F -purity of isolated log canonical singularities Shunsuke Takagi In this article, we explain the correspondence of log canonical singularities and F -pure singularities in the case of isolated singularities. This article is based on the joint work [4] with Osamu Fujino, and the reader is referred to [4] for the proofs.

1

Preliminaries

First we recall the definition of log canonical singularities. Definition 1.1. Let x ∈ X be a point of a normal Q-Gorenstein complex algebraic variety. Let f : Y → X be a resolution of singularities such that the exceptional locus Exc(f ) is a simple normal crossing divisor. Then we can write ∑ KY = f ∗ KX + ai Ei , i

where the ai are rational numbers and the Ei are f -exceptional prime divisors on Y . We say that x ∈ X is a log canonical singularity (resp. a log terminal singularity) if ai ≥ −1 (resp. ai > −1) for all i such that x ∈ f (Ei ). This definition is independent of the choice of the resolution f . Example 1.2. Let X = Spec C[X, Y, Z]/(X a + Y b + Z c ). Then X has only log canonical singularities if and only if a1 + 1b + 1c ≥ 1. Fujino [3] introduced the invariant µ(x ∈ X) of an isolated log canonical singularity x ∈ X. Definition 1.3. Let x ∈ X be an isolated log canonical singularity which is not log terminal. First we assume that x ∈ X is quasi-Gorenstein. Take a projective e such that Exc(f ) and birational morphism f : Y → X from a smooth variety X −1 Supp f (x) are simple normal crossing divisors. Then we can write KY = f ∗ KX + F − E, where E and F are effective divisors on Y and have no common irreducible components. By assumption, E is a reduced simple normal crossing divisor on Y . We define µ(x ∈ X) by µ(x ∈ X) = min{dim W | W is a stratum of E}.

This definition is independent of the choice of the resolution f . In general, we take an index one cover ρ : X 0 → X with x0 = ρ−1 (x) to define µ(x ∈ X) by µ(x ∈ X) = µ(x0 ∈ X 0 ). Since the index one cover is unique up to ´etale isomorphisms, the above definition of µ(x ∈ X) is well-defined. By definition, 0 ≤ µ(x ∈ X) ≤ dim X − 1. Remark 1.4. A Gorenstein isolated log canonical singularity x ∈ X with µ = µ(x ∈ X) is called in [9] as a purely elliptic singularity (X, x) of type (0, µ). Next we recall the definition of F -pure singularities. Definition 1.5. Let x ∈ X be a (closed) point of an F -finite integral scheme X of characteristic p > 0. (i) x ∈ X is said to be F -pure if the Frobenius map F : OX,x → F∗ OX,x

a 7→ ap

splits as an OX,x -module homomorphism. (ii) x ∈ X is said to be strongly F -regular if for every nonzero c ∈ OX,x , there exist an integer e ≥ 1 such that cF e : OX,x → F∗e OX,x

e

a 7→ cap

splits as an OX,x -module homomorphism. Example 1.6. Let X = Spec Fp [X, Y, Z]/(X 3 + Y 3 + Z 3 ). Then X has only F -pure singularities if and only if p ≡ 1 mod 3. Using reduction from characteristic zero to positive characteristic, we can define the notion of F -purity in characteristic zero. Definition 1.7. Let x ∈ X be a point of a complex algebraic variety X. Choosing a suitable finitely generated Z-subalgebra A ⊆ C, we can construct a point xA of a scheme XA of finite type over A such that (XA , xA ) ×Spec A C ∼ = (X, x). By the generic freeness, we may assume that (XA , xA ) is flat over Spec A. We refer to xA ∈ XA as a model of x ∈ X over A. Given a closed point s ∈ Spec A, we denote by xs ∈ Xs the fiber of x ∈ X over s. We say that x ∈ X is of strongly F -regular type (resp. dense F -pure type) if for a model of x ∈ X over a finitely generated Z-subalgebra A ⊆ C, there exists a dense open subset (resp. a dense subset) of closed points S ⊆ Spec A such that xs ∈ Xs is strongly F -regular (resp. F -pure) for all s ∈ S. Example 1.8. Let X = Spec C[X, Y, Z]/(X 3 + Y 3 + Z 3 ). By Lemma 1.6, X is of dense F -pure type.

Hara proved the equivalence of log terminal singularities and strongly F -regular singularities. Theorem 1.9 ([5]). Let x ∈ X be a point of a normal Q-Gorenstein complex algebraic variety X. Then x ∈ X is log terminal if and only if it is of strongly F -regular type. In this article, we will discuss an analogous statement for isolated log canonical singularities.

2

Main Theorem

In order to state our main result, we need the following conjecture. Conjecture An . Let V be an n-dimensional projective variety over an algebraically closed field k of characteristic zero with only rational singularities. Assume that KV ∼ 0. Given a model of V over a finitely generated Z-subalgebra A of k, there exists a dense subset of closed points S ⊆ Spec A such that the natural Frobenius action on H n (Vs , OVs ) is bijective for every s ∈ S. Lemma 2.1. Conjecture An is true if n ≤ 2. Proof. By an argument similar to the proof of [8, Proposition 5.3], we may assume that k = Q without loss of generality. Conjecture A0 is trivial. Conjecture A1 follows from a result of Serre [11]. So we consider the case when n = 2. Let π : Ve → V be a minimal resolution. Ve is an abelian surface or a K3 surface. Suppose given a model of π over a finitely generated Z-subalgebra A of k. Then there exists a dense subset of closed points S ⊆ Spec A such that the Frobenius action on H 2 (Ves , OVes ) is bijective for every s ∈ S (the abelian surface case follows from a result of Ogus [10] and the K3 surface case follows from a result of Bogomolov–Zarhin [2] or that of Joshi and Rajan [7]). Since X has only rational singularities, we may assume that H 2 (Vs , OVs ) ∼ = H 2 (Ves , OVes ) as κ(s)[F ]-modules for all s ∈ S. Thus, we obtain the assertion. The recent progress in the minimal model program [1] allows us to prove the following theorem. Theorem 2.2 ([4, Theorem 3.3]). Let x ∈ X be an isolated log canonical singularity. If Conjecture Aµ holds where µ = µ(x ∈ X), then x ∈ X is of dense F -pure type. In particular, if µ(x ∈ X) ≤ 2, then x ∈ X is of dense F -pure type. Sketch of Proof. Let d = dim X. After passing through an index one cover, we may assume that x ∈ X is quasi-Gorenstein. We take a dlt blow-up g : Z → X of x ∈ X. That is, g is a projective birational morphism satisfying the following properties: (i) KZ + D = f ∗ KX , where D is a reduced divisor on X, (ii) (Z, D) is a Q-factorial dlt pair,

(iii) g is an isomorphism outside x, (iv) Z has only canonical singularities. Then we can take a minimal log canonical center V of (Z, D) such that (v) V is a projective variety with only rational singularities, (vi) dim V = µ = µ(x ∈ X), (vii) KV ∼ 0, (viii) H µ (V, OV ) ∼ = H d−1 (D, OD ). Applying Conjecture Aµ to this V and running a KZ -minimal model program with scaling over X, we can obtain the assertion. Corollary 2.3. Let x ∈ X be an isolated singularity of a normal Q-Gorenstein complex algebraic variety of dimension ≤ 3. Then x ∈ X is log canonical if and only if it is of dense F -pure type. Proof. The if part follows from [6, Theorem 3.9]. The only if part follows from Theorem 2.2, because µ(x ∈ X) ≤ dim X − 1 = 2.

References [1] C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. [2] F. Bogomolov and Y. Zarhin, Ordinary reduction of K3 surfaces, Cent. Eur. J. Math. 7, (2009), no.3, 73–212. [3] O. Fujino, On isolated log canonical singularities with index one, J. Math. Sci. Univ. Tokyo. 18 (2011), 299–323. [4] O. Fujino, S. Takagi, On the F -purity of isolated log canonical singularities, arXiv:1112.2383, submitted. [5] N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981–996. [6] N. Hara and K.-i. Watanabe, F -regular and F -pure rings vs. log terminal and log canonical singularities, J. Algebraic. Geom. 11 (2002), no. 2, 363–392. [7] K. Joshi and C. S. Rajan, Frobenius splitting and ordinary, arXiv:math/ 0110070, Int. Math. Res. Not. 2003, no. 2, 109–121. [8] M. Mustat¸a˘ and V. Srinivas, Ordinary varieties and the comparison between multiplier ideals and test ideals, arXiv:1012.2818, to appear in Nagoya Math. J.

[9] S. Ishii, On isolated Gorenstein singularities, Math. Ann. 270 (1985), no.4, 541–554. [10] A. Ogus, Hodge cycles and crystalline cohomology, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math., vol. 900, Springer-Verlag, BerlinNew York, 1981, 357–414. [11] J. P. Serre, Groupes de Lie l-adiques attach´es aux courbes elliptiques, Les Tendances G´eom. en Alg´ebre et Th´eorie des Nombres, Editions du Centre National de la Recherche Scientique, Paris, 1966, 239–256. Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan E-mail : [email protected]