RATIONAL POINTS ON K3 SURFACES AND DERIVED EQUIVALENCE

BRENDAN HASSETT AND YURI TSCHINKEL

The geometry of vector bundles and derived categories on complex K3 surfaces has developed rapidly since Mukai’s seminal work [Muk87]. Many foundational questions have been answered:

• the existence of vector bundles and twisted sheaves with prescribed invariants;

• geometric interpretations of isogenies between K3 surfaces [Orl97, Căl00];

• the global Torelli theorem for holomorphic symplectic manifolds [Ver13, Huy12b];

• the analysis of stability conditions and its implications for birational geometry of moduli spaces of vector bundles and more general objects in the derived category [BMT14, BM14, Bri07].

Given the precision and power of these results, it is natural to seek arithmetic applications of this circle of ideas. Questions about zero cycles on K3 surfaces have attracted the attention of Beauville-Voisin [BV04], Huybrechts [Huy12a], and other authors.

Our focus in this note is on rational points over non-closed fields of arithmetic interest. We seek to relate the notion of derived equivalence to arithmetic problems over various fields. Our guiding questions are:

Question 1. Let X and Y be K3 surfaces, derived equivalent over a field F. Does the existence/density of rational points of X imply the same for Y?

Given α ∈ Br(X), let (X, α) denote the twisted K3 surface associated with α, i.e., if P → X is an étale projective bundle representing α, of relative dimension r − 1 then (X, α) = [P/ SL_{r}].

Question 2. Suppose that (X, α) and (Y, β) are derived equivalent over F. Does the existence of a rational point on the former imply the same for the latter?

Date: October 8, 2016.

Note that an F -rational point of (X, α) corresponds to an x ∈ X(F) such that α|x = 0 ∈ Br(F). After this paper was released, Ascher, Dasaratha, Perry, and Zhou [ADPZ17] found that Question 2 has a negative answer, even over local fields.

We shall consider these questions for F finite, p-adic, real, and local with algebraically closed residue field. These will serve as a foundation for studying how the geometry of K3 surfaces interacts with Diophantine questions over local and global fields. For instance, is the Hasse/Brauer-Manin formalism over global fields compatible with (twisted) derived equivalence? See [HVAV11, HVA13, MSTVA17] for concrete applications to rational points problems.

In this paper, we first review general properties of derived equivalence over arbitrary base fields. We then offer examples which illuminate some of the challenges in applying derived category techniques. The case of finite and real fields is presented first—here the picture is well developed. Local fields of equicharacteristic zero are also fairly well understood, at least for K3 surfaces with semistable or other mild reduction. The analogous questions in mixed characteristic remain largely open, but comparison with the geometric case suggests a number of avenues for future investigation.

Acknowledgments: We are grateful to Jean-Louis Colliot-Thélène, Daniel Huybrechts, Christian Liedtke, Johannes Nicaise, Sho Tanimoto, Anthony Várilly-Alvarado, Olivier Wittenberg, and Letao Zhang for helpful conversations. We thank Vivek Shende for pointing out the application of A’Campo’s Theorem. We have benefited enormously from the thoughtful feedback of the referees. The first author was supported by NSF grants 0901645, 0968349, and 1148609; the second author was supported by NSF grants 0968318 and 1160859. We are grateful to the American Institute of Mathematics for sponsoring workshops where these ideas were explored.

1. Generalities on derived equivalence for K3 surfaces

1.1. Definitions. Let X and Y denote K3 surfaces over a field F. Let p and q be the projections from X × Y to X and Y respectively.

Let E ∈ D^{b}(X × Y) be an object of the bounded derived category of coherent sheaves, which may be represented by a perfect complex of locally free sheaves. The Fourier-Mukai transform is defined

\begin{matrix}Φ_{E} : D^{b}(X) & → & D^{b}(Y) \\ F & ↦ & q_{∗}(E ⊗ p^{∗}F), \end{matrix}

where push forward and tensor product are the derived operations.

We say X and Y are derived equivalent if there exists an equivalence of triangulated categories over F

Φ : D^{b}(X) →^{∼} D^{b}(Y).

A fundamental theorem of Orlov [Orl97] implies that Φ arises as the Fourier-Mukai transform Φ_{E} for some perfect complex E ∈ D^{b}(X × Y). When we refer to Fourier-Mukai transforms below they will induce derived equivalences.

1.2. Mukai lattices. Suppose X is a K3 surface defined over ℂ and consider its Mukai lattice

H̃(X, ℤ) = H̃(X, ℤ) := H^{0}(X, ℤ)(−1) ⊕ H^{2}(X, ℤ) ⊕ H^{4}(X, ℤ)(1),

where we apply Tate twists to get a Hodge structure of weight two. Mukai vectors refer to type (1, 1) vectors in H̃(X, ℤ). Let (,) denote the natural nondegenerate pairing on H̃(X, ℤ); this coincides with the intersection pairing on H^{2}(X, ℤ) and the negative of the intersection pairing on the other summands. There is an induced homomorphism on the level of cohomology:

\begin{matrix}ϕ_{E} : H̃(X, ℤ) & → & H̃(Y, ℤ) \\ η & ↦ & q_{∗}(\sqrt{Td_{X×Y}} · ch(E) ∪ p^{∗}η). \end{matrix}

Note that

ϕ_{E} ch(F) = ch(Φ_{E}(F)).

For X defined over a non-closed field F, there are analogous constructions in ℓ-adic and other flavors of cohomology [LO15, § 2]. When working ℓ-adically, we interpret H̃(X, ℤ_{ℓ}) as a Galois representation rather than a Hodge structure. Observe that ϕ_{E} is defined on Hodge structures, de Rham cohomologies, and ℓ-adic cohomologies—and these are all compatible.

1.3. Characterizations over the complex numbers.

Theorem 3. [Orl97, §3] Let X and Y be K3 surfaces over ℂ, with transcendental cohomology groups

T (X) := Pic(X)^{⊥} ⊂ H^{2}(X, ℤ), T (Y) := Pic(Y)^{⊥} ⊂ H^{2}(Y, ℤ).

The following are equivalent • there exists an isometry of Hodge structures H̃(X, ℤ) ≃ H̃(Y, ℤ);

• there exists an isometry of Hodge structures T (X) ≃ T (Y);

• X and Y are derived equivalent;

• Y is isomorphic to a moduli space of stable vector bundles over X, admitting a universal family E → X × Y, i.e., Y = M_{v}(X) for a Mukai vector v such that there exists a Mukai vector w with (v, w) = 1.

See [Orl97, §3.8] and [Huy08, §3] for discussion of the fourth condition; for our purposes, we need not distinguish among various notions of stability. This has been extended to arbitrary fields as follows: Theorem 4. [LO15, Th. 1.1] Let X and Y be K3 surfaces over an algebraically closed field F of characteristic ≠ 2. Then the third and the fourth statements are equivalent.

Morover, a derived equivalence between X and Y induces Galoiscompatible isomorphisms between their ℓ-adic Mukai lattices. See [Huy16, §16.4] for more discussion.

1.4. Descending derived equivalence. Let F be a field of characteristic zero with algebraic closure F̄. Given Y smooth and projective over F, let Ȳ denote the corresponding variety over F̄. We say Y is of K3 type if Ȳ is deformation equivalent to the Hilbert scheme of a K3 surface. Lemma 5. Suppose Y_{1} and Y_{2} are of K3 type over F and there exists an isomorphism ι : Ȳ_{1} →^{∼} Ȳ_{2} defined over F̄ inducing

ι^{∗} : H^{2}(Ȳ_{2}, ℤ_{ℓ}) →^{∼} H^{2}(Ȳ_{1}, ℤ_{ℓ})

compatible with the action of Gal(F̄/F). Then Y_{1} ≃ Y_{2} over F. Proof. In characteristic zero, Galois fixed points of Isom(Ȳ_{1}, Ȳ_{2}) correspond to isomorphisms between the varieties defined over F. Automorphisms of Ȳ_{1} (resp. Ȳ_{2}) act transitively and faithfully on this set by pre-composition (resp. post-composition).

The Torelli theorem implies that the automorphism group of a K3 surface has a faithful representation in its second cohomology. This holds true for manifolds of K3 type as well—see [Mar10, Prop. 1.9] as well as previous work of Beauville and Kaledin-Verbitsky. The Galois invariance of ι^{∗} implies that ι is Galois-fixed, hence defined over F. □

Suppose that X and Y are K3 surfaces over F. Given a derived equivalence

Φ : D^{b}(X) →^{∼} D^{b}(Y)

over F the induced

ϕ : H̃(X̄, ℤ_{ℓ}) →^{∼} H̃(Ȳ , ℤ_{ℓ})

is compatible Gal(F̄/F) actions. We consider whether the converse holds.

We use the notation M_{v}(X) for the moduli stack of vector bundles/complexes over X; this is typically a 𝔾_{m}-gerbe over M_{v}(X) due to homotheties.

Proposition 6. Suppose that X and Y are K3 surfaces over F. Let Φ : D(X̄) →^{∼} D(Ȳ) be a derived equivalence over F̄ such that the induced

ϕ : H̃(X̄, ℤ_{ℓ}) →^{∼} H̃(Ȳ , ℤ_{ℓ})

is compatible with Galois actions. Write v = ϕ^{−1}(1, 0, 0) and w = ϕ(1, 0, 0).

Then we have isomorphisms Y ≃ M_{v}(X) and X ≃ M_{w}(Y) inducing the derived equivlance over F̄. Moreover, there exist 𝔾_{m}-gerbes X → X and Y → Y and isomorphisms Y ≃ M_{v}(X) and X ≃ M_{w}(Y) such that X → X and Y → Y admit trivializations over F̄.

Proof. Choosing an appropriate polarization on X, the moduli space M_{v}(X̄) is a K3 surface. This follows from [Muk87, Prop. 4.2]; Mukai’s argument shows that the requisite polarization can be found over a non-closed field, as we just have to avoid walls orthogonal to certain Mukai vectors.

There exists a Mukai vector v′ ∈ H̃(X̄, ℤ_{ℓ}) that is Galois invariant and satisfies (v, v′) = 1, e.g., v′ = ϕ^{−1}(0, 0, −1). Thus the universal sheaf

E → X × M_{v}(X)

has the following property: on basechange to F̄ it may be obtained as the pull-back of a universal sheaf on the coarse space

Ē → X̄ × M_{v}(X̄).

Indeed, the universal sheaf exists wherever M_{v′}(X) admits a rational point (see [LO15, Th. 3.16]).

We apply Lemma 5 to Y and M_{v}(X). The Torelli Theorem implies Ȳ ≃ M_{v}(X̄) with the induced isomorphism compatible with the Galois actions on ℓ-adic cohomology. Thus it descends to give Y ≃ M_{v}(X) defined over F. □

Remark 7. Can X and Y fail to be derived equivalent over F? How do the pull-back homomorphisms

Br(F) → Br(X), Br(F) → Br(Y)

compare? They must have the same kernel if X and Y are derived equivalent over F.

Remark 8. Sosna [Sos10] has given examples of complex K3 surfaces that are derived equivalent but not isomorphic to their complex conjugates.

1.5. Cycle-theoretic invariants of derived equivalence.

Proposition 9. Let X and Y be derived equivalent K3 surfaces over a field F of characteristic ≠ 2. Then Pic(X) and Pic(Y) are stably isomorphic as Gal(F̄/F)-modules, and Br(X)[n] ≃ Br(Y)[n] provided n is not divisible by the characteristic.

Even over ℂ, this result does not extend to higher dimensional varieties [Add17].

Proof. The statement on the Picard groups follows from the Chow realization of the Fourier-Mukai transform—see [LO15, §2.7] for discussion. We write

H̃_{ét}(X, μ_{n}) = H_{ét}^{0}(X, ℤ/nℤ) ⊕ H_{ét}^{2}(X, μ_{n}) ⊕ H_{ét}^{4}(X, μ_{n}^{⊗2}).

The Chow-theoretic interpretation of the Fourier-Mukai kernel gives the realization

ϕ : H̃_{ét}(X, μ_{n}) → H̃_{ét}(Y, μ_{n}),

compatible with cycle class maps [LO15, Prop. 2.10]. Modding out by the images of the cycle class maps we get the desired equality of Brauer groups. □

Recall that the index ind(X) of a smooth projective variety X over a field F is the greatest common divisor of the degrees of field extensions F′/F over which X(F′) ≠ ∅, or equivalently, the lengths of zero-dimensional subschemes Z ⊂ X.

Given a bounded complex of locally free sheaves on X

E = {E_{−M} → E_{−M+1} → ⋯ → E_{N}}

we may define the Chern character

ch(E) = ∑_{j}(−1)^{j} ch(E_{j})

in Chow groups with ℚ coefficients. The degree zero and one pieces yield the rank and first Chern class of E, expressed as alternating sums of the ranks and determinants of the terms, respectively. Similarly, we may define

c_{2}(E) = − ch_{2}(E) + ch_{1}(E)^{2}/2,

a quadratic expression in the Chern classes of the E_{j} with integer coefficients. Modulo the ℤ algebra generated by the first Chern classes of the E_{j}, we may write

c_{2}(E) ≡ ∑_{j}(−1)^{j}c_{2}(E_{j}).

Lemma 10. If (S, h) is a smooth projective surface over F then

ind(S) = gcd{c_{2}(E) : E vector bundle on S}
= gcd{c_{2}(E) : E ∈ D^{b}(S)}.

Proof. Consider the ‘decomposible index’

inddec(S) := gcd{D_{1} · D_{2} : D_{1}, D_{2} very ample divisors on S}

which is equal to

gcd{D_{1} · D_{2} : D_{1}, D_{2} divisors on S},

because for any divisor D the divisor D + Nh is very ample for N ≫ 0. All three quantities in the assertion divide inddec(S), so we work modulo this quantity.

By the analysis of Chern classes above, the second and third quantities agree. Given a reduced zero-dimensional subscheme Z ⊂ S we have a resolution

0 → E_{−2} → E_{−1} → O_{S} → O_{Z} → 0

with E_{−2} and E_{−1} vector bundles. This implies that

gcd{c_{2}(E) : E vector bundle on S}| ind(S).

Conversely, given a vector bundle E there exists a twist E ⊗ O_{S}(Nh) that is globally generated and

c_{2}(E ⊗ O_{S}(Nh)) ≡ c_{2}(E) (mod inddec(S)).

Thus there exists a zero-cycle Z with degree c_{2}(E ⊗ O_{S}(Nh)) and

ind(S)| gcd{c_{2}(E) : E vector bundle on S}.

□ Proposition 11. If X is a K3 surface over a field F then

ind(X)| gcd{24, D_{1} · D_{2} where D_{1}, D_{2} are divisors on X}.

This follows from Lemma 10 and the fact that c_{2}(T_{X}) = 24. Beauville-Voisin [BV04] and Huybrechts [Huy10] have studied the corresponding subgroup of CH_{0}(XF̄).

Proposition 12. Let X and Y be derived equivalent K3 surfaces over a field F. Then ind(X) = ind(Y).

The proof will require the notion of a spherical object on a K3 surface. This is an object S ∈ D^{b}(X) with

Ext^{0}(S, S) = Ext^{2}(S, S) = F, Ext^{i}(S, S) = 0, i ≠ 0, 2.

These satisfy the following • (v(S), v(S)) = −2;

• rigid simple vector bundles are spherical;

• each spherical object S has the associated spherical twist [Huy16,

16.2.4]:

T_{S} : D^{b}(X) → D^{b}(X),

an autoequivalence such that the induced homomorphism on the Mukai lattice is the reflection associated with v(S);

• each spherical object S̄ on XF̄ is defined over a finite extension F′/F [Huy10, 5.4];

• over ℂ, each v = (r, D, s) ∈ H̃(X, ℤ) ∩ H^{1,1} with (v, v) = −2 arises from a spherical object, which may be taken to be a rigid vector bundle E if r > 0 [Kul89];

• under the same assumptions, for each polarization h on X there is a unique h-slope stable vector bundle E with v(E) = v [Huy12a, 5.1.iii].

Proof. Proposition 6 allows us to express Y = M_{v}(X) for v = (r, ah, s) where h is a polarization on X and a^{2}h^{2} = 2rs. It also yields a Mukai vector w = (r′, bg, s′) ∈ H̃(X, ℤ_{ℓ}) with

(v, w) = abg · h − rs′ − sr′ = 1.

Thus we have

(1.1)  ⟨r, s⟩ = ⟨1⟩ (mod g · h).

Consider a Fourier-Mukai transform realizing the equivalence

Φ : D^{b}(X) → D^{b}(Y)

and the induced homomorphism ϕ on the Mukai lattice. Note that

ϕ(v) = (0, 0, 1)

reflecting the fact that a point on Y corresponds to a sheaf on X with Mukai vector v.

Suppose that Y has a rational point over a field of degree n over F; let Z ⊂ Y denote the corresponding subscheme of length n. Applying Φ^{−1} to O_{Z} gives an element of the derived category with Mukai vector (nr, nah, ns) and

c_{2}(Φ^{−1}(O_{Z})) = \frac{c_{1}(Φ^{−1}(O_{Z}))^{2}}{2} − χ(Φ^{−1}(O_{Z})) + 2 rank(Φ^{−1}(O_{Z}))

which equals n(nrs + r − s). Following the proof of Lemma 10, we compute

c_{2}(Φ^{−1}(O_{Z})) (mod inddec(X)).

First suppose that r and s have different parity, so that

gcd(nrs + r − s, 2rs) = gcd(nrs + r − s, rs).

Then using (1.1) we obtain

⟨nrs + r − s, rs⟩ = ⟨r − s, rs⟩ = ⟨r − s, r⟩ ⟨r − s, s⟩
= ⟨−s, r⟩ ⟨r, s⟩ = ⟨r, s⟩^{2} = ⟨1⟩ (mod g · h).

If r and s are both even then g · h must be odd and

⟨nrs + r − s, 2rs⟩ = ⟨nrs + r − s, rs⟩ (mod g · h)

and repeating the argument above gives the desired conclusion.

Now suppose that r and s are both odd. It follows that h^{2} ≡ 2 (mod 4) and we write h^{2} = 2γ − 2 for some even integer γ. Let S denote the spherical object associated with h so that v(S) = (1, h, γ). Applying T_{S} to the Mukai vector

(r, ah, s) ↦ (r, ah, s) + ((r, ah, s), (1, h, γ)) (1, h, γ),

we obtain a new vector with rank r + (ah^{2} − s − rγ), which is even. This reduces us to the previous situation.

In each case, we find

c_{2}(Φ^{−1}(O_{Z})) ≡ n (mod inddec(X)),

whence ind(X)|n. Varying over all degrees n, we find

ind(X)| ind(Y)

and the Proposition follows. □

The last result raises the question of whether spherical objects are defined over the ground field:

Question 13. Let X be a K3 surface over a field F. Suppose that S̄ is a spherical object on XF̄ such that c_{1}(S̄) ∈ Pic(XF̄) is a divisor defined over X. When does S̄ come from an object S on X?

Kuleshov [Kul89, Kul90] gives a partial description of how to generate all exceptional bundles on K3 surfaces of Picard rank one through ‘restructuring’ operations and ‘dragons’. It would be worthwhile to analyze which of these operations could be defined over the ground field.

Example 14. We give an example of a K3 surface X over a field F with

Pic(X) = Pic(XF̄) = ℤh

and a rigid sheaf E over XF̄ that fails to descend to F.

Choose (X, h) to be a degree fourteen K3 surface defined over ℝ with X(ℝ) = ∅. This may be constructed as follows: Fix a smooth conic C and quadric threefold Q with

C ⊂ Q ⊂ ℙ^{4}, Q(ℝ) = ∅.

Let X′ denote a complete intersection of Q with a cubic containing C; we have X′(ℝ) = ∅ and X′ admits a lattice polarization

\begin{aligned}& g & C \\ g & 6 & 2 \\ C & 2 & −2 \end{aligned} .

Write h = 2g − C so that (X′, h) is a degree 14 K3 surface containing a conic. Let X be a small deformation of X′ with Pic(X_{ℂ}) = ℤh.

The K3 surface X is Pfaffian if and only if it admits a vector bundle E with v(E) = (2, h, 4) corresponding to the classifying morphism X → Gr(2, 6). However, note that

c_{2}(E) = 5

which would mean that ind(X) = 1. On the other hand, if X(ℝ) = ∅ then ind(X) = 2.

2. Examples of derived equivalence

2.1. Elliptic fibrations. The paper [AKW14] has a detailed discussion of derived equivalences among genus one curves over function fields.

In this section we work over a field F of characteristic zero.

A K3 surface X is elliptic if it admits a morphism X → C to a curve of genus zero with fibers of genus one. We allow C = ℙ^{1} or a non-split conic over F.

Lemma 15. A K3 surface X is elliptic if and only if it admits a nontrivial divisor D with D^{2} = 0.

Proof. If X is elliptic then the pull back of a non-trivial divisor from C gives a square-zero class; we focus on the converse.

This is well-known if F is algebraically closed [PŠŠ71, §6, Th. 1]. Indeed, let C_{+} ⊂ H^{2}(X, ℝ) denote the component of the positive cone {η : (η, η) > 0} containing an ample divisor and Γ ⊂ Aut(H^{2}(X, ℤ)) the group generated by Picard-Lefschetz reflections ρ_{R} associated with (−2)-curves R ∈ Pic(X). Then the Kähler cone of X is a fundamental domain for the action of Γ on C_{+}, in the sense that no two elements of the cone are in the same orbit and each orbit in C_{+} meets the closure of the Kähler cone. Cohomology classes in the interior of the cone have trivial Γ-stabilizer; classes on walls of the boundary associated with (−2)-curves are stabilized by the associated reflections.

Thus if Pic(X) represents zero then there exists a non-zero divisor D in the closure of the Kähler cone with D^{2} = 0, which induces an elliptic fibration X → ℙ^{1}. Given a divisor D ∈ C_{+}, we can be a bit more precise about the γ ∈ Γ required to take D to a nef divisor. We can write

γ = ρ_{R_{1}} ⋯ ρ_{R_{m}}

where each R_{j} is the class of an irreducible rational curve contained in the fixed part of the linear series |D| (cf. proof in [PŠŠ71, §6], [SD74, §2]).

Now suppose F is not algebraically closed and D is defined over F. The group Gal(F̄/F) acts on the (−2)-curves on X̄ and thus on Γ and the orbit Γ · D. The Galois action on R_{1}, …, R_{m} may be nontrivial. However, the fundamental domain description guarantees a unique f ∈ Γ · D in the nef cone of X, which is necessarily Galois invariant; some multiple of this divisor is defined over F. This semiample divisor induces our elliptic fibration. □

Proposition 16. Let X be an elliptic K3 surface over F and Y another K3 surface derived equivalent to X over F. Then Y is elliptic over F.

Proof. By Proposition 9, the Picard groups of X and Y are stably isomorphic as lattices

Pic(X) ⊕ U ≃ Pic(Y) ⊕ U, U = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) .

In particular, Pic(X) and Pic(Y) share the same discriminant and padic invariants. A rank-two indefinite lattice represents zero if and only if its discriminant is a square. In higher ranks, an indefinite lattice represents zero if and only if it represents zero p-adically for each p.

Thus Pic(Y) admits a square-zero class and is elliptic by Lemma 15. □

An elliptic K3 surface J → C is Jacobian if it admits a section C → J. It has geometric Picard group containing

(2.1)  \begin{aligned}& f & Σ \\ f & 0 & 1 \\ Σ & 1 & −2 \end{aligned}

where f is a fiber and Σ is the section. Jacobian elliptic surfaces admit numerous autoequivalences [Bri98, §5]. Let a, b ∈ ℤ with a > 0 and (a, b) = 1. The moduli space of rank a degree b indecomposable vector bundles on fibers of J → C is an elliptic fibration with section, isomorphic to J over C. This reflects Atiyah’s classification of vector bundles over elliptic curves [Ati57, Th. 7]. The associated Fourier-Mukai transform induces an autoequivalence of J acting on H̃(J, ℤ) via an element of

(\begin{matrix}c & a \\ d & b \end{matrix}) ∈ SL_{2}(ℤ)

obtained as in [Bri98, Th. 3.2,5.3]. This acts on the Mukai vectors (or their ℓ-adic analogues)

(1, 0, 0), (0, 0, 1) ∈ H^{0}(J, ℤ)(−1) ⊕ H^{2}(J, ℤ) ⊕ H^{4}(J, ℤ)(1)

by the formula

(1, 0, 0) ↦ (0, af, c), (0, 0, 1) ↦ (b, d(f + Σ), 0).

The action is transitive on the (−2)-vectors in the lattice of algebraic classes

(2.2) H^{0}(J, ℤ)(−1) ⊕ H^{4}(J, ℤ)(1) ⊕ ⟨f, Σ⟩ ≃ (\begin{matrix}0 & −1 \\ −1 & 0 \end{matrix}) ⊕ (\begin{matrix}0 & 1 \\ 1 & −2 \end{matrix}) .

Thus we have established the following:

Proposition 17. Let J → C be a Jacobian elliptic K3 surface over F. Then autoequivalences of J defined over F induce a representation of SL_{2}(ℤ) on (2.2) as above.

The point is that here autoequivalences are defined over the ground field.

Over ℂ, the availability of autoequivalences has strong consequences for Jacobian elliptic K3 surfaces. For instance, orientation preserving automorphisms of the Mukai lattice of a complex K3 surface arise from autoequivalences [HLOY04a, Th. 1.6]. This implies that derived equivalent Jacobian elliptic K3 surfaces are isomorphic [HLOY04b, Cor. 2.7.3]. The autoequivalences are generated via spherical twists associated with rigid objects.

As we have seen (e.g., in Example 14), rigid objects over non-closed fields need not descend. It is natural to ask the following question:

Question 18. Let J_{1} and J_{2} denote Jacobian elliptic K3 surfaces over a field F of characteristic zero. If J_{1} and J_{2} are derived equivalent does it follow that J_{1} ≃ J_{2}?

If the geometric Picard group has rank two then the isomorphism follows from Proposition 17.

We recall the classical Ogg-Shafarevich theory for elliptic fibrations, following [Căl00, 4.4.1,5.4.5]: Let F be algebraically closed of characteristic zero and J → ℙ^{1} a Jacobian elliptic K3 surface. We may interpret

�(J/ℙ^{1}) ≃ Br(J)

and each α in this group may be realized by an elliptically fibered K3 surface X → ℙ^{1} with Jacobian fibration J → ℙ^{1}. If α has order n then we have natural exact sequences

0 → ℤ/nℤ → Br(J) → Br(X) → 0

and

0 → T (X) → T (J) → ℤ/nℤ → 0.

Note that if Y = Pic^{c}(X/ℙ^{1}) then [Y] = c[X] ∈ �(J/ℙ^{1}). If c is relatively prime to n, so there exists an integer b with bc ≡ 1 (mod n), then we have Pic^{b}(Y) = X as well. It follows that X and Y are derived equivalent with the universal sheaves inducing the Fourier-Mukai transform [Bri98, Th. 1.2].

Proposition 19. Let F be algebraically closed of characteristic zero. Let ϕ : X → ℙ^{1} be an elliptic K3 surface with Jacobian fibration J(X) → ℙ^{1}. Let α ∈ Br(J(X)) denote the Brauer class associated with [X] in the Tate-Shafarevich group of J(X) → ℙ^{1}. Then X is derived equivalent to the pair (J(X), α).

This follows from the proof of Căldăraru’s conjecture; see [HS06, 1.vi] as well as [Căl00, 4.4.1] for the fundamental identification between the twisting data and the Tate-Shafarevich group.

Question 20. Let X and Y be K3 surfaces derived equivalent over a field F. Suppose we have an elliptic fibration X → C over F. Does Y admit a fibration Y → C such that

X →^{∼} Pic^{b}(Y/C)
↘  ↙
C

for some integer b?

By symmetry we have X ≃ Pic^{b}(Y/C) and Y ≃ Pic^{c}(X/C) for b, c ∈ ℤ. As before, if n = ord([X]) = ord([Y]) in the Tate-Shafarevich group then bc ≡ 1 (mod n).

A positive answer to Question 20 would imply:

• X dominates Y over F and vice versa.

• If X and Y are elliptic K3 surfaces derived equivalent over a field F of characteristic zero then X(F) ≠ ∅ if and only if Y (F) ≠ ∅.

2.2. Rank one K3 surfaces. We recall the general picture:

Proposition 21. [Ogu02, Prop. 1.10], [Ste04] Let X/ℂ be a K3 surface with Pic(X) = ℤh, where h^{2} = 2n. Then the number m of isomorphism classes of K3 surfaces Y derived equivalent to X is given by

m = 2^{τ(n)−1}, where τ(n) = number of prime factors of n.

Example 22. The first case where there are multiple isomorphism classes is degree twelve. Let (X, h) be such a K3 surface and Y = M_{(2,h,3)}(X) the moduli space of stable vector bundles E → X with

rk(E) = 2, c_{1}(E) = h, χ(E) = 2 + 3 = 5,

whence c_{2}(E) = 5. Note that if Y (F) ≠ ∅ then X admits an effective zero-cycle of degree five and therefore a zero-cycle of degree one. Indeed, if E → X is a vector bundle corresponding to [E] ∈ Y (F) then a generic σ ∈ Γ(X, E) vanishes at five points on X. As we vary σ, we get a four-parameter family of such cycles. Moreover, the cycle h^{2} has degree twelve, relatively prime to five.

Is X(F) ≠ ∅ when Y (F) ≠ ∅?

2.3. Rank two K3 surfaces. Exhibiting pairs of non-isomorphic derived equivalent complex K3 surfaces of rank two is a problem on quadratic forms [HLOY04b, §3]. Suppose that Pic(X_{ℂ}) = Π_{X} and Pic(Y_{ℂ}) = Π_{Y} and X and Y are derived equivalent. Orlov’s Theorem implies T (X) ≃ T (Y) which means that Π_{X} and Π_{Y} have isomorphic discriminant groups/p-adic invariants. Thus we have to exhibit padically equivalent rank-two even indefinite lattices that are not equivalent over ℤ. (Proposition 9 asserts this is a necessary condition over general fields of characteristic zero.)

Example 23. We are grateful to Sho Tanimoto and Letao Zhang for assistance with this example. Consider the lattices

Π_{X} = \begin{aligned}& C & f \\ C & 2 & 13 \\ f & 13 & 12 \end{aligned} Π_{Y} = \begin{aligned}& D & g \\ D & 8 & 15 \\ g & 15 & 10 \end{aligned}

which both have discriminant 145. Note that Π_{X} represents −2

(2f − C)^{2} = (25C − 2f)^{2} = −2

but that Π_{Y} fails to represent −2.

Let X be a K3 surface over F with split Picard group Π_{X} over a field F. We assume that C and f are ample. The moduli space

Y = M_{(2,C+f,10)}(X)

has Picard group

\begin{aligned}& 2C & (C + f)/2 \\ 2C & 8 & 15 \\ (C + f)/2 & 15 & 10 \end{aligned} ≃ Π_{Y}

while M_{(2,D,2)}(Y) has Picard group

\begin{aligned}& D/2 & 2g \\ D/2 & 2 & 15 \\ 2g & 15 & 40 \end{aligned} ≃ Π_{X}

and is isomorphic to X.

These surfaces have the following properties:

• X and Y admit decomposable zero cycles of degree one over F; • X(F) ≠ ∅: the rational points arise from the smooth rational curves with classes 2f − C and 25C − 2f, both of which admit zero-cycles of odd degree and thus are ≃ ℙ^{1} over F;

• Y (F′) is dense for some finite extension F′/F, due to the fact

that | Aut(Y_{ℂ})| = ∞. We do not know whether

• X(F′) is dense for any finite extension F′/F;

• Y (F) ≠ ∅.

3. Finite and real fields

The ℓ-adic interpretation of the Fourier-Mukai transform yields

Theorem 24. [LO15] [Huy16, 16.4.3] Let X and Y be K3 surfaces derived equivalent over a finite field F. Then for each finite extension F′/F we have

|X(F′)| = |(Y (F′)|.

For the case of general surfaces see [Hon15].

We have a similarly complete picture over the real numbers. We review results of Nikulin [Nik79, §3] [Nik08, §2] on real K3 surfaces.

Let X be a K3 surface over ℝ, X_{ℂ} the corresponding complex K3 surface, and φ the action of the anti-holomorphic involution (complex conjugation) of X_{ℂ} on H^{2}(X_{ℂ}, ℤ). Let Λ_{±} ⊂ H^{2}(X_{ℂ}, ℤ) denote the eigenlattices where φ acts via ±1. If D is a divisor on X defined over ℝ then

φ([D]) = −D;

the sign reflects the fact that complex conjugation reverses the sign of (1, 1) forms. In Galois-theoretic terms, the cycle class of a divisor lives naturally H^{2}(X_{ℂ}, ℤ(1)) and twisting by −1 accounts for the sign change. Let Λ̃_{±} denote the eigenlattices of the Mukai lattice; note that Λ̃_{−} contains the degree zero and four summands. Again, the sign change reflects the fact that these are twisted in the Mukai lattice.

We introduce the key invariants: Let r denote the rank of Λ_{−}. The discriminant groups of Λ_{±} are two-elementary groups of order 2^{a} where a is a non-negative integer. Note that Λ̃_{±} have discriminant groups of the same order. Finally, we set

δ_{φ} = \begin{cases}0 if (λ, φ(λ)) ≡ 0(mod 2) for each λ ∈ Λ \\ 1 otherwise. \end{cases}

Note that δ_{φ} can be computed via the Mukai lattice

δ_{φ} = 0 iff (λ, φ(λ)) ≡ 0(mod 2) for each λ ∈ Λ̃,

as the degree zero and four summands always give even intersections. We observe the following:

Proposition 25. Let X and Y be K3 surfaces over ℝ, derived equivalent over ℝ. Then

(r(X), a(X), δ_{φ,X}) = (r(Y), a(Y), δ_{φ,Y}).

Proof. The derived equivalence induces an isomorphism

H̃(X_{ℂ}, ℤ) ≃ H̃(Y_{ℂ}, ℤ)

compatible with the conjugation actions. Since (r, a, δ_{φ}) can be read off from the Mukai lattice, the equality follows. □

The topological type of a real K3 surface is governed by these invariants. Let T_{g} denote a compact orientable surface of genus g.

Proposition 26. [Nik79, Th. 3.10.6] [Nik08, 2.2] Let X be a real K3 surface with invariants (r, a, δ_{φ}). Then the manifold X(ℝ) is orientable and

X(ℝ) = \begin{cases}∅ & if (r, a, δ_{φ}) = (10, 10, 0) \\ T_{1} ⊔ T_{1} & if (r, a, δ_{φ}) = (10, 8, 0) \\ T_{g} ⊔ (T_{0})^{k} & otherwise, where \\ & g = (22 − r − a)/2, k = (r − a)/2. \end{cases}

Corollary 27. Let X and Y be K3 surfaces defined and derived equivalent over ℝ. Then X(ℝ) and Y (ℝ) are diffeomorphic. In particular, X(ℝ) ≠ ∅ if and only if Y (ℝ) ≠ ∅.

The last statement also follows from Proposition 12: A variety over ℝ has a real point if and only if its index is one. (This was pointed out to us by Colliot-Thélène.)

Example 28. Let X and Y be derived equivalent K3 surfaces, defined over ℝ; assume they have Picard rank one. Then Y = M_{v}(X) for some isotropic Mukai vector v = (r, f, s) ∈ H̃(X(ℂ), ℤ) with (r, s) = 1. For a vector bundle E of this type note that

c_{2}(E) = c_{1}(E)^{2}/2 + rχ(O_{X}) − χ(E) = rs + r − s,

which is odd as r and s are not both even. Then a global section of E gives an odd-degree cycle on X over ℝ, hence an ℝ-point.

4. Geometric case: local fields with complex residue field

We start with a general definition: Let R be a discrete valuation ring with quotient field F. Given a K3 surface X over F, a model is a flat proper algebraic space X → Δ = Spec(R) with generic fiber X. We say X has good reduction if there exists a smooth model and ADE reduction if there is a model with (at worst) rational double points in the central fiber X_{0}.

Let F = ℂ((t)) with algebraic closure F̄, obtained by adjoining all the roots of t. Let X be a projective K3 surface over F = ℂ((t)); write X̄ for its base-change to F̄. Consider the monodromy action

T : H^{2}(X̄, ℤ) → H^{2}(X̄, ℤ)

associated with a counterclockwise loop about t = 0. This is quasiunipotent, i.e., there exist e, f ∈ ℕ such that (T^{e} − I)^{f} = 0 [Lan73]; we choose e, f minimal with this property.

If X and Y are derived equivalent over F then Orlov’s Theorem and the discussion preceding Proposition 6 implies that their Mukai lattices admit a monodromy equivariant isomorphism.

Write Δ = Spec ℂ[[t]] and fix a projective model

X → Δ

and a resolution

ϖ : X′ → Δ

such that the central fiber ϖ^{−1}(0) is a normal crossings divisor, perhaps with multiplicities along some components. Let X′_{∘} ⊂ ϖ^{−1}(0) denote the smooth locus of the central fiber, i.e., the points of mulitiplicity one. A’Campo [A’C75, Th. 1] proved that

χ(X′_{∘}) = 2 + trace(T).

This may be interpreted as the alternating sum of the traces of the monodromy matrices on all the cohomology groups of X.

An application of Hensel’s Lemma yields

Proposition 29. If trace(T) ≠ −2 then X → Δ admits a section, i.e., X(F) ≠ ∅.

This result was previously obtained by Nicaise [Nic11, Cor. 6.6]; his techniques are also applicable in mixed characteristic under appropriate tameness assumptions.

Proposition 29 applies when T is unipotent (e = 1). This is the case when there exists a resolution X′ → Δ with central fiber reduced normal crossings. (See the Appendix and Theorem 38 for further analysis.)

If X and Y are derived equivalent over F then Orlov’s Theorem and the discussion preceding Proposition 6 implies that their Mukai lattices admit a monodromy equivariant isomorphism. Thus the characteristic polynomials of their monodromy matrices are equal.

Corollary 30. Suppose that X and Y are derived equivalent K3 surfaces with monodromy satisfying trace(T) ≠ −2. Then both X(F) and Y (F) are nonempty.

Let X → Δ denote a nonsingular model with central fiber X_{0}. Inclusion X̄ ↪ X and retraction of X to X_{0} induce

ϱ : H_{2}(X̄, ℤ) → H_{2}(X , ℤ) → H_{2}(X_{0}, ℤ);

elements of the kernel are called vanishing cycles. The local invariant cycle theorem [CEZGT14, 5.3.4] gives an exact sequence

H^{2}(X_{0}, ℚ) ⟶^{ϱ^{∗}} H^{2}(X̄, ℚ) ⟶^{T −I} H^{2}(X̄, ℚ).

The kernel of ϱ is dual to the cokernel of ϱ^{∗}; we may identify this with the image of (T − I). Thus we define the lattice of vanishing classes of X as saturation of the image of (T − I) in H^{2}(X̄, ℤ). This carries the structure of a representation of a cyclic group, i.e., the action of T. Since (T −I) acts trivially on H^{0} and H^{4}, the lattice of vanishing cycles may also be in interpreted as the saturation of the image of (T − I) on the full Mukai lattice H̃(X, ℤ). Again, Orlov’s Theorem implies

Proposition 31. For K3 surfaces over F = ℂ((t)), lattices of vanishing classes are a derived invariant.

We have results in some cases where T is semisimple, i.e., when T^{e} = I for some e ∈ ℕ.

Definition 32. Let X be a K3 surface over F = ℂ((t)) with monodromy T. It is of ADE type if

• the lattice of vanishing classes admits a finite-index sublattice isomorphic to a direct sum of negative definite lattices associated with root systems of types A_{n}, D_{n}, E_{6}, E_{7}, or E_{8};

• the action of T on the vanishing classes may be expressed as a product of reflections in roots in these lattices.

Roughly, the monodromy is in the product of Weyl groups associated with the vanishing classes.

Since the lattice of vanishing classes is a derived invariant, being of ADE type is as well.

Proposition 33. X is of ADE type if and only if it has ADE reduction.

An application of Hensel’s Lemma gives

Corollary 34. If X is of ADE type then X(F) is non-empty.

Proof. Suppose X admits a model X → Δ with ADE singularities in the central fiber. Replacing X with a small modification, we may assume that it has ℚ-factorial terminal singularities [KM98, Th. 6.25]. Such a model is maximal among small partial resolutions of the original family; see [Rei83, §1,8] and [KM92] for interpretations in the context of families of surfaces with ADE singularities. The point of passing to this model is to ensure that the cycles collapsed by the specialization X̄ ⇝ X_{0} are in fact vanishing cycles.

After a suitable basechange

Δ_{1} → Δ t = t_{1}^{e},

our family admits a simultaneous resolution [Art74, Bri71]

X̃ → X ×_{Δ} Δ_{1},

i.e., ρ_{0} : X̃_{0} → X_{0} is the minimal resolution. As X̃_{0} is a smooth degeneration of X it is also a K3 surface. Moreover, the vanishing cycles of X are spanned (over ℚ) by ρ_{0}-exceptional curves and the monodromy of X is a product of reflections in the product of Weyl groups indexed by the singularities of X_{0}. Thus X is of ADE type.

Conversely, assume that X satisfies the requisite monodromy condition. After a basechange Δ_{1} → Δ as above, the Torelli Theorem gives a smooth (Type I Kulikov) model

X̃ → Δ_{1}.

We analyze the central fiber X̃_{0}. The lattice of vanishing classes specializes to the saturation of a direct sum of ADE lattices in the Picard group of X̃_{0}. Moreover, we may interpret T as acting birationally on X̃, resolved after a flop of vanishing smooth rational curves in X̃_{0} [Rei83, Thm. 8.2]. This action is over the action t_{1} → ζt_{1} with ζ = exp(2πi/e).

Contracting the vanishing (−2)-curves, we obtain a model

X̃  →  Y
↘  ↙
Δ_{1}

with ADE singularities in the central fiber Y_{0}, over which T is regular. The quotient

X := Y/T → Δ_{1}/T = Δ

is the desired model of X. □

5. Semistable models over p-adic fields

Let F be a p-adic field with ring of integers R.

We start with the case of good reduction, which follows from Theorem 24 and Hensel’s Lemma:

Corollary 35. Let X and Y be K3 surfaces over F, which are derived equivalent and have good reduction over F. Then X(F) ≠ ∅ if and only if Y (F) ≠ ∅.

We can extend this as follows:

Proposition 36. Assume that the residue characteristic p ≥ 7. Let X and Y be K3 surfaces defined and derived equivalent over F. Assume X and Y admit models

X , Y → Spec(R)

that are regular and have ADE singularities in the central fiber. Then X(F) ≠ ∅ if and only if Y (F) ≠ ∅.

Proof. Let k be the finite residue field, X_{0} and Y_{0} the resulting reductions, and X̃_{0} and Ỹ_{0} their minimal resolutions over k̄.

Our assumptions on p guarantee that the classification and deformations of rational double points over k coincides with the classification in characteristic 0 [Art77]. Applying Artin’s version of Brieskorn simultaneous resolution [Art74, Th. 2], there exists a finite extension

Spec(R_{1}) → Spec(R)

and proper models

X̃ → X ×_{Spec(R)} Spec(R′) → Spec(R′),
Ỹ → Y ×_{Spec(R)} Spec(R′) → Spec(R′),

in the category of algebraic spaces, with central fibers X̃_{0} and Ỹ_{0}.

The Fourier-Mukai transform induces isomorphisms

H̃_{ét}(X̄, ℤ_{ℓ}) → H̃_{ét}(Ȳ , ℤ_{ℓ})

compatible with monodromy and the action of Frobenius. Specializing yields an isomorphism

ψ : H_{ét}^{2}(X̃_{0}, ℤ_{ℓ}) → H_{ét}^{2}(Ỹ_{0}, ℤ_{ℓ}).

Note that since X̃, Ỹ are not projective over Spec(R′), there is not an evident interpretation of this as a derived equivalence over Spec(R′). (See [BM02] for such interpretations for K3 fibrations over complex curves.) Furthermore ψ is far from unique, as we may compose with reflections arising from exceptional curves in either X̃_{0} → X_{0} or Ỹ_{0} → Y_{0} associated with vanishing cycles of X or Y.

Let L (resp. M) denote the lattice of vanishing classes in H^{2}(X̄, ℤ_{ℓ}) (resp. H^{2}(Ȳ , ℤ_{ℓ})). As in the proof of Proposition 31, L ≃ M, compatibly with the monodromy and Frobenius actions. (Since X and Y have ADE reduction, the local invariant cycle theorem still applies to their models.) Their orthogonal complements in the Mukai lattice L^{⊥} and M^{⊥} are isomorphic as well.

The central fibers X_{0} and Y_{0} are obtained from X̃_{0} and Ỹ_{0} by blowing down the (−2)-curves classes associated with L and M respectively. Let X_{∘} ⊂ X_{0} and Y_{∘} ⊂ Y_{0} denote the smooth loci, i.e., the complements of the rational curves associated with L and M respectively.

We can relate compactly supported cohomology to our lattices:

L^{⊥} ⊗ ℚ_{ℓ} ⊃ H_{c,ét}^{2}(X̄_{∘}, ℚ_{ℓ}), M^{⊥} ⊗ ℚ_{ℓ} ⊃ H_{c,ét}^{2}(Ȳ_{∘}, ℚ_{ℓ}),

with the difference reflecting contributions from H^{0} and H^{4}. Thus ψ induces an isomorphism

H_{c,ét}^{2}(X̄_{∘}, ℚ_{ℓ}) ≃ H_{c,ét}^{2}(Ȳ_{∘}, ℚ_{ℓ}),

compatible with Galois actions. The Weil conjectures yield then that

|X_{∘}(k)| = |Y_{∘}(k)|.

Rational points of X and Y over F correspond to sections of X → Spec(R) and Y → Spec(R). Since X and Y are regular, these reduce to points of X_{∘}(k) and Y_{∘}(k). Hensel’s Lemma then implies our result. □

Question 37. Is admitting a model with good or ADE reduction a derived invariant?

Y. Matsumoto and C. Liedtke have recently addressed this. Having potentially good reduction is governed by whether H_{ét}^{2}(X̄, ℚ_{ℓ}) is unramified, under technical assumptions [Mat15, Th. 1.1]. These are satisified if there exists a Kulikov model after some basechange [ML14, p. 2]. The ramification condition depends on the ℓ-adic cohomology and thus only on the derived equivalence class. Proposition 33 suggests a monodromy characterization of ADE reduction in mixed characteristic; under technical assumptions, there exists such a model if the cohomology is unramified [ML14, Th. 5.1].

Appendix: Semistability and derived equivalence

To understand the implications of derived equivalence for rational points over local fields, we must first describe how monodromy governs the existence of models with good properties. For K3 surfaces, it is widely expected that unipotent monodromy should suffice to guarantee the existence of a Kulikov model. Over ℂ this is widely known to experts; the referee advised us that this was addressed in correspondence among R. Friedman, D. Morrison, and F. Scattone in 1983. As we are unaware of a published account, we offer an argument:

Theorem 38. Let X be a K3 surface over F = ℂ((t)). Then X admits a Kulikov model if and only if its monodromy is unipotent.

Corollary 39. Let X and Y be derived equivalent K3 surfaces over F. Then X admits a Kulikov model if and only if Y admits a Kulikov model.

As we have seen, if X and Y are derived equivalent over F then their Mukai lattices admit a monodromy-equivariant isomorphism; thus the characteristic polynomials of their monodromy matrices are equal.

The remainder of this section is devoted the proof of Theorem 38. We start with a review of basic results on Kulikov models.

Let R = ℂ[[t]], Δ = Spec(R), and Δ^{∘} = Spec(F). The monodromy T of X over ℂ((t)) satisfies

(T^{e} − I)^{f} = 0

for some e, f ∈ ℕ. We take e and f minimal with this property.

The semistable reduction theorem [KKMSD73] implies there exists an integer n ≥ 1 such that after basechange to

R_{2} = ℂ[[t_{2}]], F_{2} = ℂ((t_{2})), t_{2}^{n} = t,

there exists a flat proper

π_{2} : X_{2} → Δ_{2} = Spec(R_{2})

such that • the generic fiber is the basechange of X to F_{2};

• the central fiber π_{2}^{−1}(0) is a reduced normal crossings divisor. We call this a semistable model for X. It is well-known that semistable reductions have unipotent monodromy so e|n.

By work of Kulikov and Persson-Pinkham [Kul77, PP81], there exists a semistable modification of X_{2}

ϖ : X̃ → Δ_{2}

with trivial canonical class, i.e., there exists a birational map X_{2} −−→ X̃ that is an isomorphism away from the central fibers. We call this a Kulikov model for X. Furthermore, the structure of the central fiber X̃_{0} can be described in more detail: Type I X̃_{0} is a K3 surface and f = 1. Type II X̃_{0} is a chain of surfaces glued along elliptic curves, with rational

surfaces at the end points and elliptic ruled surfaces in between; here f = 2.

Type III X̃_{0} is a union of rational surfaces and f = 3. We will say more about the Type III case: It determines a combinatorial triangulation of the sphere with vertices indexed by irreducible components, edges indexed by double curves, and ‘triangles’ indexed by triple points [Mor84]. We analyze this combinatorial structure of X̃_{0} in terms of the integer m.

Let X̃_{0} = ∪_{i=1}^{n}V_{i} denote the irreducible components, Ṽ_{i} their normalizations, and D_{ij} ⊂ Ṽ_{i} the double curves over V_{i} ∩ V_{j}.

Definition 40. X̃_{0} is in minus-one form if for each double curve D_{ij} we have (D′_{ij}^{2})_{V_{i}} = −1 if D′_{ij} is a smooth component of D_{ij} and (D_{ij}^{2})_{V_{i}} = 1 if D_{ij} is nodal.

Miranda-Morrison [MM83] have shown that after elementary transformation of X̃, we may assume that X̃ is in minus-one form.

The following are equivalent [Fri83, §3],[FS85, 0.5,7.1]:

• the logarithm of the monodromy is m times a primitive matrix;

• X̃_{0} admits a ‘special μ_{m} action’, i.e., acting trivially on the sets of components, double/triple points, and Picard groups of the irreducible components;

• X̃_{0} admits ‘special m-bands of hexagons’, i.e., the triangulation coming from the components of X̃_{0} arises as a degree m refinement of another triangulation.

In other words, X̃_{0} ‘looks like’ it is obtained from applying semistable reduction to the degree m basechange of a Kulikov model. Its central fiber X̃′_{0} can readily be described [Fri83, 4.1]—its triangulation is the one with refinement equal to the triangulation of X̃_{0}, and its components are contractions of the corresponding components of X̃_{0}. (When we refer to X̃′_{0} below in the Type III case, we mean the surface defined by this process.)

For Type II we can do something similar [Fri84, §2]. After elementary modifications, we may assume the elliptic surfaces are minimal. Then following are equivalent:

• the logarithm of the monodromy is m times a primitive matrix;

• X̃_{0} = V_{0} ∪_{E} … ∪_{E} V_{m} is a chain of m + 1 surfaces glued along copies of an elliptic curve E, where V_{0} and V_{m} are rational and V_{1}, …, V_{m−1} are minimal surfaces ruled over E.

Again X̃_{0} ‘looks like’ it is obtained from applying semistable reduction to another Kulikov model with central fiber X̃′_{0} = V_{0} ∪_{E} V_{m}. Moreover (E^{2})_{V_{0}} + (E^{2})_{V_{m}} = 0 and X̃′_{0} is d-semistable in the sense of Friedman [Fri84, 2.1]. (When we refer to X̃′_{0} below in the Type II case, we mean the surface defined by this process.)

There are refined Kulikov models taking into account polarizations: Let (X, g) be a polarized K3 surface over F of degree 2d. Shepherd-Barron [SB83] has shown there exists a Kulikov model ϖ : X̃ → Δ_{2} with the following properties:

• there exists a specialization of g to a nef Cartier divisor on the central fiber X̃_{0};

• g is semi-ample relative to Δ_{2}, inducing

X̃  →^{|g|}  Z
↘  ↙
Δ_{2}

where X̃_{0} → Z_{0} is birational and Z_{0} has rational double points, normal crossings, or singularities with local equations

xy = zt = 0.

These will be called quasi-polarized Kulikov models and their central fibers admissible degenerations of degree 2d.

Recall the construction in sections five and six of [FS85]: Let D denote the period domain for degree 2d K3 surfaces and Γ the corresponding arithmetic group—the orientation-preserving automorphisms of the cohomology lattice H^{2}(X, ℤ) fixing g. Fix an admissible degeneration (Y_{0}, g) of degree 2d and its image (Z_{0}, h), with deformation spaces Def(Y_{0}, g) → Def(Z_{0}, h); the morphism arises because g is semiample over the deformation space. Let

\overline{Γ\D}_{N_{Y_{0}}} ⊃ Γ\D

denote the partial toroidal compactification parametrizing limiting mixed Hodge structures with monodromy weight filtration given by a nilpotent N_{Y_{0}} associated with Y_{0} (see [FS85, p.27]). We do keep track of the stack structure. Given a holomorphic mapping

f : {t : 0 < |t| < 1} → Γ\D,

that is locally liftable (lifting locally to D), with unipotent monodromy Γ-conjugate to N_{Y_{0}}, then f extends to

f : \{t : |t| < 1\} → \overline{Γ\D}.

The period map extends to an étale morphism [FS85, 5.3.5,6.2]

Def(Y_{0}, g) → \overline{Γ\D}.

Thus the partial compactification admits a (local) universal family.

Theorem 38 thus boils down to The smallest positive integer n for which we have a Kulikov model equals the smallest positive integer e such that T^{e} is unipotent.

Proof. We show that a Kulikov model exists provided the monodromy is unipotent. Suppose we have unipotent monodromy over

R_{1} = ℂ[[t_{1}]], t_{1}^{e} = t,

and semistable reduction

X_{2} → Δ_{2} = Spec(R_{2}), R_{2} = Spec(ℂ[[t_{2}]]), t_{2}^{me} = t.

Let X̃ → Δ_{2} denote a Kulikov model, obtained after applying elementary transformations as specified above. Write

mN = log(T^{e}) = (T^{e} − I) − \frac{1}{2}(T^{e} − I)^{2}

where m ∈ ℕ and N is primitive (cf.[FS85, 1.2] for the Type III case).

Let X̃′_{0} be the candidate for the ‘replacement’ Kulikov model, i.e., the central fiber of the Kulikov model we expect to find

X̃′ → Δ_{1}.

In the Type I case X̃′_{0} = X̃_{0} by Torelli, so we focus on the Type II and III cases.

Lemma 41. Suppose that X̃_{0} admits a degree 2d semiample divisor g. Then X̃′_{0} admits one as well, denoted by g′.

Proof. In the Type II case, this follows from [Fri84, Th. 2.3]. The discussion there shows how (after elementary modification) the divisor can be chosen to induce the morphism X̃_{0} → X̃′_{0} collapsing V_{1} ∪_{E} …∪_{E} V_{m−1} to E, interpreted as the normal crossings locus of X̃′_{0}.

For Type III, we rely on Proposition 4.2 of [Fri83], which gives an analogous process for modifying the coefficients of h so that it is trivial or a sum of fibers on the special bands of hexagons. However, Friedman’s result does not indicate whether the resulting line bundle is nef. This can be achieved after birational modifications of the total space [SB83, Th. 1]. □

We can apply the Friedman-Scattone compactification construction to both (X̃_{0}, g) and (X̃′_{0}, g′), with N = NX̃′_{0} and mN = NX̃_{0}. Thus we obtain two compactifications

\overline{Γ\D}_{mN} → \overline{Γ\D}_{N} ⊃ Γ\D,

both with universal families of degree 2d K3 surfaces and admissible degenerations.

To construct X̃′ → Δ_{1} = Spec(R_{1}) we use the diagram

Δ_{2} → \overline{Γ\D}_{mN}
↓  ↓
Δ_{1}  \overline{Γ\D}_{N}.

The liftability criterion for mappings to the toriodal compactifications gives an arrow

Δ_{1} → \overline{Γ\D}_{N}

making the diagram commute. The induced universal family on this space induces a family

X̃′ → Δ_{1},

agreeing with our original family for t_{1} ≠ 0 by the Torelli Theorem. More precisely, the monodromies of the two families over Δ_{1}^{∘} = Spec(R_{1}) \ {0} are identified and automorphisms of K3 surfaces act faithfully on cohomology, so they are isomorphic over Δ_{1}^{∘}. Thus X̃′ → Δ_{1} is the desired model. □

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Department of Mathematics, Rice University, MS 136, Houston, TX 77251-1892, USA

Current address: Department of Mathematics, Brown University, Box 1917, 151 Thayer Street Providence, RI 02912, USA

E-mail address: bhassett@math.brown.edu

Courant Institute, New York University, New York, NY 10012, USA

E-mail address: tschinkel@cims.nyu.edu

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