K-polystability of 3-dimensional log Fano pairs of Maeda type

Abstract

Using the Abban-Zhuang theory and the classification of three-dimensional log smooth log Fano pairs due to Maeda, we prove that threefold log Fano pairs (X, D) of Maeda type with reducible boundary D are K-unstable, with four exceptions. We also correct several inaccuracies in Maeda's classification.K-POLYSTABILITY OF 3-DIMENSIONAL LOG FANO PAIRS OF MAEDA TYPE 3 PreliminariesWe work over algebraically closed field of characteristic 0. For the standard definitions of the minimal model program (MMP for short) we refer to [KM98]. We recall the notion of a pair.Definition 2.1. Let X be a normal projective variety, and let ∆ be an effective Weil Q-divisor whose coefficients are between 0 and 1. We say that (X, ∆) is asmooth, and Supp(∆) has simple normal crossings.Definition 2.2. Log Fano pair is a log canonical pair (X, ∆) such that −K X − ∆ is ample. In this situation, X is called a log Fano variety. If, moreover, the pair (X, ∆) is log smooth, then we say that X is a log Fano manifold.Throughout the paper, we will assume that the divisor ∆ is integral. Definition 2.3 ([Fu20a]). Log Fano pair of Maeda type is a klt log Fano pair (X, D) with D = 0 such that there exists a a log smooth log Fano pair (X, ∆) with integral ∆, and Supp(D) ⊆ Supp(∆).We discuss some generalities on log smooth log Fano pairs. Let (X, ∆) be such pair. By the main result in [Zha06] applied to the pair (X, (1 − ǫ)∆) for some 0 < ǫ ≪ 1, the variety X is rationally connected. Let ∆ = k i=1 D i where D i are prime divisors. By [Lo19, 3.3], ∆ has no more than dim X components. In [LM20] it is proven that if this bound is attained, then the pair (X, ∆) is toric. By section 2 in [M83], we have Pic(X) ≃ H 2 (X, Z) and h i (O X ) = 0 for i > 0. Also, Pic(X) is torsion free. Moreover, Pic(X)⊗ R = N 1 (X) since the linear equivalence of divisors coincides with the numerical equivalence. The Mori cone NE(X) is polyhedral and (K X + ∆) is negative on NE(X) \ {0}. In particular, by Mori theory for any extremal face of NE(X) there exists the corresponding contraction which contracts precisely the curves whose classes belong to that extremal face. For more details on the geometry of log smooth log Fano pairs, see [Fu14b].There exists a description of the Picard group of a log smooth log Fano pair. For the definition of the Picard group of a reducible variety, we refer to [Fu14b].