofs of all these heavily rely on analytics input e.g. [DS14,CDS15,Tia15]. To proceed, naturally people were trying to construct the moduli space purely algebraically, and therefore remove the `smoothable' assumption. However, this only became plausible after the progress of the foundation theory of K-stability described in the previous paragraph. Indeed, together with major results from birational geometry (e.g.