Stable Maps and Chow Groups

D. Huybrechts and M. Kemeny

Received: April 26, 2012 Revised: May 15, 2012

Communicated by Thomas Peternell

Abstract. According to the Bloch–Beilinson conjectures, an automorphism of a K3 surface X that acts as the identity on the transcendental lattice should act trivially on CH^{2}(X). We discuss this conjecture for symplectic involutions and prove it in one third of all cases. The main point is to use special elliptic K3 surfaces and stable maps to produce covering families of elliptic curves on the generic K3 surface that are invariant under the involution.

2010 Mathematics Subject Classification: 14J28 (14J50)

0.1. Let X be a complex projective K3 surface with an automorphism f : X [figure: ∼] X. According to the general philosophy of the Bloch–Beilinson conjectures, the induced action of f on the kernel of the cycle map CH^{∗}(X) [figure] H^{∗}(X, ℤ) should be determined by the action of f on the cokernel of the cycle map. More precisely, one expects the following to be true:

Conjecture 0.1. f^{∗} = id on CH^{2}(X)_{0} if and only if f^{∗} = id on T (X).

Here, CH^{2}(X)_{0} ⊂ CH^{2}(X) is the degree zero part, i.e. the kernel of the cycle map CH^{2}(X) [figure] H^{4}(X, ℤ) ≅ ℤ, and T (X) ⊂ H^{2}(X, ℤ) is the transcendental lattice which can be described as the orthogonal complement of the Néron– Severi group NS(X) ⊂ H^{2}(X, ℤ). Alternatively, T (X) ⊂ H^{2}(X, ℤ) is the smallest sub-Hodge structure such that H^{2,0}(X) ⊂ T (X) ⊗ ℂ. Thus, f^{∗} = id on T (X) if and only if f acts trivially on H^{2,0}(X). The latter is spanned by the unique (up to scaling) regular two-form σ ∈ H^{0}(X, Ω_{X}^{2}), which we think of as a holomorphic symplectic structure. For this reason, an automorphism f : X [figure: ∼] X with f^{∗} = id on T (X) is called a symplectomorphism. It is well known that f^{∗} = id on CH^{2}(X)_{0} implies that f acts trivially on T (X) (see e.g. [16, Ch. 23]). Appropriately rephrased, this holds for arbitrary smooth projective varieties and for arbitrary correspondences. It is the converse of the statement that is difficult and that shall be discussed here for symplectic involutions of K3 surfaces, i.e. automorphisms f of order two with f^{∗}σ = σ.

1This work was supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation). The second author is supported by a PhD-scholarship of the Bonn International Graduate School in Mathematics.

0.2. K3 surfaces X endowed with a symplectic involution f : X [figure: ∼] X come in families. As shown by van Geemen and Sarti in [4], the moduli space of such (X, f) has one resp. two connected components in each degree 2d > 0 depending on the parity of d. To be more precise, let Λ_{d} be the lattice ℤℓ ⊕ E_{8}(−2) with (ℓ.ℓ) = 2d and denote for d ≡ 0(2) by Λ̃_{d} the unique even lattice containing Λ_{d} with Λ̃_{d}/Λ_{2} ≅ ℤ/2ℤ and such that E_{8}(−2) ⊂ Λ̃_{d} is primitive (see [4, Prop. 2.2]). Then for generic (X, f) one has NS(X) ≅ Λ_{d} or NS(X) ≅ Λ̃_{d}. The class ℓ corresponds under this isomorphism to an ample line bundle L on X which spans the f-invariant part of NS(X). If (X, f) is not generic, then one still finds Λ_{d} or Λ̃_{d} as a primitive sublattice in NS(X) with E_{8}(−2) as the orthogonal complement of the invariant part. Conversely, by the Global Torelli theorem any X parametrized by the (non-empty and in fact 11-dimensional) connected moduli spaces 𝔐_{Λ_{d}} or 𝔐Λ̃_{d} of Λ_{d} resp. Λ̃_{d}-lattice polarized K3 surfaces comes with a symplectic involution f that is determined by its action = −id on E_{8}(−2) and = id on its orthogonal complement. In other words, for each d ≡ 1(2) the moduli space of K3 surfaces X with a symplectic involution f and an invariant polarization of degree 2d has one connected component 𝔐_{Λ_{d}}, whereas for d ≡ 0(2) it has two connected components, 𝔐_{Λ_{d}} and 𝔐Λ̃_{d}. Thus the following theorem, the main result of the present paper, proves Conjecture 0.1 in one third of all possible cases.

Theorem 0.2. Let d ≡ 0(2) and (X, f) ∈ 𝔐Λ̃_{d}. Then f^{∗} = id on CH^{2}(X).

0.3. For d = 1 (double covers of ℙ^{2}) and d = 2 (quartics in ℙ^{3}) the conjecture is known to hold, see [2, 14, 15]. For d = 3 (complete intersection of a cubic and a quadric in ℙ^{4}) an interesting approach is outlined in [6]. Theorem 3.2 in [17] proves the conjecture for equivariant complete intersections in varieties with trivial Chow groups. In [7] the conjecture has been proven for (X, f) in dense subsets of 𝔐_{Λ_{d}} and 𝔐Λ̃_{d}. The proof there relies on Fourier–Mukai equivalences of the bounded derived category of coherent sheaves on X and it is not clear how to push the techniques further to cover generic and hence arbitrary (X, f). The techniques to prove Theorem 0.2 can be applied to symplectic automorphisms f : X [figure: ∼] X of order > 2. If the order of f is a prime p, then p = 2, 3, 5, or 7 (cf. [13]), and the results of [4] have in [3] been successfully generalized to cover also the cases p = 3, 5, and 7. Our methods prove Conjecture 0.1 for many components of the moduli space of (X, f) in these cases too, see Section 5.

Theorem 0.3. For p = 3, 5, or 7 and d = ep, there exists one component of the moduli space of polarized K3 surfaces (X, L) with a symplectic automorphism f : X [figure: ∼] X of order p and L^{2} = 2d such that Conjecture 0.1 holds true. □

0.4. The proof of Theorem 0.2 neither uses derived categories as in [7] nor any deep cycle arguments as e.g. in [6]. As we shall explain in Section 1, it is enough to find a dominating family of integral genus one curves on X that are invariant under f and avoid the fixed points of f. The conjecture is then deduced from the absence of torsion in CH^{2}(X). It is not clear whether the existence of such a family should be expected in general, but it will be shown here for generic K3 surfaces parametrized by points in 𝔐Λ̃_{d}. This is done in two steps. Firstly, we construct a family of genus one (reducible) curves on a particular elliptic K3 surface for which f is given by translation by a two-torsion section, see Section 3. Then, the theory of stable maps is applied to obtain the desired family for generic X. The missing piece to prove Conjecture 0.1 in full generality, or at least for symplectic involutions, is the lack of special K3 surfaces in 𝔐_{Λ_{d}} for which appropriate families of genus one curves can be described explicitly. Acknowledgments: We thank Richard Thomas for a useful discussion concerning Section 2. Addendum: The results of this article have meanwhile been improved by C. Voisin and the first author. In [18] Voisin replaces singular elliptic curves in the quotient X/⟨f⟩ and their étale cover in X, as used in this paper, by smooth ample curves of high genus. A dimension estimate for the corresponding Prym variety yields finite dimensionality of the ani-invariant part of the Chow group and eventually f^{∗} = id on CH^{2}(X) for symplectic involutions. In particular, neither degeneration to special elliptic K3 surfaces nor the deformation theory of stable maps, which we believe to be of independent interest, enter her arguments. In [8] we use instead arguments involving the derived category D^{b}(Coh(X)) and lattice theory as developed in [7] to prove the result for symplectic automorphisms of prime order p ≠ 2. Both results taken together prove f^{∗} = id on CH^{2}(X) for symplectic automorphisms of arbitrary finite order.

1. Covering families of elliptic curves

Let f : X [figure: ∼] X be a symplectic automorphism of finite order and denote its quotient by X̄ := X/⟨f⟩, which is a singular K3 surface. If f has prime order p, then p = 2, 3, 5, or 7 (see [13]). In order to prove Conjecture 0.1 for symplectic automorphisms of finite order (and we do not have anything to say for automorphisms of infinite order), one can restrict to those. The number of fixed points of f, all isolated, can be determined by the Lefschetz fixed point formula. E.g. for a symplectic involution, i.e. p = 2, there are exactly eight fixed points. In the following, a family C_{t} ⊂ X of curves given by C ⊂ S × X is called dominating if the projection C [figure] X is dominant, i.e. if the curves C_{t} parametrized by the closed point t ∈ S cover a Zariski open subset of X.

Proposition 1.1. Let f : X [figure: ∼] X be a symplectic automorphism. Assume there exists a dominating family of integral f-invariant curves C_{t} ⊂X of geometric genus one with C_{t} ∩ Fix(f) = ∅ for generic t. Then f^{∗} = id on CH^{2}(X).

Proof. It suffices to prove that for generic x ∈ X the points x and y := f(x) are rationally equivalent, i.e. [x] = [y] in CH^{2}(X). Since by Roitman’s theorem CH^{2}(X) is torsion free (see e.g. [16, Ch. 22]), the latter is equivalent to [x]−[y] being torsion. For any morphism g : C [figure] X from a smooth irreducible curve C the induced g_{∗} : Pic(C) = CH^{1}(C) [figure] CH^{2}(X) is a group homomorphism. Thus, if there exist lifts x̃, ỹ ∈ C of x resp. y such that O(x̃ − ỹ) ∈ Pic^{0}(C) is a torsion line bundle, then automatically [x] = [y] in CH^{2}(X). By assumption, a generic closed point x ∈ X lies on one of the curves C_{t}. Since the curves C_{t} are assumed to be f invariant, y = f(x) is contained in the same curve and f lifts to an automorphism f̃ of the normalization C := C̃_{t}, which is a smooth integral curve of genus one. As C_{t} avoids the fixed points of f, the automorphism f̃ : C [figure: ∼] C is fixed point free and hence D := C/⟨f̃⟩ is also smooth of genus one. After choosing origins for C and D appropriately, C [figure] D is a morphism of elliptic curves which can be viewed as a quotient of C by a finite subgroup Γ ⊂ C ≅ Pic^{0}(C). Hence points in the same fibre of C [figure] D differ by elements of Γ. In particular, O_{C}(x̃ − ỹ) ∈ Γ ⊂ Pic^{0}(C) is a torsion line bundle. □

The problem now becomes to construct a family of genus one curves as required. We do not know how to do this directly. On the special elliptic surface considered in Section 3 a family of genus one curves is constructed, but the curves are not integral. They become integral only after deformations to the generic case.

2. Stable maps to K3 surfaces

Let X [figure] S be an irreducible family of K3 surfaces with a global line bundle L. Consider the moduli stack M_{g}(X , L) [figure] S of stable maps h : D [figure] X_{t} to fibres of X [figure] S such that D is of arithmetic genus g with h_{∗}(D) ∈ |L_{t}|. As the stack structure of M_{g}(X , L) is of no importance to us, we shall ignore it and treat M_{g}(X , L) as a moduli space. If we do not want to fix the linear equivalence class of the image curves, we will simply write M_{g}(X). The following fact has been used in various contexts in the literature, but mostly for g = 0 (see e.g. [1, 12]). We shall need the following statement for g = 1.

Proposition 2.1. Every irreducible component of M_{g}(X , L) is of dimension at least g + dim(S).

Proof. The starting point is [10, Thm. 2.17]: For simplicity let π : X [figure] S be a smooth projective family over an irreducible base S and let D [figure] S be a flat and projective family of curves. Every irreducible component of Mor_{S}(D, X) containing a morphism h : D := D_{0} [figure] X := X_{0} is of dimension at least

(1)  χ(D, h^{∗}T_{X}) + dim(S).

The first term χ(D, h^{∗}T_{X}) = h^{0}(D, h^{∗}T_{X}) − h^{1}(D, h^{∗}T_{X}) reflects the usual deformation-obstruction theory for the morphism h : D [figure] X. A priori, the obstructions to deforming the morphism h : D [figure] X are contained in H^{1}(D, h^{∗}T_{X}), which is part of an exact sequence

… [figure] H^{1}(D, h^{∗}T_{X}) [figure] H^{1}(D, h^{∗}T_{X}) [figure] H^{1}(D, h^{∗}π^{∗}T_{S}) [figure] 0.

Since the morphism D [figure] X ⊂ X [figure] S is constant, there are no obstructions to deforming it sideways at least when S is smooth. In other words, the obstructions to deforming h : D [figure] X are contained in the image of H^{1}(D, h^{∗}T_{X}) which leads to the stronger bound in (1). A similar argument allows one to treat the case of varying domain D. The usual obstruction theory for stable maps shows that M_{g}(X, L) in [h : D [figure] X] is of dimension at least χ(D, (h^{∗}Ω_{X} [figure] Ω_{D})^{∗}), where the two term complex h^{∗}Ω_{X} [figure] Ω_{D} is concentrated in degree −1 and 0, see [5]. For X = X_{0} in a family X [figure] S, the analogue of (1) then says that M_{g}(X) in a point corresponding to a stable map h : D [figure] X is of dimension at least

(2)  χ(D, (h^{∗}Ω_{X}  [figure] Ω_{D})^{∗}) + dim(S) = g − 1 + dim(S)

The last equation follows from a standard Riemann–Roch calculation. The remaining issue is to increase the bound by restricting to families X [figure] S which come with a deformation L of L := O(h_{∗}(D)). One can either invoke reduced deformation theory for K3 surfaces as developed recently in [11] in great detail or use the following trick. Any given family (X , L) [figure] S with a polarization L can be thickened to a family X̃ [figure] S̃ with dim S̃ = dim S + 1 such that transversally to S ⊂ S̃ the line bundle L is obstructed (even to first order). More precisely, for t ∈ S the line bundle L_{t} on X_{t} deforms to first order in the direction of v ∈ T_{S̃,t} if and only if v ∈ T_{S,t} ⊂ T_{S̃,t}. If L is fibrewise ample, then the thickening X̃ [figure] S̃ can be explicitly described by using the twistor space construction for each fibre X_{t} and the Kähler class given by c_{1}(L_{t}). (Note that in particular, X̃ [figure] S̃ will in general not be projective.) Otherwise, one uses the standard deformation theory of K3 surfaces to produce such a family at least locally, which is enough for the following dimension count. By the discussion above, M_{g}(X̃) is in [h : D [figure] X] of dimension

g − 1 + dim(S̃) = g + dim(S).

On the other hand, h : D [figure] X cannot deform sideways in a tangent direction v ∈ T_{S̃,0} that is not contained in T_{S,0}, because O(h_{∗}(D)) = L_{0}. This shows that the two moduli spaces M_{g}(X̃) and M_{g}(X) coincide near the point given by [h : D [figure] X]. □

Corollary 2.2. Suppose the fibre M_{0} of an irreducible component M ⊂ M_{g}(X , L) is of dimension ≤ g for some 0 ∈ S. Then M dominates S. □

In other words, if the moduli space M_{g}(X_{0}, L_{0}) of stable maps to one fibre X_{0} has the expected dimension g in [h : D [figure] X_{0}], then h can be deformed to a stable map h_{t} : D_{t} [figure] X_{t} to the generic fibre. To ensure that the condition is met, we shall later use the following criterion, c.f. [9, Cor. 1.2.5] and [12, Lem. 2.6].

Proposition 2.3. Suppose the stable map h : D [figure] X satisfies the following conditions:

i) If D_{1}, D_{2}, …, D_{n} are the components of D, then D_{2}, …, D_{n} are smooth and rational. ii) The first component D_{1} is smooth of genus g and h|_{D_{1}} : D_{1} [figure] X is an embedding. iii) The morphism h is unramified. iv) Two components D_{i} and D_{j} intersect transversally in one point if |i−j| = 1 and not at all otherwise. Then M_{g}(X) is of dimension g in [h : D [figure] X].

Proof. We copy the argument from [1, Lem. 2.7]. First of all, since h is unramified, the complex h^{∗}Ω_{X} [figure] Ω_{D} is a locally free sheaf of rank one concentrated in degree −1, the dual of which is denoted N_{h}. Then, one proceeds by induction over n and uses the exact sequence

0 [figure] N_{h}(−x)|_{D′}  [figure] N_{h}  [figure] N_{h}|_{D_{n}}  [figure] 0,

where D′ := D_{1} ∪ … ∪ D_{n−1} and {x} = D_{n−1} ∩ D_{n}. From the exact sequence

0 [figure] N_{h}^{∗}|_{D_{n}}  [figure] h^{∗}Ω_{X}|_{D_{n}}  [figure] Ω_{D}|_{D_{n}}  [figure] 0

and Ω_{D}|_{D_{n}} ≅ O(−1), one deduces N_{h}|_{D_{n}} ≅ O(−1). Thus, H^{i}(N_{h}) ≅ H^{i}(N_{h}(−x)|_{D′}). On the other hand, N_{h}(−x)|_{D′} = N_{h′}, where h′ := h|_{D′} : D′ [figure] X. By induction this eventually yields H^{i}(N_{h}) ≅ H^{i}(N_{D_{1}/X}). But clearly, h^{0}(N_{D_{1}/X}) = h^{0}(D_{1}, ω_{D_{1}}) = g and the deformations of D_{1} ⊂ X are unobstructed. □

Remark 2.4. Maybe more geometrically, the arguments show that deformations of h : D [figure] X are all given by deforming D_{1} ⊂ X.

3. Special elliptic surfaces

We follow [4, Sect. 4] for the construction of an elliptic K3 surface X [figure] ℙ^{1} with a symplectic involution given by a two-torsion section. Deformations of X will lead to K3 surfaces with Néron–Severi group Λ̃_{2d} with d = 2e > 2. The elliptic K3 surface X [figure] ℙ^{1} is described by an equation of the form

(3)  y^{2} = x(x^{2} + a(t)x + b(t))

with general a(t) and b(t) of degree 4 resp. 8. Then the fibration has two obvious sections: The section at infinity σ given by x = z = 0, which will serve us as the zero section, and a disjoint section τ given by x = y = 0. Using the explicit equation, one finds that τ has order two. Thus, translation by τ defines an involution f : X [figure: ∼] X which is symplectic. Still following [4], one computes the singular fibres of X [figure] ℙ^{1}: There are eight fibres of type I_{1} (a rational curve with one node) and eight fibres of type I_{2} (the union of two copies of ℙ^{1} intersecting transversally in two points). They can be found over the zeroes of b ∈ H^{0}(ℙ^{1}, O(8)) resp. a^{2} − 4b ∈ H^{0}(ℙ^{1}, O(8)). The fixed points of f are the nodes of the eight I_{1}-fibres which are all avoided by σ and τ. Moreover, f interchanges the two components of each I_{2}-fibre.

[figure: …; …; σ; τ; N_{1}; N_{8}]

The components of the I_{2}-fibres not meeting σ are denoted N_{1}, …, N_{8}. Then N̂ = (1/2) ∑ N_{i} ∈ NS(X). Moreover, if F denotes the class of a generic fibre (and by abuse also a generic fibre itself), then σ and F span a hyperbolic plane and τ = σ + 2F − N̂. The Néron–Severi group of X (for general a and b) is thus ⟨σ, F ⟩ ⊕ ⟨N_{1}, …, N_{8}, N̂⟩, which is of rank 10. Next consider a curve of the form C = eN +F +σ+τ, where N is one of the I_{2}- fibres, and let L := O(C). Then L is big and nef. Indeed, (L.L) = 4e > 0 and C intersects all its irreducible components positively, e.g. (C.σ) = e − 1 > 0. In fact, L is ample as it clearly intersects all horizontal curves positively and has also positive intersection with all (−2)-curves (e.g. the two components of the I_{2}-fibres). Moreover, L is primitive, as (C.N_{i}) = 1. Since f respects the fibration and interchanges σ and τ, the curve C is f-invariant and disjoint from Fix(f).

[figure: σ; τ; eN; F; C :]

Let us now consider the quotient X̄ := X/⟨f⟩ which is a singular K3 surface with eight ordinary double points. Its minimal resolution Y [figure] X̄ comes with a natural elliptic fibration Y [figure] ℙ^{1}. Note that the fibres of type I_{1} and I_{2} are interchanged when passing from X to Y.

The quotient C̄ := C/⟨f⟩ ⊂ X̄ avoids the singular locus of X̄ and thus can also be viewed as a curve in Y. For the same reason, the line bundle L descends to an ample line bundle L̄ on Ȳ. Note that C̄ decomposes as C̄ = eN̄ + F̄ + σ̄, where N̄ is an I_{1}-fibre of Y [figure] ℙ^{1}, F̄ is a smooth fibre, and σ̄ is a section.

[figure: σ̄; eN̄; F̄; C̄ :]

Lemma 3.1. There exists a stable map h : D [figure] X̄ of arithmetic genus one with image C̄ and such that M_{1}(X̄, L̄) is one-dimensional in h.

Proof. Since C̄ avoids the singularities of X̄, we can equally work with C̄ ⊂ Y. The curve D shall have components D_{1}, D_{2}, …, D_{n}, n = e + 2, with D_{1} [figure: ∼] F̄,

D_{2} [figure: ∼] σ̄, and D_{i} [figure] N̄, i ≥ 2, being the normalization. The gluing is defined according to the picture (cf. [1, 9, 12]):

[figure: D_{1}; D_{2}; D_{3}; D_{n}; …; σ̄; eN̄; F̄]

Obviously, D is of arithmetic genus one and h_{∗}(D) = C̄. Moreover, the assumptions of Proposition 2.3 are satisfied and hence M_{1}(Y) is of dimension one in [h : D [figure] Y]. □

Now consider a generic deformation

(4)  (X , L) [figure] S

of (X, L, f), i.e. (X_{0}, f_{0}) = (X, f) for a distinguished 0 ∈ S and for generic t ∈ S the fibre NS(X_{t}) has rank ρ = 9 with f_{t}-invariant part spanned by L_{t}. Taking quotients, one obtains a family of singular K3 surfaces X̄ [figure] S. Clearly, L = O(C) descends to the quotient X̄, for C is f-invariant and avoids the fixed points. Hence the line bundle L also descends to a relatively ample line bundle L̄. (The obstructions to deforming L resp. L̄ sideways are the same.) Note that for generic t ∈ S the line bundle L̄_{t} generates Pic(X̄_{t}). Let us apply the discussion of Section 2 to h : D [figure] X̄_{0} = X̄. So, consider the relative moduli space of stable maps of genus one M_{1}(X̄, L̄) [figure] S.

Corollary 3.2. The stable map h : D [figure] X̄ thus constructed deforms sideways to stable maps h_{t} : D_{t} [figure] X̄_{t}. Moreover, for generic t ∈ S the curve h_{t∗}(D_{t}) ⊂ X̄_{t} and its preimage in X_{t} are integral and disjoint from the singular locus resp. the fixed point set of f_{t}.

Proof. The existence of the deformation to the nearby fibres follows directly from Proposition 2.3 and Corollary 2.2. Since C̄ = h_{∗}(D) avoids the singularities of X̄, this will hold for generic t. Clearly, h_{t∗}(D_{t}) ∈ |L̄_{t}|. Therefore, since L_{t} generates the invariant part of NS(X_{t}) and hence L̄_{t} generates NS(X̄_{t}), the curve h_{t∗}(D_{t}) must be integral. Suppose the preimage C_{t} of h_{t∗}(D_{t}) were not integral for t generic, i.e. C_{t} = C′_{t} +C′′_{t} with f_{t}(C′_{t}) = C′′_{t}. (Use that f_{t} is an involution.) The two components would then specialize to C′ resp. C′′ on X with C = C′ + C′′ and f(C′) = C′′. We may assume that F ⊂ C′. But then also F = f(F) ⊂ f(C′) = C′′ which eventually yields the contradiction that F appears with multiplicity at least two in C. □

Remark 3.3. In fact, since the stable map h : D [figure] X deforms with the fibre component F in a one-dimensional family, also the deformations h_{t} : D_{t} [figure] X̄_{t} come in a family dominating X̄_{t}. Thus, one obtains a dominating family of integral genus one curves in the generic deformation X_{t} that are f_{t}-invariant and avoid the fixed points of f_{t}.

4. Proof of the main theorem

The outcome of the above construction are generic K3 surfaces X_{t} ∈ 𝔐Λ̃_{d} with a symplectic involution f_{t} such that X̄_{t} = X_{t}/⟨f_{t}⟩ contains a one-dimensional family of integral curves of geometric genus one that avoid the singular locus. This immediately leads to a proof of our main result.

Theorem 4.1. For all (X, f) ∈ 𝔐Λ̃, the symplectic involution f : X [figure: ∼] X acts as id on CH^{2}(X).

Proof. The case d = 2 follows from [14]. So we assume d = 2e > 2, i.e. e > 1. We first show that the above discussion combined with Proposition 1.1 proves the assertion for generic (X, f) ∈ 𝔐Λ̃_{d}. Consider a deformation (4) of the special elliptic K3 surface (3). Then for generic t ∈ S one has NS(X_{t}) = Λ̃_{d}. Indeed, by [4, Prop. 2.7] only in this case do all f_{t}-invariant line bundles actually descend to the quotient X̄_{t}. Hence the elliptic K3 surfaces described by (3) can be connected to the generic K3 surface parametrized by 𝔐Λ̃_{d}. Here we use that 𝔐Λ̃_{d} is connected. The generic fibre of the family (4) satisfies the assumption of Proposition 1.1. Indeed, by Corollary 3.2 and Remark 3.3 there exists a dominating family of integral curves of arithmetic genus one on the generic fibre X_{t} that are invariant under the involution and avoid the fixed points. Now consider an arbitrary (X, f) ∈ 𝔐Λ̃_{d}. Then any x ∈ X can be viewed as a specialization of points x_{t} in generic deformations (X_{t}, f_{t}) ∈ 𝔐Λ̃_{d}. Clearly, the points f_{t}(x_{t}) then specialize to f(x). For generic X_{t} we have already shown [x_{t}] = [f_{t}(x_{t})] in CH^{2}(X_{t}) and thus specialization yields [x] = [f(x)] in CH^{2}(X) for all x ∈ X. □

5. Further comments

We briefly outline how to adapt our techniques to the case of symplectic automorphisms of prime order. For p = 3, 5, and 7, Garbagnati and Sarti describe in [3, Thm. 4.1] lattices Ω_{p} of rank 12, 16, resp. 18 that are isomorphic to the anti-invariant part of f^{∗} acting on H^{2}(X, ℤ). Similar to the case p = 2, the generic polarized K3 surface (X, L) of degree 2d with a symplectic automorphism f : X [figure: ∼] X of order p leaving L fixed has Picard group isomorphic to Λ_{p,d} := ℤL ⊕ Ω_{p} or possibly, if d ≡ 0(p), isomorphic to a lattice Λ̃_{p,d} that contains Λ_{p,d} as a primitive sublattice of index p. In fact, the case Λ_{7,d} is not realized if d ≡ 0(7) (cf. [3, Prop. 5.2]), but unfortunately it is not known whether the lattices Λ̃_{p,d} are unique for given p and d ≡ 0(p) (see [3, Sec. 6]). The moduli spaces are of dimension 7, 3, resp. 1. Examples of symplectic automorphisms of order 3, 5, and 7 have been described in [3, Sec. 3.1]. They are again given by translation by a torsion section. The Picard numbers in these examples are 14, 18, resp. 20 and in each case they correspond to points in (at least) one component of the moduli space of polarized K3 surfaces (X, L) with a symplectic automorphism f of degree L^{2} = 2d. This leads to the following result:

Theorem 5.1. For p = 3, 5, or 7 and d = ep, there exists one component of the moduli space of polarized K3 surfaces (X, L) with a symplectic automorphism f : X [figure: ∼] X of order p and L^{2} = 2d such that Conjecture 0.1 holds true. □

It is very likely that for p = 7 and d ≡ 0(7) the result can be strengthened to cover all K3 surfaces, as we would expect that Λ̃_{7,d} is in fact unique.

References

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Daniel Huybrechts Mathematisches Institut Universität Bonn Endenicher Allee 60 53115 Bonn Germany huybrech@math.uni-bonn.de Michael Kemeny Mathematisches Institut Universität Bonn Endenicher Allee 60 53115 Bonn Germany michael.kemeny@gmail.com 518

Documenta Mathematica 18 (2013)