Reductivity of the automorphism group of K-polystable Fano varieties

Abstract

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and Θ-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space. Contents 1. Introduction 1 2. Preliminaries 4 3. S-completeness 11 4. Reductivity of the automorphism group 19 5. Θ-reductivity 23 References 29Throughout, we work over an algebraically closed field k of characteristic 0.

Del Pezzo surfaces with infinite automorphism groups

“…Thus, the surfaces listed in this corollary are not K-polystable [ABHX21,Mat57], which is known (see [OSS16]).…”

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

“…This is a straightforward application of Theorem 1.2 and the classification of GIT polystable hypersurfaces. For (n, d) = (2, 3), (2,4), (3,3), see [44,Examples 7.12…”

“…For example, when we consider the log pair (P 2 , (1 − β 4 )H 4 ), it follows by the study of del Pezzo surfaces of degree 2 [48] that for β = 1 2 , the GIT picture above is not accurate any more. More precisely: (P(1, 1, 4), 1 2 H), where H is the hyperelliptic curve z 2 3 = f 8 (z 1 , z 2 ), where f 8 is a polystable binary octic.…”

“…If is smooth or a Gromov–Hausdorff degeneration, this follows from by Matsushima's obstruction [43] and Chen–Donaldson–Sun [12]. Otherwise, this has been very recently established in [2] via purely algebro‐geometric techniques…”

“…The paper [3] also includes the description of a natural algebraic structure on the moduli space of general smoothable K‐polystable Fano pairs, using analytic techniques, in analogy to the absolute case established in [55], [47] and [38]. Regarding this last point, it is worth mentioning that recently there has been several important advances in establishing properties of the moduli of (non‐necessarily smoothable) Fano (pairs) via purely algebro‐geometric techniques, such as its separatedness [6], its existence [2] (but not yet properness) and the general openness of K‐semistability [57] and [5].…”