We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using Geometric Invariant Theory and the anticanonical polarization. The construction depends on a weight on the divisor. For smaller weights the stable pairs consist of mildly singular surfaces and very singular divisors. Conversely, a larger weight allows more singular surfaces, but it restricts the singularities on the divisor. The one-dimensional space of stability conditions decomposes in a wall-chamber structure. We describe all the walls and relate their value to the worst singularities appearing in the compactification locus. Furthermore, we give a complete characterization of stable and polystable pairs in terms of their singularities for each of the compactifications considered.

We study the GIT compactifications of pairs formed by a hypersurface and a hyperplane. We provide a general setting to characterize all polarizations which give rise to different GIT quotients. Furthermore, we describe a finite set of one-parameter subgroups sufficient to determine the stability of any GIT quotient. We characterize all maximal orbits of non stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. Our construction gives natural compactifications of the space of log smooth pairs for Fano and Calabi-Yau hypersurfaces.GIT n,d,t corresponding to values t = t i and to any t ∈ (t i , t i+1 ).Theorem 1.1. Let S n,d be the set of one-parameter subgroups in Definition 3.1. All GIT walls {t 0 , . . . , t n,d } correspond to a subset of the finite set − m, λ x i , λ m is a monomial of degree d, 0 i n + 1, λ ∈ S n,d (1)and they are contained in the interval [0, t n,d ] where t n,d = d n+1 . Every pair (X, H) has an interval of stability [a, b] with a, b ∈ {t 0 , . . . , t n,d }. Namely, (X, H) is t-semistable if and only if t ∈ [t i , t j ] for some walls t i , t j . If (X, H) is t-stable for some t then (X, H) is t-stable if and only if t ∈ (t i , t j ).Corollary 1.2. Assume that the ground field is algebraically closed with characteristic 0 and that the locus of stable points is not empty and d 3. Then dim M GIT n,d,t =

Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation families. This book determines whether the general element of each family admits a Kähler–Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to prove the existence of a Kähler–Einstein metric, containing many additional relevant results such as the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.

We provide new examples of K-unstable polarized smooth del Pezzo surfaces using a flopped version first used by Cheltsov and Rubinstein of the test configurations introduced by Ross and Thomas. As an application, we provide new obstructions for the existence of constant scalar curvature Kähler metrics on polarized smooth del Pezzo surfaces.All varieties are assumed to be algebraic, projective and defined over C. IntroductionK-stability is an algebraic notion of polarized varieties which has been of great importance in the study of the existence of canonical metrics on complex varieties. This is mainly because of the following Conjecture (Yau-Tian-Donaldson). Let X be a smooth variety, and let L be an ample line bundle on it. Then X admits a constant scalar curvature Kähler (cscK) metric in c 1 (L) if and only if the pair (X, L) is K-polystable.It is known in different degrees of generality that K-polystability is a necessary condition for the existence of a cscK metric, with the most general result due to Berman, Darvas and Lu [4] following work of Darvas and Rubinstein [9]. For smooth Fano varieties polarized by anticanonical line bundles, Conjecture 1 was recently proved by Chen, Donaldson and Sun in [7].In spite of the above (conjectural) characterizations, deciding whether a given polarized variety is K-stable is a problem of considerable difficulty. In this paper, we study this problem for del Pezzo surfaces polarized by ample Q-divisors. Using Q-divisors does not affect the original problem, since K-stability is preserved when we scale the polarization positively.Let S be a smooth del Pezzo surface, and let L be an ample Q-divisor on it. Recall that S is toric if and only if K 2 S 6. In this case, the problem we plan to consider is completely solved. In the non-toric case, few results in this direction are known. For instance, if S is not toric, then it admits a Kähler-Einstein metric by Tian's theorem [25], so that (S, −K S ) is K-stable. Moreover, a result of LeBrun and Simanca [16] implies that Date: 18/01/2018.Remark 2.4. The K-polystability of the pair (X, L) implies its K-semistability. Similarly, the K-stability of the pair (X, L) implies its K-polystability. Moreover, if the group Aut(X, L) is finite, then all product test configurations of the pair (X, L) are trivial, so that (X, L) is K-stable if and only if it is K-polystable.The pair (X, L) is K-polystable (respectively, K-stable or K-semistable) if and only if the pair (X, L ⊗k ) is K-polystable (respectively, K-stable or K-semistable) for some positive integer k. Thus, we can adapt both Definitions 2.1 and 2.3 to the case when L is an ample Q-divisor on the variety X. This gives us notions of K-polystability, K-stability, K-semistability and K-instability for varieties polarized by ample Q-divisors. Similarly, we can assume that L in the test configuration (X , L, p) is a p-ample Q-divisor on X . Because of this, we can assume that r = 1 in the formula (2.2) for the Donaldson-Futaki invariant. Slope stability and Atiyah flopsLet S be a smooth ...

We give a simple sufficient condition for K-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kähler metric in the Kähler class corresponding to the polarization.