Applications of the moduli continuity method to log K‐stable pairs

Abstract

The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from Geometric Invariant Theory. More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients. GIT d of degree d hypersurfaces in P n. The first statement of the above result is a special case of the following natural expectation, at least when the automorphism group has no non-trivial characters. Conjecture 1.3. Let X be a K-polystable Fano variety. Then for any sufficiently large and divisible l ∈ N, there exists β 0 = β 0 (l, X) such that for all D ∈ | − lK X | and β ∈ (β 0 , 1), the pair (X, (1 − β)D) is log Kpolystable if and only if [D] ∈ P(H 0 (−lK X)) is GIT-polystable for the natural representation of Aut(X) on H 0 (−lK X). Moreover, the Gromov-Hausdorff compactification M K X,l,β of the above pairs (X, (1 − β)D) is homeomorphic to the GIT quotient P(H 0 (−lK X)) ss //Aut(X). Note that the automorphism group Aut(X) of a K-polystable Fano is always reductive. If X is smooth or a Gromov-Hausdorff degeneration, this follows from by Matsushima's obstruction [Mat57] and Chen-Donaldson-Sun [CDS15]. Otherwise, this has been very recently established in [ABHLX20] via purely algebro-geometric techniques Once its different steps are established, the application of the moduli continuity method (the proof of theorems 1.1 and 1.2) is just a couple of paragraphs. However, in order to keep the reader focused on the main goal of the paper, we recall here what the different steps of the moduli continuity method are, noting that each of them requires involved technical proofs to accomplish them. Suppose that we have a Gromov-Hausdorff

K-stability and birational models of moduli of quartic K3 surfaces

“…That is, by varying the coefficient c, the K-moduli spaces M K c provide a natural interpolation between the GIT quotient for quartic surfaces and the Baily-Borel compactification, and explicitly resolve the period map. We note that a special case of Theorem 1.1(1) was proved earlier in [36,Theorem 1.2] and [3,Theorem 1.4] (see also [103]). We also note that the two walls which are divisorial contractions are actually weighted blowups of Kirwan type (see Remarks 5.17 and 6.10).…”

“…We sketch the proofs of Theorems 1.1 and 1.2. First of all, by [3,36] we know that M K ∼ = M GIT . If S is a quartic surface in P 3 with semi-log canonical (slc) singularities (also called insignificant limit singularities), then (P 3 , S) is a Ksemistable log Calabi-Yau pair, and P 3 is K-polystable.…”

Log K-stability of GIT-stable divisors on Fano manifolds

“…For example, in the case X = P n , the work [ADL19, Theorem 1.4] establishes such a correspondence between GIT-stability and K-stability. In this paper, we apply moduli continuity method to prove the following theorem, which is conjectured in [GMGS,Conjecture 1.3].…”

The moduli space of cubic surface pairs via the intermediate Jacobians of Eckardt cubic threefolds

“…Moduli spaces of cubic surfaces with boundary divisors have been studied before in a number of contexts, for example, [19, 26]. More recently, in the context of log K‐stability, some compactifications of the moduli space were described in [22], by reducing to the GIT compactifications of cubic surface pairs analyzed in [21]. Cubic surface pairs also provide examples of log Calabi–Yau surfaces, which arise in the study of mirror symmetry [24]; the mirror family to a particular cubic surface pair was recently given in [25].…”