Degeneration of K3 surfaces with non-symplectic automorphisms

We prove that a K3 surface with an automorphism acting on the global $2$-forms by a primitive $m$-th root of unity, $m \neq 1,2,3,4,6$, does not degenerate (assuming the existence of the so-called Kulikov models). A key result used to prove this is the rationality of the actions of automorphisms on the graded quotients of the weight filtration of the $l$-adic cohomology groups of the surface.


Introduction
Let O K be a Henselian discrete valuation ring (DVR) with fraction field K and residue field k.We consider the problem of degeneration of K3 surfaces: Given a K3 surface X over K, we investigate its possible extensions X over O K and their reductions X 0 over k.
The so-called Kulikov models, which are semistable models of X over O K with nice properties, is a standard tool to discuss degeneration of K3 surfaces.The special fibers of Kulikov models are classified into three types: Type I, smooth K3 surfaces, and Types II and III, which are reducible surfaces satisfying certain conditions.It is conjectured that any K3 surface over K admits a Kulikov model after replacing K by a finite extension, but not yet proved in general.See Section 4 for details.
In this paper we relate the properties of Kulikov models with (nonsymplectic) automorphisms of X: Theorem 1.1.Assume char K = 2. Let X be a Kulikov model over O K of a K3 surface X over K and X 0 its special fiber over k.Denote by m = m(X) the order of the image of ρ : Aut K (X) → GL(H 0 (X, Ω 2 X/K )) (which is finite).
(2) Assume m = 1, 2. Then X 0 is either of Type I or II.
The key idea of the proof is describing the action of Aut(X) on the n-th graded quotients gr W n of the weight filtration of H 2 ét (X K , Q l ) in terms of ρ (Lemma 5.1).We also show the rationality (and l-independence) of such action (Theorem 3.3), and using this we derive some restrictions on dim gr W n .
This work was supported by JSPS KAKENHI Grant Numbers 15H05738 and 16K17560.
By calculating dim gr W n (using the classification of the special fibers of Kulikov models, see Section 4) we can exclude certain types of degeneration, and then the remaining possibilities are as stated in the theorem.
The assumption on m in Theorem 1.1 is optimal: see Examples 7.1-7.2.
In Section 6 we give an application on K3 surfaces with non-symplectic automorphisms of prime order p ≥ 5: we can show that the moduli space of such K3 surfaces (in characteristic 0) is compact and that any such surface defined over a number field has everywhere potential good reduction.
Characteristic p > 0, finite height: Let W = W (k) be the ring of Witt vectors over k and let K = Frac W . Lieblich-Maulik [LM18, Corollary 4.2] showed that there exists a lifting X over W such that the specialization morphism NS( XK ) → NS( Xk ) is an isomorphism.They also showed [LM18, Theorem 2.1] that for such X the restriction map Aut( X) → Aut(X K ) is an isomorphism, and the same assertion holds for XR := X ⊗ R for any finite extension R of W .They also showed [LM18, Section 6] that for such X the specialization map Aut( XK ) → Aut(X k ), defined as the limit of Aut( XFrac R ) ∼ ← Aut( XR ) → Aut(X k ), has finite cokernel.Comparing the actions on a 2-form on XR and its mod p reduction, we observe that this specialization map is compatible with ρ.The assertion is reduced to the characteristic 0 case.Remark 2.2.We cited a theorem of Nygaard [Nyg80, Theorem 2.1], which is stated for p = 2.We show that this is still true if p = 2: this argument is due to Kazuhiro Ito.The only step where the assumption p = 2 is used is the inductive step of [Ogu79, Lemma 3.14].If p = 2, we can argue as follows.If there exists x ∈ Γ with p ∤ x, x then we argue as in [Ogu79].Suppose there is no such x.There are still x 1 , x 2 ∈ Γ with p ∤ x 1 , x 2 , and then the matrix ( x i , x j ) 2 i,j=1 is invertible (since p | x i , x i ), hence Γ decomposes to the sum of the subspace generated by (x 1 , x 2 ) and its orthogonal complement.Apply the induction hypothesis to the complement.
We recall the transcendental lattices of K3 surfaces.Let T l (X) be the orthogonal complement of NS(X k ) ⊗ Z l (−1) in H 2 ét (X k , Z l ) and denote by χ l : Aut(X) → GL(T l (X)) the natural action.If k = C we define T (X) ⊂ H 2 (X, Z) and χ : Aut(X) → GL(T (X)) similarly.If char k > 0 we define T crys (X) ⊂ H 2 crys (X k /W (k)) and χ crys : Aut(X) → GL(T crys (X)) similarly, where W (k) is the ring of Witt vectors over k.
In characteristic 0 the following relationship between χ and ρ is wellknown (for example see [Sch16,Remark 3.4]).We include the proof for the reader's convenience.
Lemma 2.3.Assume char k = 0. Then the characteristic polynomial of χ l (g) belongs to Z[x] and is independent of l.It is a power of the m(g)-th cyclotomic polynomial Φ m(g) .If k = C then it is equal to that of χ(g).
Proof.We may assume k = C.By the comparison of Betti and étale cohomology groups, χ(g) and χ l (g) have the same characteristic polynomial P ∈ Z[x].Since H 0 (X, Ω 2 ) ⊂ T (X) ⊗ C (by Hodge decomposition), P has ρ(g) as a root, and hence is divisible by Φ m(g) .Note that T (X) Q is irreducible as a rational Hodge structure (if it admits a decomposition T (X) Q = T 1 ⊕ T 2 , then since dim(T (X) C ) 2,0 = h 2,0 (X) = 1 there exists i ∈ {1, 2} for which T i ⊂ H 1,1 , but then by the Lefschetz (1, 1)-theorem we have We also need a positive characteristic version.We say that g ∈ Aut(X) in positive characteristic is liftable to characteristic 0 if there exists a pair ( X, g) of a proper smooth scheme X over a DVR V that is finite over W (k) and an automorphism g ∈ Aut( X) satisfying ( X, g) ⊗ V k = (X, g) ⊗ k k. (By [Del81, proof of Corollaire 1.10], the generic fiber of X is then automatically a K3 surface.)Lemma 2.4.Assume char k = p > 0.
(1) The characteristic polynomial P of χ l (g) belongs to Z[x], is independent of l, and is equal to that of χ crys (g).
(2) If g is liftable to characteristic 0, then P is a power of Φ m(g)p e for some integer e ≥ 0.
(3) If p > 2 and X is of finite height (we no longer assume liftability), then P is a product of cyclotomic polynomials of the form Φ m(g)p e i for some integers e i ≥ 0.
Proof.We may assume k is algebraically closed.
(1) This follows from the corresponding assertions for the actions on H 2 ét and H 2 crys (showed in [Ill75,3.7.3]) and on their subspaces generated by NS (clear).
(2) Let ( X, g) be a lifting over V and let K = Frac V .Comparing the actions of g on a 2-form on X and its mod p reduction, we observe that ρ(g| XK ) maps to ρ(g) under the map µ m (K) → µ m (k), where m = m(g| XK ).Since the kernel of this map is precisely the elements whose order is a power of p, we obtain m(g| XK ) = m(g)p e for some e ≥ 0. Since NS( XK ) ֒→ NS(X k ), we have an equivariant injection T l (X) ֒→ T l ( XK ), and the assertion follows from Lemma 2.3.
(3) The images of χ l and χ crys are finite (this can be showed by reducing to characteristic 0 as in the proof of Lemma 2.1).By replacing g with its p N -th power for some N we may assume that the order of χ crys (g) is prime to p; this does not change m(g) because p ∤ m(g) (since there are no primitive p-th roots of 1 in k).By [Jan16, Theorem 3.2], an automorphism of a K3 surface of finite height in characteristic p > 2 with this property is liftable to characteristic 0. Hence the assertion is reduced to (2).
In general e and e i in (2),(3) may be nonzero: see Example 7.3.We do not know whether P in (3) can have more than one different factors.

Action of correspondences on the weight spectral sequence
Let l be an arbitrary prime different from char k.In this section we study the actions of automorphisms, and more generally of algebraic correspondences, on the l-adic cohomology groups of varieties X over the fraction field K of a Henselian DVR O K .We show that they act on the graded quotients gr W n of the weight filtration and that for certain n these actions are rational, i.e. their characteristic polynomials have coefficients in Q.
In this paper, we call an algebraic space X over O K to be a strictly semistable model of its generic fiber X if it is regular and flat over O K , its generic fiber X over K is a smooth scheme, and its special fiber X 0 over k is a simple normal crossing divisor that is a scheme.(X itself is not assumed to be a scheme.) We review the following results on the weight spectral sequence.
Theorem 3.1.Let X be a strictly semistable model over O K of a variety X.Then one can attach to X a spectral sequence where are the disjoint unions of (p + 1)-fold intersections of the irreducible components of X 0 := X 0 ⊗ k k.The spectral sequence is compatible with automorphisms of X .The spectral sequence degenerates at E 2 .
Under the assumption that X is a scheme, the first assertion is proved by Rapoport-Zink [RZ82, Satz 2.10] and also by Saito [Sai03, Corollary 2.8], and E 2 -degeneration is proved by Nakayama [Naka00, Proposition 1.9].Compatibility with automorphisms follows from the proof of Saito.In [Mat15, Proposition 2.3] we removed the assumption that X is a scheme.
The filtration induced by this spectral sequence is called the weight filtration and denoted This filtration is independent of the choice of a proper strictly semistable model X (use the argument in [Ito05, Section 2.3]).We let gr More generally, we can define the weight filtration on H i ét (X K , Q l ) in the following way without assuming the existence of a proper strictly semistable model of X.It is known [dJ96, Theorem 6.5] that, after replacing O K by a finite extension of its completion, there exists an alteration Y of X (i.e. a proper surjective generically-finite morphism f : Y → X from a variety Y ) that admits a proper strictly semistable model that is a scheme.The composite defined by using a proper strictly semistable model of Y ).Again using the argument in [Ito05, Section 2.3], we can show that this filtration is independent of the choice of Y and a strictly semistable model of Y .In particular, if X itself admits a strictly semistable model, then this filtration coincides with the one defined in the previous paragraph.
We also recall the monodromy filtration M r H i ét on is the inertia subgroup.The monodromy operator N is the unique nilpotent map N : H i ét (1) → H i ét such that there exists an open subgroup J ⊂ I K such that any element σ ∈ J acts on H i ét by exp(t l (σ)N ).Then the monodromy filtration is defined as the unique increasing filtration satisfying In particular, since N acts by zero on the classes of algebraic cycles, we have NS(X) ⊂ Ker N ⊂ M 0 H 2 ét (1).The weight-monodromy conjecture states that M r H i ét = W r+i H i ét for any X and any i.
Theorem 3.2.If dim X = 2 then the weight-monodromy conjecture for H 2 ét is true, i.e. we have M r H 2 ét = W r+2 H 2 ét .This is proved by Rapoport-Zink [RZ82, Satz 2.13] in the case X admits a strictly semistable model that is a scheme.The general case is reduced to this case by de Jong's alteration (see Saito [Sai03, Lemma 3.9]).Now we consider the actions of algebraic correspondences on the l-adic cohomology groups.
Theorem 3.3.Let X be a proper smooth variety over K of dimension d, and Γ ∈ CH d (X × X) an algebraic correspondence (i.e.Γ is a Z-linear combination of codimension d subvarieties of X × X).Then, (1) For each integer 0 ≤ i ≤ 2d, the action of Γ * on H i ét (X K , Q l ) preserves the weight filtration.Hence it acts on gr To the author's knowledge, this result is not previously known.We use the following lemma, which follows from [Kle68, Lemma 2.8] (cf.[Sai03, Lemma 3.4]).
Lemma 3.4.Let F be a field of characteristic 0 and f an F -linear endomorphism of an F -vector space of finite dimension.If there exists a nonzero integer N such that any power of f has trace in N −1 Z, then the characteristic polynomial of f has coefficients in Z.
Proof of Theorem 3.3.First we prove the assertions (1) and (2) under the assumption that X admits a proper strictly semistable model X that is a scheme.
By [Sai03, Proposition 2.20] there exists a collection of algebraic cycles such that there is an endomorphism of the weight spectral sequence that acts on and on This complex is naturally isomorphic to the base change V (•) ⊗ Z Q l of the complex V (•) of the Z-modules freely generated by the components of X . By [Sai03, Lemma 3.6], this complex is naturally isomorphic to T l (A (•−1) ) ⊗ Q l (−1) induced by the complex A (•−1) of the Picard varieties of X (•−1) 0 . As above, Γ (p) induce an endomorphism of this complex of abelian varieties that is independent of l.Hence the action of Γ on gr W 1 H i ét is (d!) −1 times the l-adic realization of an endomorphism of an abelian variety, and therefore its characteristic polynomial has coefficients in (d!) −1 Z and is independent of l.Again by Lemma 3.4 the coefficients lie in Z.
The cases of n = 2d−1, 2d are similar to the cases of n = 1, 0 respectively.Consider the latter assertion of (2).Since gr W n is 0 outside the range 0 ≤ n ≤ 2d, we are done if d ≤ 1.If d = 2, then the only remaining case of n = 2 follows from the assertions for H i ét ([Sai03, Corollary 0.2]) and for gr W n for the other n's (n = 0, 1, 3, 4).Now we show that the assertions for the general case (i.e.not assuming the existence of a semistable model that is a scheme) can be reduced to this special case.Take an alteration Y → X as above (after replacing for Γ and its powers, we reduce assertion (2) for Γ to the case of Y .
For n = 1, 2d − 1 we actually proved: Corollary 3.5.Let X and Γ be as in Theorem 3.3.Then for each integer n is an abelian variety with 1 is obtained as the i-th "cohomology" of the complex A (•−1) above, and Remark 3.6.We can also show Theorem 3.3 without using alterations if X admits a strictly semistable model that is not necessarily a scheme, which is the case in the setting of our main theorems.For this, we need to generalize [Sai03, Proposition 2.20] to the case of algebraic spaces.Most of its proof is étale-local and hence can be reduced to the scheme case, and the only nontrivial part is the construction ([Sai03, Lemma 2.17]) of cycle classes Γ (p) on X (p) 0 for a cycle class Γ on the generic fiber X.Although we cannot mimic the construction of Saito (which uses locally-free resolution of coherent sheaves), we can take an algebraic cycle on X (p) 0 to be the intersection with the closure of (a cycle representing) Γ, and then check the required properties étalelocally.We omit the details.

Kulikov models
Definition 4.1.Let X be a K3 surface or an abelian surface over K.An algebraic space X over O K with generic fiber X is a Kulikov model of X if it is a proper strictly semistable model (in the sense of the previous section) and its relative canonical divisor K X /O K is trivial.
Remark 4.2.The standard, but conditional, recipe to construct a Kulikov model of a K3 (or an abelian) surface X is the following.Take a proper strictly semistable model of X after extending K, then apply a suitable MMP to get a log terminal model with nef canonical divisor (then the canonical divisor is in fact trivial), and then apply Artin's simultaneous resolution to make it semistable.If the residue field k is of characteristic 0 this is unconditional.If char k = p > 0 then the existence of a semistable model is still conjectural and the MMP is proved only for p ≥ 5 [Kaw94].
Classification of the special fibers of Kulikov models is given by Kulikov [Kul77, Theorem II] in characteristic 0 and Nakkajima [Nakk00, Proposition 3.4] in characteristic > 0. Then we can compute dim gr W n using the weight spectral sequence.We summarize: Proposition 4.3.Let X be a K3 surface and X a Kulikov model of X.
(1) The geometric special fiber X 0 = X 0 ⊗ k is one of the following.Type I: a smooth K3 surface.
Type II: , Z 1 and Z N are rational and the others are elliptic ruled (i.e.birational to a P 1 -bundle over an elliptic curve), Z i ∩ Z j is an elliptic curve if |i − j| = 1, and there are no other intersections.(The components form a "chain".)Type III: X 0 is a union of rational surfaces, each double curve is rational, and the dual graph of the components forms a triangulation of S 2 (the 2-dimensional sphere).
(2) dim Q l gr W n H 2 ét (X K , Q l ) (n = 0, 1, 2, 3, 4) of the generic fiber X depends only on the type, and are given by the following.
We will simply say that X 0 is of Type I, II, or III if X 0 is so.

Proof of the main theorems
Lemma 5.1.Let K, X, and X be as in Theorem 1.1 (so char K = 2).Assume either char K = p > 2 and ht(X) < ∞, or char K = 0. Assume X 0 is of Type II or III, and let n = 3 or n = 4 respectively.Then any eigenvalue of the action of g ∈ Aut(X) on gr W n = gr W n H 2 ét (X K , Q l ) is a primitive m ′ -th root of 1, where m ′ = m(g)p e for some integer e ≥ 0 (possibly depending on g and the eigenvalue) if char K = p > 0, and m ′ = m(g) if char K = 0.

Proof. By Proposition 4.3 we have gr
The assertions follow from Lemmas 2.3-2.4.
Proof of Theorem 1.1.If char K > 0 and ht(X) ≥ 3, then it is proved in [RZS82, Section 2, Corollary] (without the condition on automorphisms) that X 0 is of Type I: Since the height is upper semi-continuous and Type II (resp.III) surfaces have height ≤ 2 (resp.= 1), X 0 cannot be of Type II nor III.Now assume either char K = p > 2 and ht(X) < ∞, or char K = 0. We can apply Lemma 5.1 to ψ l : Aut(X) → GL(gr W n ).Assume that X 0 is of Type II.Let n = 3.By Corollary 3.5, ψ l factors through (End(C) Q ) * , where C is the elliptic curve appearing as the intersection of two components of X 0 .For any g ∈ Aut(X), ψ l (g) belongs to a (commutative) Q-subalgebra of End(C) Q generated by a single element (ψ l (g) itself), and such a subalgebra is either Q or an imaginary quadratic field.Hence m ′ ∈ {1, 2, 3, 4, 6} and so is m(g).
Corollary 5.2.Assume that X 0 is of Type III.Let g ∈ Aut(X), and suppose g extends to an automorphism of X , so that g acts on X 0 and on the set of the irreducible components of X 0 .Then the induced action of g on the (2element) set of orientations of S 2 (which the dual graph of X 0 triangulates) coincides with the image ρ(g) ∈ {±1} of g by ρ : Aut(X) → GL(H 0 (X, Ω 2 )).
Proof.Indeed, we have and the two generators of H 2 (S 2 , Z) corresponds to the two orientations.
Remark 5.3.Actually it was this assertion (Corollary 5.2), proposed by Yuji Odaka, that led the author to the study of this paper.The author first looked for an example of a Type III degeneration with m(X) = |Im ρ| ≥ 3, which would be a counterexample to the assertion, but failed to find one.It turned out that such examples do not exist!Next we prove Theorem 1.3.First we define the Hodge endomorphism field.
Let X be a K3 surface over a field F of characteristic 0. Let X C be a K3 surface over C isomorphic to X over some field F ′ containing both F and C (such X C always exists).We call E = End HS (T (X C ) Q ) the Hodge endomorphism field of X, where End HS denotes the endomorphisms of a rational Hodge structure.It is known that E is either a totally real field or a CM field ([Zar83, Theorem 1.5.1]).
This definition of E depends a priori on the choice of X C .However, if the Hodge conjecture for the self-product of a K3 surface holds, then every element of E can be realized as the action of an algebraic cycle on X C × X C , hence of an algebraic cycle on (X × X) F , and therefore E (up to isomorphism) depends only on X.The conjecture for X C × X C is proved by Ramón Marí [RM08, Theorem 5.4] under the assumption that E (of X C ) is a CM field.Thus, for a CM field E, the statement "E is the Hodge endomorphism field of X" is well-defined, and the statement of Theorem 1.3 should be understood accordingly.
Proof of Theorem 1.3.Assume Type II or III, and let n = 3 or n = 4 respectively.We have two homomorphisms of Q-algebras Since we are assuming that the Hodge conjecture for X × X is true, the E-action is realized as algebraic correspondences, i.e. χ is surjective.Since T (X C ) ⊗ Q l → gr W n is surjective, we have a surjection Im χ ։ Im ψ l .Since Im χ ∼ = E is a field and Im ψ l is nonzero, this surjection is an isomorphism.
Assume that X 0 is of Type II.Let n = 3.By Corollary 3.5, Im ψ l is contained in End(C) Q for some elliptic curve C. A (commutative) subfield of End(C) Q is either Q or an imaginary quadratic field.
Assume that X 0 is of Type III.Let n = 4. Similarly by Theorem 3.3,

Application: moduli spaces of K3 surfaces with non-symplectic automorphisms of prime order
We apply the main theorem to obtain a compactification of the moduli spaces of K3 surfaces with non-symplectic automorphisms of fixed prime order ≥ 5.
For a moment we work over C. In this section a lattice is a free Z-module of finite rank equipped with a Z-valued symmetric bilinear form.Let U be the hyperbolic plane (i.e. the rank 2 lattice with Gram matrix 0 1 1 0 ) and E 8 the (negative definite) root lattice of type E 8 .Then is isometric to H 2 (X, Z) for any K3 surface X over C, and is called the K3 lattice.
We recall the notation of [AST11].Fix a prime p ≤ 19 and a primitive p-th root ζ p of 1. Fix an isometry σ ∈ O(L K3 ) of order p, and denote by [σ] its conjugacy class.We write S(σ) = (L K3 ) σ=1 .Let M σ be the moduli space of a pair (X, g) consisting of a complex K3 surface X and an automorphism g of X of order p with ρ(g) = ζ p and acting on H 2 (X, Z) by σ via some marking (i.e.isometry) L K3 ∼ → H 2 (X, Z).We call such (X, g) to be a [σ]-polarized K3 surface.Such X is automatically algebraic and has an ample class in S(g) = H 2 (X, Z) g=1 .The marking induces an isometry S(σ) ∼ = S(g).
In fact, the moduli space M σ can be defined algebraically over Q(ζ p ). Indeed, by [AST11, Proposition 9.3], if p ≥ 3, then [σ] is uniquely determined by the topology of Fix(g) ⊂ X, and if p = 2, then [σ] is uniquely determined by the parameters r, a, δ of the lattice S(σ) ∼ = S(g) which can be read off from the action of g on the 2-adic étale cohomology group H 2 ét (X, Z 2 ).Now we consider the compactification of M σ .Theorem 1.1 does not imply that M σ is compact, since the action on the K3 lattice may change by specialization.So we need to attach some M τ 's on the boundary.
Consider the space Γ σ \D σ .The points on the boundary correspond to pairs (X, g) with non-symplectic automorphism of order p but acting on H 2 by some τ = σ.We can translate this into an algebraic construction of M σ by attaching M τ 's to M σ , and we have M σ (C) = Γ σ \D σ .Then we have the following.Proposition 6.1.Let σ be an isometry of L K3 of order p ≥ 5. Then M σ is proper.
Proof.It suffices to show that any K-rational point, after replacing K by a finite extension C((t 1/n )).So let (X, g) ∈ M(K) be a pair defined over K. Then by Theorem 1.1 there exists, after extending K, a smooth proper algebraic space X over ] with K3 fibers, and g extends to a birational map g : X X .The period map gives an extension of the morphism Spec C((t X is defined over the complement of a closed subspace of X of codimension ≥ 2, and the birational map g 0 := g| X 0 extends to an automorphism g0 of X 0 .This g0 also satisfy ρ(g 0 ) = ζ p but the action on H 2 is possibly different from σ. Example 6.2.Let p = 11.By [AST11, Theorem 7.3] there are three non-conjugate isometries σ 1 , σ 2 , σ 3 ∈ O(L K3 ) of order 11, respectively with S(σ 1 ) ∼ = U , S(σ 2 ) ∼ = U (11), S(σ 3 ) ∼ = U ⊕ A 10 , and the geometric points of M σ 1 consists of elliptic K3 surfaces y 2 = x 3 + ax + t 11 − b with an automorphism g(x, y, t) = (x, y, ζ 11 t), parametrized by {(a 3 : b 2 ) ∈ P 1 : 4a 3 + 27b 2 = 0}.This elliptic surface has 1 singular fiber of type II at t = ∞ and 22 of type I 1 at 4a 3 + 27(t 11 − b) 2 = 0, unless a = 0, in which case it has 12 singular fiber of type II.(This is Kodaira's notation of singular fibers of elliptic surfaces, and is not related to the classification of Kulikov models.)The boundary of the compactification M σ 1 consists of one point (4a 3 + 27b 2 = 0), and this is M σ 3 .At this point the elliptic surface has 1 singular fiber of type II, 11 of type I 1 , and one of type I 11 .
The following is also a direct consequence of Theorem 1.1.Proposition 6.3.Assume the existence of a Kulikov model (after field extension) for any K3 surface defined over a number field.Take an isometry σ with order p ≥ 5. Then any K3 surface corresponding to a point of M σ (Q) has everywhere potential good reduction.
Previously the author proved (conditionally) everywhere potential good reduction of K3 surfaces with complex multiplications [Mat15, Theorem 6.3], which however gives only isolated examples.To the contrary, Proposition 6.3 can be applied to positive dimensional family: for p = 5, 7, 11 there exists σ ∈ O(L K3 ) of order p with dim M σ = 4, 2, 1 respectively.Example 6.4.Again consider a K3 surface of the form y 2 = x 3 +ax+t 11 −b, this time over defined a number field K. Then by the previous proposition it has potential good reduction at any prime of K. We can also show potential good reduction directly.If the residue characteristic is equal to 11 then this is done in [Mat21,Example 6.8].In other characteristics we proceed as follows.After extending K, we can find a i ∈ O K such that is a minimal Weierstrass model over O K of the elliptic curve (y 2 = x 3 + ax − b), and that the special fiber E 0 = E ⊗ O K k of E is either an elliptic curve or a nodal curve.Then the special fiber of is smooth or has one A 10 singularity, according to E 0 being smooth or nodal respectively.Applying Artin's simultaneous resolution (after extending K) we achieve good reduction.Example 7.1 (Type III).Assume char k = 2. Consider the family X ′ over O K of quartic surfaces given by t(w 4 + x 4 + y 4 + z 4 ) + wxyz = 0, where t is a uniformizer of O K .This is not a Kulikov model, since it has non-regular points, but we can perform small blow-ups to obtain a Kulikov model X .
(For example, we can resolve the singularity at t = w = x = y 4 + z 4 by blowing-up either the ideal (t, w) or (t, x) at a neighborhood.)Then the special fiber is of Type III (whose dual graph is a tetrahedron).The symmetric group S 4 acts on X (over K) naturally and its action on the 2-forms is given by sgn : S 4 ։ {±1}.Hence m(X) is divisible by 2 (and by Theorem 1.1 we have m(X) = 2).
Example 7.2 (Type II).Again assume char k = 2. Let E be an imaginary quadratic field.Let C 1 be an elliptic curve with complex multiplication by an order of E, and C 2 an elliptic curve with multiplicative reduction.Let C i be the minimal regular models of C i over O K .By extending K, we may assume that (C 1 ) 0 is smooth and that (C 2 ) 0 has an even number of components.Let ι be the multiplication-by-(−1) map on In particular, if we take C 1 to be the elliptic curve with an automorphism of order 4 (resp.6), then we have an example of Type II degeneration with m(X) divisible by 4 (resp.6).We give examples for p e = 2, 2 2 , 3, 5, 7, 11.All examples are automorphisms of finite order.For the other cases we do not know whether examples exist.
Let p e be one of 2, 2 2 , 3, 5, 7, 11.Define an integer n as in the table below 2, 2 2 , 3, 7, 11, let X (over O K ) be the elliptic K3 surface defined by the equation below (with the E 8 singularity at t = ∞ resolved in the standard way if p e = 2, 2 2 , 7).If p e = 5, let X (over O K ) be the blow-up of the double sextic surface defined by the equation below at the non-smooth locus (w = x = G 5 (y) = 0).Define g 1 , g 2 ∈ Aut(X ) as below.
Here G p and H are defined as follows: ) Then X is smooth proper over O K with K3 fibers, g 1 , g 2 ∈ Aut(X ) commute and are of orders p e and n respectively, and ρ(g 1 | X K ) = ζ p e and ρ(g 2 | X K ) = ζ n .Hence g := g 1 g 2 is of order p e n, and m(g| X K ) = p e n, m(g| X 0 ) = n.Therefore g acts on T l (X K ) by a power of Φ p e n , hence also on T l (X 0 ).
We can moreover determine the characteristic polynomial of g on H 2 ét completely (although we do not need this).For p e = 2, 4, 3, 7, one observes that the sublattice of NS(X K ) generated by the zero section and the components of the singular fibers already has rank 22 − φ(p e n), and hence H 2 ét of X K is generated up to torsion by these curves and T l (X K ).Since the action of g on the classes of those curves are trivial, we obtain the characteristic polynomial Φ 22−φ(p e n) 1 Φ p e n .For p e = 5, similarly H 2 ét is generated by the pullback of O P 2 (1) and the five exceptional curves and T l (X K ).Since g 1 and g 2 respectively act on the classes of the exceptional curves transitively and trivially, we obtain the characteristic polynomial Φ 2 1 Φ 5 Φ 40 .Finally, for p e = 11, it is proved by by Dolgachev-Keum [DK09, Lemma 2.3(i)] that the characteristic polynomial of an order 11 automorphism of a K3 surface in characteristic 11 is always Φ 2 1 Φ 2 11 .It remains to show that X 0 is not supersingular.For the case p e = 11 this is checked in [Sch13, Section 3.2].Assume p e = 11.By [Nyg80, Theorem 2.1] (see Remark 2.2), if g is an automorphism of a supersingular K3 surface Y then m(g) should divide p σ 0 + 1, where σ 0 is the Artin invariant of Y (which is a positive integer ≤ 10).For the cases p e = 11 we observe that no such integer σ 0 exist, and hence X 0 are not supersingular.
Case p e = 2 gives order 42 in characteristic 2, and by replacing 7 with 11 we obtain order 66.However these two examples are almost written in [Keu16,Example 3.6].
Case p e = 3 gives an automorphism of order 66 of a K3 surface in characteristic 3.By replacing 11 with 8 and 10 (and then resolving the singularity at t = ∞ in the standard way) we also obtain automorphisms of order 42, 48 and 60.
(•) 0 .The algebraic correspondences Γ (p) induce an endomorphism of this complex of Z-modules, independent of l.Hence the coefficients of the characteristic polynomial of Γ * | gr W 0 lie in (d!) −1 Z, and the same assertion holds for any power of Γ. Hence by Lemma 3.4 the coefficients actually lie in Z. Assume n = 1.Then gr W 1 7. Examples7.1.Theorems 1.1 and 1.3 are optimal.The following two examples show that we cannot weaken the assumptions on m and E in Theorems 1.1 and 1.3.