K-stability of Fano varieties
In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where it is the correct stability condition to allow the formation of moduli spaces, and where it precisely characterises the existence of Kähler–Einstein metrics. K-stability was first defined for Fano manifolds by Gang Tian in 1997 in response to a conjecture of Shing-Tung Yau from 1993 that there should exist a stability condition which characterises the existence of  a Kähler–Einstein metric on a Fano manifold. It was defined in reference to the "K-energy functional" previously introduced by Toshiki Mabuchi. Tian's definition of K-stability was reformulated by Simon Donaldson in 2001 in a purely algebro-geometric way. K-stability has become an important notion in the study and classification of Fano varieties. In 2012 Xiuxiong Chen, Donaldson, and Song Sun and independently Gang Tian proved that a smooth Fano manifold is K-polystable if and only if it admits a Kähler–Einstein metric. This was later generalised to singular K-polystable Fano varieties due to  the work of Berman–Boucksom–Jonsson and others. K-stability is important in constructing moduli spaces of Fano varieties, where observations going back to the original development of geometric invariant theory show that it is necessary to restrict to a class of stable objects to form good moduli. It is now known through the work of Chenyang Xu and others that there exists a projective coarse moduli space of K-polystable Fano varieties of finite type. This work relies on Caucher Birkar's proof of boundedness of Fano varieties, for which he was awarded the 2018 Fields medal. Due to  the reformulations of the K-stability condition by Fujita–Li and Odaka, the K-stability of Fano varieties may be explicitly computed in practice. Which Fano varieties are K-stable is well understood in dimension one, two, and three. Definition and characterisations. The notion of K-stability for Fano manifolds was originally specified using differential geometry by Tian, who extended the purely analytical notion of the Futaki invariant of a vector field to the case of certain normal varieties with orbifold singularities. This was later reformulated in a purely algebro-geometric form by Donaldson, but this general definition lost a  direct link to the geometry of Fano varieties, instead making sense for the broader class of all projective varieties. Work of Tian shows that the Donaldson–Futaki invariant specifying the weight of the formula_1-action on the central fibre of a test configuration can be computed in terms of certain intersection numbers (corresponding to the weight of an action on the so-called CM line bundle). In the Fano case these intersection numbers, which involve the anticanonical divisor of the variety and its test configuration, can be given powerful alternative characterisations in terms of the algebraic and birational geometry of  the Fano variety. Thus in the case of Fano varieties, there are many different but equivalent characterisations of K-stability, and some of these characterisations lend themselves to explicit calculation or easier proofs of results. In this section all definitions are stated in the generality of a formula_2-Fano variety, which is a Fano variety with ample formula_2-Cartier anticanonical divisor and at worst Kawamata log terminal (klt) singularities. The definitions of K-stability can be made for any formula_2-Gorenstein Fano variety (that is, any Fano variety where the anticanonical divisor is formula_2-Cartier), however it was proven by  Odaka that every K-semistable Fano variety has at worst klt singularities, so for the purpose of studying K-stability it suffices to assume at worst klt singularities. Every definition can be extended in a straightforward way to formula_2-log Fano pairs, a pair formula_7 of a klt variety X and klt divisor such that formula_8 is ample and formula_2-Cartier. Traditional definition. The definition of K-stability for a Fano manifold, or more generally a formula_2-Fano variety, can be given in many forms. The general definition of K-stability in terms of test configurations (see K-stability for  more details) can be simplified if the type of test configuration one considers can be simplified. For example, in the case of toric varieties, one may always take test configurations which are also toric, and this leads to a recharacterisation of K-stability in terms of convex functions on the moment polytope of the toric variety, as was observed by Donaldson in his first paper on K-stability. In the case of Fano manifolds, it was already implicit in the work of Tian that one may restrict to test configurations with a simplified central fibre, in that case where the  central fibre is a normal variety. In this case there exists an intersection-theoretic formula for the Donaldson–Futaki invariant of a normal test configuration formula_11 for formula_12. Explicitly, one extends the test configuration formula_13 to a test configuration over the complex projective line formula_14 trivially at the point formula_15, one has a formula With respect to this invariant, if formula_17 is a formula_2-Fano variety, we say formula_17 is According to the above definitions, there are implications The above definitions are not well-suited to the situation where the Fano variety formula_17 has automorphisms. When the space of automorphisms  is positive-dimensional formula_36, it was observed by Akito Futaki that there are certain test configurations constructed out of the automorphisms of formula_17 which are "trivial" for the perspective of testing K-stability. In this case one should restrict to those test configurations which are equivariant with respect to the action of a maximal torus formula_38, and this leads to the notion of K-polystability or reduced K-stability. We say formula_17 is As for the case where the automorphism group is not positive-dimensional, we have implications The condition of "uniform" stability is "a priori" stronger than stability, because  it assumes a "uniform" bound above zero for the Donaldson–Futaki invariant of the test configuration. However it turns out in the case of formula_2-Fano varieties uniform stability is actually equivalent to stability.Theorem (Liu–Xu–Zhuang): (Reduced) uniform K-stability is equivalent to K-(poly)stability for formula_2-Fano varieties.Many results can be proved more easily for uniform K-stability because a uniform bound is stronger than a non-uniform bound, so often one works with this definition as opposed to the traditional K-(poly)stability. In the more general case of a polarised variety considered in the article on K -stability it is still an open and important problem to characterise how (reduced) uniform K-stability relates to K-(poly)stability. Special test configurations. As mentioned above, sometimes the type of test configuration to be considered can be simplified. In the case of Fano varieties, a special test configuration is a test configuration formula_11 such that we have a rational equivalence of divisors formula_51 for some formula_52, and the central fibre formula_53 is also a formula_2-Fano variety. One may prove that given any test configuration formula_11, there exists a "special" test configuration formula_56 such that This implies that for the purposes  of testing the K-stability of formula_17, it suffices restrict to just looking at the above definitions of K-stability for special test configurations. The fact that one may assume the central fibre of the test configuration is also Fano leads to strong links with birational geometry and the minimal model program, providing a number of alternative characterisations of K-stability described in the following sections. The main alternative characterisation is in terms of a different notion of Ding stability, which is a variation of the K-stability condition for the Ding invariant where one adds on the log canonical  threshold of the test configuration. The Ding invariant can only be defined in the setting of Fano varieties. Using this new invariant instead of formula_60, one can define every notion of Ding stability exactly as above, leading to Ding (semi/poly)stability and uniform versions. The Ding invariant has better formal properties with respect to algebraic geometry than the Donaldson–Futaki invariant. It is known that when a test configuration is special, the Ding invariant agrees with the Donaldson–Futaki invariant up to a constant factor, and so for Fano varieties Ding stability is equivalent to K-stability. Alpha invariant. The  first known effective criteria to test for K-stability was developed by Tian. Originally Tian's work was designed to directly provide a criterion for the existence of a Kähler–Einstein metric on a Fano manifold, and by later work it is known that every Kähler–Einstein Fano manifold is K-polystable. Tian's original definition of the alpha invariant was analytical in nature, but can be used to verify the existence of a Kähler–Einstein metric in practice. The alpha invariant of Tian can be defined relative to a group of automorphisms formula_61, and the alpha invariant formula_62 corresponds  to the concept of reduced K-stability or K-polystability above. Fix a formula_61-invariant Kähler metric formula_64 on a Fano manifold. Define a special class of Kähler potentials by Then the alpha invariant is defined by The importance of this invariant is as follows: Theorem: (Tian) Let formula_17 be a smooth Fano manifold of dimension formula_68. If the alpha invariant formula_69 then formula_17 admits a formula_61-invariant Kähler–Einstein metric. It was later observed by Odaka–Sano that the alpha invariant can be given a purely algebro-geometric definition in terms of an infimum of the log canonical threshold  over all formula_61-invariant linear systems contained inside formula_73. Precisely, Demailly showedformula_74This allows purely algebro-geometric proofs of the existence of Kähler–Einstein metrics. Beta invariant. The beta invariant formula_75 makes close contact with birational geometry. This invariant was developed by Fujita and Li in an attempt to discover a characterisation of K-stability in terms of divisors or valuations of the Fano variety formula_17. This work was inspired by earlier ideas of Ross–Thomas which attempted to describe K-stability in terms of algebraic invariants coming out of subschemes of the variety formula_17. Whilst it is not possible to  show that this "slope" K-stability is equivalent to K-stability, by passing not just to divisors inside formula_17 but divisors inside any birational model over formula_17, one obtains "enough" objects to accurately test for K-stability. In particular Fujita realised that Ross–Thomas's notion of slope K-stability was limited by only integrating up to the Seshadri constant of the subscheme, where the natural divisor on the blow-up becomes ample. By contract the formula_80-invariant integrates up to the pseudoeffective threshold where the natural divisor has positive volume (since every ample divisor has positive volume, the pseudoeffective  threshold goes beyond the Seshadri constant). This extra information gives Fujita and Li's valuative criterion enough information to fully characterise K-stability. Suppose formula_17 is a normal variety with formula_2-Cartier canonical divisor formula_83. One says formula_84 is a "divisor over formula_17" if formula_84 is a divisor contained inside some normal variety formula_87 such that there exists a proper birational morphism formula_88 (for example given by a blow up of formula_17). One defines the "log discrepancy" of a divisor formula_84 over formula_17 as where formula_93 is the discrepancy of the divisor formula_84 in the sense of birational geometry (see  canonical singularity). The discrepancy of a divisor formula_84 over formula_17 is defined as follows. Away from the exceptional locus of the birational morphism formula_88, the canonical divisors of formula_87 and formula_17 agree. Therefore, their difference is given by some sum of prime divisors formula_100 contained in the exceptional locus of formula_101. That is, where formula_103. By definition formula_104 and formula_105 when formula_84 is not one of the prime divisors formula_100 in the exceptional locus. The log discrepancy of formula_17 measures the singularities of the Fano variety. In particular X is Kawamata log terminal if and only if formula_109 for any  formula_84 over formula_17. To define the beta invariant, one also needs the a volume term. For a divisor formula_84 over formula_17, define Here the volume of a divisor measures the rate at which its space of sections grows in comparison to the expected dimension. Namely, where formula_68. Finally, the beta invariant was defined by Fujita and Li as Despite the complicated definition, due to the powerful tools of birational geometry, this invariant may be explicitly computed in practice for many classes of Fano varieties where the structure of divisors in their birational models is known. This can often be achieved  with the use of computational algebraic geometry or by-hand calculation. The relevance of the formula_80-invariant is in the following characterisation of K-stability first observed (in one direction) by Fujita and Li independently. Theorem: (Fujita–Li, Blum–Xu) A formula_2-Fano variety formula_17 is K-semistable if and only if formula_121 for all divisors formula_84 over formula_17. Furthermore formula_17 is K-stable if and only if formula_125 for all divisors over formula_17. Delta invariant. The delta invariant can be defined as a "multiplicative" version of the "additive" beta invariant. The "delta invariant of a divisor formula_84 over formula_17  is defined by The delta invariant of formula_17 is then given by a uniform measurement of the delta invariants of all divisors over formula_17. The delta invariant of a divisor is conceptually similar to the beta invariant, however it was observed by Fujita–Odaka that one can compute the delta invariant as a limit of quantized" delta invariants formula_133 as formula_134. The quantized delta invariants can be computed in terms of "m-basis type divisors" which are given by choices of bases in the fixed finite-dimensional vector space formula_135. Thus the delta invariant is generally more computable and more  theoretically powerful than its predecessors, and much progress on the explicit computations of K-stability for Fano varieties, and in the theory of moduli of Fano varieties has occurred since its introduction. Its initial importance to the theory of K-stability is captured in the following characterisation. Theorem: (Fujita–Odaka, Blum–Xu) A formula_2-Fano variety formula_17 is K-semistable if and only if formula_138. Furthermore it is uniformly K-stable if and only if formula_139. The algebraic formula_140-invariant can make contact with the explicit analytical properties of Kähler–Einstein metrics. In particular, one may define the "greatest Ricci  lower bound" formula_141 as the supremum of all formula_142 such that there exists a Kähler metric formula_143 such that formula_144. This is the limit of how far one can traverse the natural continuity method to solve the Kähler–Einstein equation. If the greatest Ricci lower bound takes the value formula_145 then one can complete the continuity method to derive the existence of a Kähler–Einstein metric. It turns out that precisely how far you can go along this continuity method, the greatest Ricci lower bound, is exactly given by the formula_140-invariant. That is, In the case of toric Fano  manifolds an even more geometric interpretation of the delta invariant was derived by Li. For such a toric Fano formula_148, the origin formula_149 is always contained in the interior of the moment polytope formula_150. If formula_151 denotes the barycentre of the polytope formula_150 and formula_153 denotes the point on the boundary of the polytope intersecting the ray formula_154, then Li showed that the greatest Ricci lower bound is given by the ratio formula_155. In particular the toric Fano has formula_156 if and only if its barycentre is the origin. Interpreted using the delta invariant (and indeed using earlier results), one  concludes that a toric Fano manifold is K-stable if and only if the barycentre of its polytope formula_150 is the origin. Existence of Kähler–Einstein metrics. From its initial introduction, the notion of K-stability has been intimately linked to the existence of Kähler–Einstein metrics on Fano manifolds. There are now many theorems which relate certain K-stability assumptions to the existence of solutions. These conjectures fall broadly under the title of the Yau–Tian–Donaldson conjecture. In the case of Fano varieties this conjecture asserts:Conjecture (Yau–Tian–Donaldson): A Fano manifold admits a Kähler–Einstein metric  if and only if it is K-polystable.For Fano manifolds this conjecture was originally proposed by Yau and Tian, and a more general form was stated by Donaldson which extends beyond just the case of Fano manifolds. Nevertheless, the conjecture even in the case of Fanos has come to be known as the Yau–Tian–Donaldson conjecture. See K-stability for more discussion of the general conjecture. In the case of Fano manifolds, the YTD conjecture admits generalisations beyond the case of smooth varieties and forms of the conjecture are now known for singular Fanos and log Fanos. Smooth  Fano varieties. The forward direction of the conjecture, that a Fano manifold with a Kähler–Einstein metric is K-polystable, was proven by Tian in his original paper when the Fano manifold has a discrete automorphism group, that is, formula_158. This direction was proven in full generality, removing the assumption that the automorphism group was discrete, by Berman. The reverse direction of the Yau–Tian–Donaldson conjecture was first resolved in the smooth case as stated above by Chen–Donaldson–Sun, and at the same time by Tian. Chen, Donaldson, and Sun have alleged that Tian's claim to equal  priority for the proof is incorrect, and they have accused him of academic misconduct. Tian has disputed their claims. Chen, Donaldson, and Sun were recognized by the American Mathematical Society's prestigious 2019 Veblen Prize as having had resolved the conjecture. The Breakthrough Prize has recognized Donaldson with the Breakthrough Prize in Mathematics and Sun with the New Horizons Breakthrough Prize, in part based upon their work with Chen on the conjecture.Theorem: If a Fano manifold is K-polystable, then it admits a Kähler–Einstein metric.The proofs of Chen–Donaldson–Sun and Tian were based on a delicate  study of Gromov–Hausdorff limits of Fano manifolds with Ricci curvature bounds. More recently, a proof based on the "classical" continuity method was provided by Ved Datar and Gabor Székelyhidi, followed by a proof by Chen, Sun, and Bing Wang using the Kähler–Ricci flow. Robert Berman, Sébastien Boucksom, and Mattias Jonsson also provided a proof from a new variational approach, which interprets K-stability in terms of Non-Archimedean geometry. Of particular interest is that the proof of Berman–Boucksom–Jonsson also applies to the case of a smooth log Fano pair, and does not use the notion of  K-polystability but of uniform K-stability as introduced by Dervan and Boucksom–Hisamoto–Jonsson. It is now known that uniform K-stability is equivalent to K-stability and so BBJ's proof provides a new proof of the full YTD conjecture. Building on the variational techniques Berman–Boucksom–Jonsson and the so-called "quantized" delta invariants of Fujita–Odaka, Zhang produced a short quantization-based proof of the YTD conjecture for smooth Fano manifolds. Using other techniques entirely, Berman has also produced a proof of a YTD-type conjecture using a thermodynamic approach called "uniform Gibbs stability", where a  Kähler–Einstein metric is constructed through a random point process. Singular Fano varieties and weak Kähler–Einstein metrics. The new proof of the Yau–Tian–Donaldson conjecture by Berman–Boucksom–Jonsson using variational techniques opened up the possible study of K-stability and Kähler–Einstein metrics for "singular" Fano varieties. The variational techniques used rely on "uniform" K-stability as described above. The result of Berman that a Fano manifold admitting a Kähler–Einstein metric is K-polystable was proven in the full generality of a formula_2-log Fano pair, admitting a weak Kähler–Einstein metric. A weak Kähler–Einstein  metric on a formula_2-Fano variety formula_17 is a positive formula_162-current formula_163which restricts to give a smooth Kähler–Einstein metric formula_164on the smooth locus formula_165 of formula_17. By requiring a compatibility with a divisor formula_167, this definition can be extended to a weak Kähler–Einstein metric on a pair formula_7. In this generality, the reverse direction of the YTD conjecture was proven by Li–Tian–Wang in the case where the automorphism group is discrete, and in full generality by Li.Theorem (Li–Tian–Wang, Li): A formula_2-log Fano pair formula_7 which is reduced uniformly K-stable admits  a weak Kähler–Einstein metric.By the resolution of the finite generation conjecture by Liu–Xu–Zhuang it is known that reduced uniform K-stability is equivalent to K-polystability, so combined with Berman's result the Yau–Tian–Donaldson conjecture is true in complete generality for singular Fano varieties.Theorem (Li–Tian–Wang, Li, Berman, Liu–Xu–Zhuang): A formula_2-log Fano pair formula_7 admits a weak Kähler–Einstein metric if and only if it is K-polystable. Moduli spaces of K-stable Fano varieties. The construction of moduli spaces is a central problem in algebraic geometry. The construction  of moduli of algebraic curves spurred the development of geometric invariant theory, stacks, and classification of algebraic surfaces has motivated results throughout algebraic geometry. The case of moduli spaces of canonically polarised varieties was settled using techniques arising from the minimal model program by Kollár–Shepherd-Barron leading to the so-called KSB moduli spaces of varieties of general type. A key property of varieties of general type which allow the construction of moduli is the lack of automorphisms of such varieties. This does not hold for Fano varieties, which can often have very large automorphism groups, so the minimal  model program did not directly find applications to the construction of moduli of Fano varieties, and it became clear that K-stability was the correct algebro-geometric notion to allow the formation of moduli in this case. Moduli spaces of K-stable varieties are known as "K-moduli". Smooth case. In the case of smooth Fano manifolds, one may use techniques arising out of the Yau–Tian–Donaldson conjecture to construct the moduli space analytically. In particular work of Odaka and Donaldson building upon the ideas of Gromov compactness of Kähler–Einstein Fanos used in the proof of the YTD  conjecture implies the existence of moduli spaces of smooth Fano Kähler–Einstein manifolds with discrete automorphism groups. These moduli spaces are Hausdorff and have at worst quotient singularities. By the YTD conjecture these are alternatively moduli spaces of smooth K-polystable Fano varieties with discrete automorphism groups. However, a Gromov–Hausdorff limit of smooth Fano Kähler–Einstein manifolds may lead to a singular formula_2-Fano variety, so the moduli spaces described by Odaka and Donaldson is not compact, a criterion that is often desirable in the formation of moduli spaces. One method of compactifying the moduli space of smooth K -polystable Fanos is to pass to a moduli space of "singular" K-polystable Fanos, and use algebraic geometry to prove its projectivity. The Yau–Tian–Donaldson conjecture for singular Fano varieties would give this compactification an alternative point of view as consisting of singular Fano varieties with "weak" Kähler–Einstein metrics. General case. The standard algebraic technique to construct moduli spaces utilizes geometric invariant theory. Typically to apply Mumford's geometric invariant theory to construct moduli, one must embed a family of varieties inside a fixed finite-dimensional projective space. Such a family then defines a locus of points in  the corresponding Hilbert scheme of the projective space, which is a projective scheme on which the group of projective automorphisms act. GIT stability with respect to this linearisation is called "Hilbert stability". If this locus forms an open set, then GIT may be used to construct a quotient which parametrises these objects. In good circumstances this quotient may be proper and projective. It is not always possible to embed a family of varieties inside a fixed projective space and therefore describe their moduli with geometric invariant theory, and this special property is called boundedness. A fundamental property of Fano varieties  is that they fail to be bounded, and thus their stability cannot be reasonably captured by any finite-dimensional geometric invariant theory. This explains why K-stability requires one to consider test configurations formula_11 for which the relatively ample line bundle formula_175 can correspond to some power formula_176 for formula_177 arbitrarily large. However, results of Caucher Birkar showed that certain families of Fano varieties with "volume bounded below" form bounded families, which suggests that it may be possible to study stability of volume-bounded families of Fano varieties to form moduli spaces. For this work Birkar was awarded the  Fields Medal in 2018. It was proven by Jiang that K-semistable formula_2-Fano varieties with volume bounded below form a bounded family. Thus for a given volume formula_179 there exists a uniform integer formula_180 such that every K-semistable formula_2-Fano with anticanonical volume larger than or equal to formula_182 admits an embedding inside the fixed projective space formula_183. The openness of this locus of K-semistable Fanos was proven by Blum–Liu–Xu and Xu. This implies the existence of an Artin stack of finite type denoted formula_184 parametrising K-semistable formula_2-Fano varieties with volume bounded below  by formula_182. In order to find a genuine moduli space as a projective variety or scheme, one must prove certain properties about "S"-completeness and formula_187-reductivity of K-semistable Fanos inside the stack formula_184. Using properties of K-polystability, these properties of the moduli stack are true and there exists a coarse moduli space formula_189 for the stack formula_184 which parametrises K-polystable formula_2-Fano varieties with volume bounded below by formula_182. It was proven that formula_189 is proper and that the CM-line bundle is ample, meaning the coarse moduli space is also projective. The existence result for K -moduli can be summarised in the following theorem.Theorem: There exists a separated, proper, projective, good moduli space formula_189 parametrising formula_195-dimensional K-polystable formula_2-Fano varieties with anticanonical volume bounded below by formula_179.The construction of the moduli space of K-polystable Fanos can be generalised to the setting of log Fano varieties. The case of singular formula_2-Fano varieties which are smoothable (that is they are limits of algebraic families of smooth K-polystable Fano manifolds) was solved earlier by Li–Wang–Xu using a combination of analytic techniques, also relying on the earlier work of Odaka, Donaldson, and  Codogni–Patakfalvi. There the coarse moduli space is shown to be a scheme, but in general the existence results for K-moduli only guarantee the existence of an algebraic space. Explicit K-stability of Fano varieties. The explicit study of K-stable Fano varieties precedes the algebraic notion of K-stability, and in low dimensions was of interest purely due to the study of Kähler–Einstein manifolds. For example, either by explicit construction or the use the Tian's alpha invariant, all smooth Kähler–Einstein manifolds of dimension 1 and 2 were known before the definition of K-stability was  introduced. In dimension 3 and higher explicit constructions of Kähler–Einstein metrics become more difficult, but advances arising out of the algebraic study of K-stability have enabled explicit computations of K-polystable Fano threefolds and certain families of higher dimensional varieties, and subsequently the discovery of new Kähler–Einstein manifolds. Dimension 1. In dimension one there is a unique smooth Fano variety, the complex projective line formula_199. This variety is easily seen to be K-stable due to the existence of the Fubini–Study metric, which is a Kähler–Einstein metric, implying the K-polystability of formula_199. A purely  algebro-geometric proof of the K-stability of smooth Riemann surfaces follows from the work of Ross–Thomas on slope K-stability, which is equivalent to K-stability in dimension one. In this case one may construct test configurations out of collections of points on the curve, and when the curve is smooth no points destabilise. Dimension 2. In dimension two the spaces which admit Kähler–Einstein metrics were classified by Tian. There are 10 deformation families of smooth Fano varieties in dimension two, the del Pezzo surfaces. Using the alpha invariant, Tian showed that a smooth Fano surface admits  a Kähler–Einstein metric and is K-polystable if and only if it is not the blow up of the complex projective plane formula_201 in one or two points. Thus 8 out of these 10 classes consist of K-polystable Fano surfaces. The K-moduli of Fano surfaces were studied in explicit examples by Tian and Mabuchi–Mukai. Explicit constructions of compact moduli spaces of Kähler–Einstein Fano surfaces were achieved by Odaka–Spotti–Sun. These spaces were constructed as Gromov–Hausdorff compactifications but were identified with explicit algebraic spaces of log Fano surfaces. For example, it is proven by  Odaka–Spotti–Sun that the compact moduli space of smoothable Kähler–Einstein surfaces of degree four is given by the weighted projective space formula_202 with the smooth Kähler–Einstein surfaces of degree four corresponding to the locus formula_203 where formula_204 is an ample divisor consisting of those points satisfying the equation formula_205. Dimension 3. In dimension 3 purely algebraic techniques can be used to find examples of K-stable Fano varieties which are not "a priori" known to admit Kähler–Einstein metrics. The Iskovskikh–Mori–Mukai classification of smooth Fano threefolds provides a natural way of breaking down the problem  of studying K-stable Fano threefolds into its components. It is known that there are 105 deformation families of smooth Fano threefolds, and explicit computations using Fujita–Li's beta invariant and Fujita–Odaka's delta invariant can be used to determine which deformation families contain smooth representatives. For every deformation family it is known whether the generic element of the family is K-(poly)stable. In particular it is known that 78 out of 105 families contain a K-polystable representative in their deformation class. For 71 out of 105 families, it is known for every single member of the deformation  class whether or not it is K-polystable. For many examples of the 105 deformation families, the K-stability of representative threefolds can be interpreted in terms of a natural GIT problem which describes that family, and so explicit examples of K-moduli of Fano threefolds can also be found as GIT quotients. For some classes of Fano threefolds the classification problem remains open. For example, it is known that the Mukai–Umemura threefold formula_206 in the deformation class formula_207 admits a Kähler–Einstein metric and is therefore K-polystable by work of Donaldson, who computed Tian's alpha invariant  explicitly using the criterion above. This manifold has non-discrete automorphism group formula_208 and which nearby deformations of formula_206 are also K-polystable is not known. It is conjectured that the deformations corresponding to GIT-polystable points within the versal deformation space of formula_210 should correspond to nearby K-polystable varieties. Higher dimensions. The first and simplest example of a K-polystable Fano manifold in any dimension is complex projective space, which always admits the Fubini–Study metric which is Kähler–Einstein in any dimension and therefore all projective spaces are K-polystable. In general there are not many such  obvious Kähler–Einstein metrics in higher dimensions, and one must use recent techniques of stability to find examples. For certain families of Fano varieties, K-stability can be proved in higher dimensions using either analytic techniques through the alpha invariant or purely algebro-geometric techniques with the beta or delta invariants. As an example, a Fermat hypersurface is a variety of the formformula_211These hypersurfaces are smooth Fano manifolds with discrete automorphism group for formula_212, and it was proven by Tian using the alpha invariant that formula_213 implying formula_214 admits a Kähler–Einstein for formula_215, and using more detailed arguments Tian  proved the existence of a Kähler–Einstein metric when formula_216. On the other hand, using the delta invariant Zhuang gave a completely algebraic proof that formula_214 is K-stable for formula_218 and therefore admits a Kähler–Einstein metric in these cases. Using the openness results for uniform K-stability and K-semistability, one can conclude from this that the generic smooth hypersurface of degree formula_218 inside formula_220 is K-stable. In some cases it is in fact known that "all" degree formula_221 hypersurfaces are K-stable, such as all formula_222 smooth hypersurfaces in formula_220. In addition to the study of  particular Fano varieties, in certain settings K-moduli may be explicitly described in higher dimensions. For example, when the K-moduli admits an "obvious" GIT interpretation, the algebraic tools of beta or delta invariants can be used to verify that GIT stability is equivalent to K-stability for that particular problem. For example, Liu showed that for cubic fourfold hypersurfaces in formula_224, the GIT moduli space of (possibly singular) cubic fourfolds is isomorphic to the K-moduli space, and thus one obtains an explicit description of the K-stable, K-polystable, and K-semistable cubic fourfolds in terms of their  GIT stability and singularity structure. In particular every smooth cubic fourfold is K-stable.