Symplectic Involutions of K3 Surfaces

Act Trivially on CH_{0}

Claire Voisin

Received: April 30, 2012 Revised: July 17, 2012

Communicated by Gavril Farkas

Abstract. A symplectic involution on a K3 surface is an involution which preserves the holomorphic 2-form. We prove that such a symplectic involution acts as the identity on the CH_{0} group of the K3 surface, as predicted by Bloch’s conjecture.

2010 Mathematics Subject Classification: 14C25, 14J28

Keywords and Phrases: zero cycles, Bloch’s conjecture, K3 surfaces

1 Introduction

For a smooth complex projective variety X, Mumford has shown in [9] that the triviality of the Chow group CH_{0}(X), i.e. CH_{0}(X)_{hom} = 0, implies the vanishing of holomorphic forms of positive degree on X. An immediate generalization is the fact that a 0-correspondence Γ ∈ CH^{d}(Y × X), with d = dim X, which induces the 0-map Γ_{∗} : CH_{0}(Y)_{hom} → CH_{0}(X)_{hom} has the property that the maps Γ^{∗} : H^{i,0}(X) → H^{i,0}(Y) vanish for i > 0. Bloch’s conjecture is a sort of converse to the above statement, but it needs the introduction of a certain filtration on CH_{0} groups of smooth projective varieties. The beginning of this conjectural filtration is

F^{0}CH_{0}(X) = CH_{0}(X), F^{1}CH_{0}(X) = CH_{0}(X)_{hom},  (1)
F^{2}CH_{0}(X) = CH_{0}(X)_{alb} := Ker (alb_{X} : CH_{0}(X)_{hom} → Alb(X)).

As the filtration is supposed to satisfy F^{k}CH^{0}(X) = 0 for k > dim X, we find that for surfaces, the filtration is fully determined by (1). Bloch’s conjecture for correspondences with values in surfaces is then the following:

Conjecture 1.1 Let S be a smooth projective surface, and let X be a smooth projective variety, Γ ∈ CH^{2}(X × S) be a correspondence such that the maps Γ^{∗} : H^{i,0}(S) → H^{i,0}(X) vanish for i > 0. Then

Γ_{∗} : CH_{0}(X)_{alb} → CH_{0}(S)_{alb}

vanishes.

This question can be addressed in particular in the case of self-correspondences associated to finite group actions on surfaces. A particular case of the conjecture above is the following:

Conjecture 1.2 Let G be a finite group acting on a smooth projective complex surface S with q = 0. Let χ : G → {1, −1} be a character. Assume that H^{2,0}(S)^{χ} = 0. Then CH_{0}(S)_{hom}^{χ} = 0.

Here

H^{2,0}(S)^{χ} := {ω ∈ H^{2,0}(S), g^{∗}ω = χ(g)ω, ∀g ∈ G},
CH_{0}(S)_{hom}^{χ} := {z ∈ CH_{0}(S)_{hom}, g^{∗}z = χ(g)z, ∀g ∈ G}.

This is indeed the particular case of the conjecture 1.1 applied to the 0correspondence

π_{χ} := ∑_{g∈G} χ(g)Γ_{g} ∈ CH^{2}(S × S),

where Γ_{g} ⊂ S × S is the graph of g. Conjecture 1.2 is proved in [13] in the situation where S is the zero set of a transverse section of a G-invariant vector bundle on any variety X with trivial Chow groups (that is CH^{∗}(X)_{hom} ⊗ ℚ = 0), under the assumption that E has many G-invariant sections. This generalizes our previous work in [12], where the case of the Godeaux action of ℤ/5ℤ on the CH_{0} group of invariant quintic surfaces was solved. This also covers the case (already considered in [12]) of the action of the involution i on ℙ^{3} acting with two −1 eigenvectors and two +1 eigenvectors on homogeneous coordinates, if we take for S a quartic surface defined by an i-invariant equation and we look at the antiinvariant part of CH_{0}(S). In the paper [5], Huybrechts proved that a derived autoequivalence of a K3 surface S acting as the identity on H^{∗}(S, ℤ) acts as the identity on CH_{0}(S). The next situation to consider is that of a symplectic finite order automorphism g of a K3 surface S. Thus g is by definition an automorphism of S such that g^{∗}ω = ω, where ω is the holomorphic 2-form on S. Such a g acts trivially on H^{2,0}(S) so it has trivial action on the transcendental lattice of S, so the difference

g^{∗} − Id ∈ Aut H^{∗}(S, ℤ)

Symplectic Involutions of K3 Surfaces Act … 853

is, at least over ℚ, induced by the cohomology class of a cycle of the form ∑_{i} α_{i}C_{i} ⊗ C′_{i}. where C_{i}, C′_{i} are curves on S and α_{i} are rational coefficients. It seems that if one could take the α_{i} to be integers, the above mentioned result of Huybrechts would apply to show that g_{∗} is the identity on CH_{0}(S). Still the problem remains open for these symplectic automorphisms and was explicitly raised by Huybrechts in [7]. In this note, the case of a symplectic involution i acting on a K3 surface S is considered. The fact that such symplectic involutions act trivially on CH_{0}(S) has been proved on one hand in a finite number of cases in [4], [12], [13], and on the other hand (and more significantly), it has been established in [6] for any K3 surface with symplectic involution in one of the three series introduced by van Geemen and Sarti [3] (each series contains itself an infinite number of families indexed by an integer d, and the three series differ first of all by the parity of this integer d, and secondly, when d is even, by the structure of the Néron-Severi lattice of the general such surface admitting an invariant line bundle of self-intersection 2d). The present paper solves the problem in general :

Theorem 1.3 Let S be an algebraic K3 surface, and let i : S → S be a symplectic involution. Then i_{∗} acts as the identity on CH_{0}(S).

The proof is elementary : It uses the fact that Prym varieties of étale double covers of curves of genus g are of dimension g − 1. This departure point is the obvious generalization of the starting point of Huybrechts and Kemeny’s work [6], who work with elliptic curves and their étale double covers. This observation is applied to the étale double covers of generic smooth ample curves C ⊂ S/i and allows us to prove in section 3 that the group of i-antiinvariant 0cycles on S is finite dimensional in the Roitman sense (the definition is recalled in section 2). One then uses a mild generalization (Theorem 2.3 established in section 2) of a fundamental result due to Roitman (cf. [10]) in order to conclude that the group of i-antiinvariant 0-cycles on S is in fact trivial.

2 Finite dimensionality in the sense of Roitman

Let X be a smooth (connected for simplicity) projective variety over ℂ, and let P ⊂ CH_{0}(X) be a subgroup.

Definition 2.1 We will say that P is finite dimensional in the Roitman sense if there exist a (nonnecessarily connected) smooth projective variety W, and a correspondence Γ ⊂ W ×X such that P is contained in the set {Γ_{∗}(w), w ∈ W }.

Remark 2.2 As P is a subgroup and the cycles Γ_{∗}(w) have finitely many possible degrees (depending on the connected component of W to which w belongs), we conclude that if P is finite dimensional in the Roitman sense, all elements of P have degree 0 (so P ⊂ CH_{0}(X)_{hom} as X is connected).

The following result is essentially due to Roitman. (It is in fact due to Roitman in the case where M = X and Im Z_{∗} = CH_{0}(X)_{hom}, see also [14], lecture 5). The proof we give below is slightly different, as it makes use of Proposition 2.4, while Roitman uses only elementary arguments. The proof given here also has the advantage that it does not need the torsion freeness of the group Ker (alb_{M} : CH_{0}(M)_{hom} → Alb M)). Let M and X be smooth connected projective varieties with X of dimension d. Let Z ∈ CH^{d}(M × X) be a correspondence.

Theorem 2.3 Assume that Im (Z_{∗} : CH_{0}(M) → CH_{0}(X)) is finite dimensional in the Roitman sense. Then the map Z_{∗} : CH_{0}(M)_{hom} → CH_{0}(X) factors through the Albanese morphism alb_{M} : CH_{0}(M)_{hom} → Alb M of M.

Proof. By definition, there exist a smooth projective variety W and a correspondence Γ ⊂ W ×X such that Im Z_{∗} is contained in the set {Γ_{∗}(w), w ∈ W }. Let C ⊂ M be a curve which is a very general complete intersection of sufficiently ample hypersurfaces H_{i} ⊂ M. Then by the Lefschetz theorem on hyperplane sections, the Jacobian J(C) maps surjectively to Alb(M) and the kernel K(C) is connected, hence an abelian variety. We will prove for completeness the following result:

Proposition 2.4 When the H_{i}’s are sufficiciently ample and very general, K(C) is a simple abelian variety.

We fix now C as above, satisfying the conclusion of Proposition 2.4 and let j : C → M be the inclusion, which induces the morphism j_{∗} : J(C) = CH_{0}(C)_{hom} → CH_{0}(M). We note that by taking the H_{i} sufficiently ample, the dimension of K(C) can be made arbitrarily large, so we may assume dim K > dim W. Let R ⊂ K(C) × W be the following set:

R = {(k, w) ∈ K(C) × W, Z_{∗}(j_{∗}(k)) = Γ_{∗}(w) in CH_{0}(X)}.

It is known (cf. [15, 10.1.1]) that R is a countable union of closed irreducible algebraic subsets R_{i} of K(C)×W. As Im Z_{∗} is contained in the set {Γ_{∗}(w), w ∈ W }, the union of the images of the first projections p_{|R_{i}} : R_{i} → K(C) is equal to K(C). A countability argument then shows that there exists an i such that

pr_{1|R_{i}} : R_{i} → K(C)

is dominating. It follows in particular that dim R_{i} ≥ dim K(C) > dim W. The fibers of the second projection

pr_{2|R_{i}} : R_{i} → W

are thus positive dimensional. Let w ∈ W, and F_{w} ⊂ K(C) be the fiber over w. Then F_{w} ⊂ K(C) is positive dimensional, hence it generates K(C) as a group because K(C) is simple. On the other hand, by definition of R, for any f ∈ F_{w}, we have Z_{∗}(j_{∗}(f)) = Γ_{∗}(w) in CH_{0}(X), hence the cycle Z_{∗}(j_{∗}(f)) is independent of f ∈ F_{W}. Thus for any 0-cycle z of F_{w}, we have Z_{∗}(j_{∗}(z)) = deg z Γ_{∗}(w) and it follows then from the fact that F_{w} generates K(C) as a group that Z_{∗} ∘ j_{∗} vanishes identically on K(C).

In order to conclude that Z_{∗} : CH_{0}(M)_{hom} → CH_{0}(X) factors through Alb M, we now observe the following: For any degree 0 cycle of M, which we write in the form z_{m} = z_{m}^{+} − z_{m}^{−} with z_{m}^{+} = ∑_{l≤k} m_{l}, z_{m}^{−} = ∑_{k+1≤l≤2k} m_{l}, with m = (m_{1}, …, m_{2k}) ∈ M^{2k}, we can blow up M at the points m_{i}, i = 1, …, 2k, which gives us τ : M′ → M, with exceptional divisors E_{i} over the points m_{i}. We can choose H ∈ Pic M so that L = τ^{∗}H − ∑_{i} E_{i} is ample on M′ and then apply Proposition 2.4 to an adequate multiple of L. We conclude that there is a complete intersection curve j : C ↪ M′ which satisfies the conclusion of Proposition 2.4, that is the kernel of the map

j_{∗} : J(C) → Alb M′ = Alb M

is a simple abelian variety of high dimension. Note that τ(C) contains all the points m_{i}. Assuming now that the 0-cycle z_{m} = z_{m}^{+} − z_{m}^{−} is annihilated by alb_{M}, any of its lifts z′ to M′ belongs to Ker alb_{M′}, hence choosing a lift z′ supported in C, we conclude that as a 0-cycle of C, z′ belongs to j_{∗}(K(C)). Applying the previous reasoning to the correspondence Z′ := Z ∘ τ between M′ and X, we conclude that Z_{∗}(z_{m}) = Z′_{∗}(z′) = 0 in CH_{0}(X).

Proof of Proposition 2.4. First of all, we reduce the problem to the case where M is a surface, by replacing M by a smooth complete intersection T = H_{1} ∩ … ∩ H_{m−2} of ample hypersurfaces and recalling that due to the Lefschetz theorem on hyperplane sections [15, 2.3.2], Alb M = Alb T. Now we take on T a Lefschetz pencil of very ample curves T_{t}, t ∈ ℙ^{1}. Picard-Lefschetz theory has for consequence (see [15, 3.2.3]) the irreducibility of the monodromy action ρ : π_{1}(ℙ_{reg}^{1}, t_{0}) → Aut H^{1}(T_{t_{0}}, ℚ)_{van}, where

H^{1}(T_{t_{0}}, ℚ)_{van} := Ker (H^{1}(T_{t_{0}}, ℚ) → H^{3}(T, ℚ)).

The same proof shows as well the irreducibility of the action of any finite index subgroup Γ ⊂ π_{1}(ℙ_{reg}^{1}, t_{0}). Assume by contradiction that for the general curve T_{t}, the abelian variety K(C_{t}) is not simple. Then there is a finite cover r : D → ℙ^{1}, and a proper sub-abelian fibration

A ⊂ K_{D},

where K_{D} → D_{reg} is the pull-back to D_{reg} := r^{−1}(ℙ_{reg}^{1}) of the family of abelian varieties K(C_{t}), t ∈ ℙ_{reg}^{1}. This sub-abelian fibration (taken up to isogenies) corresponds to a sub-local system 𝕃 of the pull-back to D_{reg} of the local system on ℙ_{reg}^{1} with fiber H^{1}(C_{t}, ℚ)_{van}. The monodromy action on ℙ_{reg}^{1} being irreducible on any finite index subgroup of π_{1}(ℙ_{reg}^{1}, t_{0}), it is irreducible on the image r_{∗}(π_{1}(D_{reg}, s_{0})), r(s_{0}) = t_{0}. This contradicts the existence of 𝕃.

In the next section, we will prove the following:

Proposition 2.5 Let S be an algebraic K3 surface, and let i : S → S be a symplectic involution. Then the antiinvariant part

CH_{0}(S)^{−} = {z ∈ CH_{0}(S), i_{∗}(z) = −z}

is finite dimensional in the Roitman sense.

Proof of Theorem 1.3 We apply Theorem 2.3 to the case where X = S, M = S and Z is the cycle Δ_{S} − Graph(i). Here Δ_{S} is the diagonal of S and Graph(i) is the graph of i. Proposition 2.5 says that Im Z_{∗} is finite dimensional in the Roitman sense and Theorem 2.3 tells us then that Z_{∗} : CH_{0}(S)_{hom} → CH_{0}(S)_{hom} factors through Alb S = 0. Hence Z_{∗} vanishes on CH_{0}(S)_{hom}. On the other hand, Z_{∗} is multiplication by 2 on CH_{0}(S)^{−} ⊂ CH_{0}(S)_{hom} and we thus proved that CH_{0}(S)^{−} is a 2-torsion group; as CH_{0}(S) has no torsion by [11], we conclude that CH_{0}(S)^{−} = 0. Thus Z_{∗} = Id on CH_{0}(S).

3 Proof of Proposition 2.5

We start with the following lemma: Let M, X be smooth projective varieties with dim X = d. Let Γ ∈ CH^{d}(M × X) be a correspondence. Each point (m_{1}, …, m_{k}) ∈ M^{k} determines an element ∑_{i} m_{i} ∈ CH_{0}(M). Hence we get a map

Γ_{∗} : M^{k} → CH_{0}(X).

Lemma 3.1 Assume there is a point m ∈ M such that Γ_{∗}(m) = 0 in CH_{0}(X) and for some integer g > 0, one has Γ_{∗}(M^{g−1}) = Γ_{∗}(M^{g}) as subsets of CH_{0}(X). Then Im Γ_{∗} is finite dimensional in the Roitman sense.

Proof. Since Γ_{∗}(M^{g−1}) = Γ_{∗}(M^{g}), it is obvious by induction that Γ_{∗}(M^{g−1}) = Γ_{∗}M^{k} for any k ≥ g − 1. Any cycle z ∈ CH_{0}(M) can be written as z^{+} − z^{−}, where z^{+} and z^{−} are effective cycles, of degree k^{+}, k^{−}. Up to adding the adequate multiples of m to z^{+} and z^{−}, which does not change Γ_{∗}z, we may assume that k^{+} = k^{−} ≥ g. Thus Γ_{∗}(z) = Γ_{∗}(z^{+}) − Γ_{∗}(z^{−}), where Γ_{∗}(z^{+}) and Γ_{∗}(z^{−}) belong to Γ_{∗}(M^{k}) = Γ_{∗}(M^{g−1}). Hence we proved that the correspondence Γ′ ∈ CH^{d}(M^{2g−2} × X), defined as

Γ′ = ∑_{i≤g−1}(pr_{i}, p_{X})^{∗}Γ − ∑_{g≤i≤2g−2}(pr_{i}, p_{X})^{∗}Γ

satisfies

Im Γ_{∗} = Γ′_{∗}(M^{2g−2}).

According to Definition 2.1, Im Γ_{∗} is finite dimensional in the Roitman sense.

Proof of Proposition 2.5. Let S be a K3 surface endowed with a symplectic involution i. The quotient surface Σ = S/i is a singular K3 surface. (By blowing-up its singular points, which correspond to the fixed points of i, it becomes a honest K3 surface.) The canonical bundle of Σ (or rather Σ_{reg}) is trivial. Let L ∈ Pic Σ be very ample, and let 2g − 2 = deg c_{1}(L)^{2}. By triviality of K_{Σ_{reg}}, g is the genus of the smooth curves in |L|. Furthermore, we have dim |L| = g, due to the exact sequence

0 → ℂ → H^{0}(Σ, L) → H^{0}(C, L_{|C}) = H^{0}(C, K_{C}) → 0,

which comes from the similar exact sequence on the desingularization Σ̃ of Σ, which has H^{1}(Σ̃, OΣ̃) = 0. Note also that for a smooth ample curve C ⊂ Σ, the inverse image C̃ ⊂ S is smooth, connected, and is an étale double cover of C. (Only the connectedness is to be proved, and this follows from the fact that otherwise each component C_{1}, C_{2} of C̃ ⊂ S has positive self-intersection and C_{1}·C_{2} = 0 since C̃ is smooth. This contradicts the Hodge index theorem.) Let Γ ∈ CH^{2}(S × S) be the correspondence Δ_{S} − Graph(i). We prove now the following, where c_{S} is the effective 0-cycle of degree 1 introduced in [1]:

Claim 3.2 We have Γ_{∗}(c_{S}) = 0 and Γ_{∗}(S^{g}) = Γ_{∗}(S^{g−1}).

According to Lemma 3.1, this proves Proposition 2.5, since CH_{0}(S)^{−} = Im Γ_{∗}. (The last fact follows from the fact that Γ_{∗} acts as −2 Id on CH_{0}(S)^{−}, which is a divisible group.)

Proof of the claim. The cycle c_{S} is obviously i-invariant since it is the class of any point of S belonging to a rational curve D ⊂ S, and if x ∈ D then i(x) ∈ i(D) also belongs to a rational curve in S. Let s = (s_{1}, …, s_{g}) be a general point of S^{g}. Then if we denote by σ_{i} the image of s_{i} in Σ = S/i, the g-uple (σ_{1}, …, σ_{g}) is generic in Σ^{g} and there exists a unique curve C_{s} ∈ |L| containing all the σ_{i}’s. The curve C_{s} being general in |L|, it is smooth and thus we have the étale double cover C̃_{s} → C_{s}, with C̃_{s} ⊂ S containing the points s_{i}. Consider the 0-cycle

z_{s} = ∑_{l} s_{l} − i(∑_{l} s_{l}) = Γ_{∗}(∑_{l} s_{l}) ∈ CH_{0}(S).

This cycle clearly depends only on the Abel image

albC̃_{s}(∑_{l} s_{l} − i(∑_{l} s_{l})),

which is an antiinvariant element of J(C̃_{s}) or, up to 2-torsion, an element of the Prym variety P (C̃_{s}/C_{s}) which is a g − 1-dimensional abelian variety.

In other words, we find that, on a Zariski open set U of S^{g}, the map

S^{g} → CH_{0}(S)^{−}, (s_{1}, …, s_{g}) ↦ z_{s},

factors through the morphism

f : U → P(C̃/C), (s_{1}, …, s_{g}) ↦ albC̃_{s}(s_{1} + … + s_{g} − i(s_{1}) − … − i(s_{g})),

where C → |L|_{0} is the universal smooth curve over the Zariski open set |L|_{0} of |L| parameterizing smooth curves, C̃ → |L|_{0} is the universal family of double covers, and P(C̃/C) → |L|_{0} is the corresponding Prym fibration. The total space of the Prym fibration P(C̃/C) has dimension 2g−1, while U has dimension 2g, so the morphism f has positive dimensional fibers. It follows that for s ∈ U, there is a curve F_{s} ⊂ S^{g} such that the 0-cycle z_{t} = ∑_{l} t_{l} − i(∑_{l} t_{l}) is rationally equivalent to z_{s} in S for any (t_{1}, …, t_{g}) ∈ F_{s}. Choose an ample curve D ⊂ S whose irreducible components are rational (the existence of such a curve is well-known and due to Mori-Mukai, cf. [8]). The curve F_{s} meets the ample divisor ∑_{l} pr_{l}^{−1}(D), where pr_{l} : S^{g} → S is the l-th projection. Hence the 0-cycle z_{s} is rationally equivalent to a 0-cycle of the form z_{t} = ∑_{l} t_{l} − i(∑_{l} t_{l}), where we have t_{l_{0}} ∈ D for some l_{0}. We have seen already that the 0-cycle t_{l_{0}} − i(t_{l_{0}}) vanishes in CH_{0}(S) and it follows that z_{s} is rationally equivalent to the cycle ∑_{l≠l_{0}} t_{l} − i(∑_{l} t_{l≠l_{0}}). Thus z_{s} ∈ Γ_{∗}(S^{g−1}) for s = (s_{1}, …, s_{g}) ∈ U. To conclude the proof, we have to show that the above result is true for any (s_{1}, …, s_{g}) ∈ S^{g}. This follows from the following statement :

Fact 3.3 Let Y be a connected complex projective variety. Let U ⊂ Y be the complement of a countable union of proper closed algebraic subsets. Then any 0-cycle z of Y is rationally equivalent in Y to a 0-cycle supported on U.

A proof of Fact 3.3 is as follows: there exists a curve C ⊂ Y which is irreducible, contains Sup z and intersects U non-trivially. Then C \ C ∩ U is countable. It suffices to prove that there exists a 0-cycle z′ of C supported on C ∩ U which is rationally equivalent to z on C. We may assume that C is smooth by taking normalization if necessary. Then we write z = z_{1} − z_{2} in Pic C, where z_{1} and z_{2} are very ample divisors on C. Since [z_{1}| and |z_{2}| are base-point free, there exist members z′_{1} ∈ |z_{1}|, z′_{2} ∈ |z_{2}| which avoid the countably many points in C \C ∩U, hence are supported on C ∩U. Then z = z′_{1} −z′_{2} in Pic C = CH_{0}(C).

We apply this statement to Y = S^{g} to conclude that the cycles z_{s} for s = (s_{1}, …, s_{g}) ∈ U fill-in the image Γ_{∗}(S^{g}). We thus conclude that z_{s} ∈ Γ_{∗}(S^{g−1}) for any s = (s_{1}, …, s_{g}) ∈ S^{g} since we already know the result for s ∈ U. Proposition 2.5 is thus proved.

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Claire Voisin Institut de Mathématiques de Jussieu Equipe Topologie et Géométrie algébriques Case 247, 4 Place Jussieu, 75005 Paris, France voisin@math.jussieu.fr