#  Bloch’s conjecture for Campedelli and Barlow surfaces

Claire Voisin
CNRS and École Polytechnique

###### Abstract

Campedelli surfaces are regular surfaces of general type with $p_{g}=0$. They
specialize to double covers of Barlow surfaces. We prove that the $CH_{0}$
group of a Campedelli surface is equal to $\mathbb{Z}$, which implies the same
result for the Barlow surfaces.

À la mémoire de Friedrich Hirzebruch

##  0 Introduction

In this paper, we establish an improved version of the main theorem of [27]
and use it in order to prove the Bloch conjecture for Campedelli surfaces. We
will also give a conditional application (more precisely, assuming the
variational Hodge conjecture) of the same method to the Chow motive of low
degree $K3$ surfaces.

Bloch’s conjecture for $0$-cycles on surfaces states the following:

######  Conjecture 0.1.

(cf. [7]) Let $\Gamma\in CH^{2}(Y\times X)$, where $Y$ is smooth projective
and $X$ is a smooth projective surface. Assume that
$[\Gamma]^{*}:H^{2,0}(X)\rightarrow H^{2,0}(Y)$ vanishes. Then

$\Gamma_{*}:CH_{0}(Y)_{alb}\rightarrow CH_{0}(X)_{alb}$

vanishes.

Here $[\Gamma]\in H^{4}(X\times X,\mathbb{Q})$ is the cohomology class of
$\Gamma$ and

$CH_{0}(Y)_{alb}:={\rm Ker}\,(CH_{0}(Y)_{hom}\stackrel{{\scriptstyle
alb_{Y}}}{{\rightarrow}}Alb(Y)).$

Particular cases concern the case where $\Gamma$ is the diagonal of a surface
$X$ with $q=p_{g}=0$. Then $\Gamma_{*}=Id_{CH_{0}(X)_{alb}}$, so that the
conjecture predicts $CH_{0}(X)_{alb}=0$. This statement is known to hold for
surfaces which are not of general type by [9], and for surfaces of general
type, it is known to hold by Kimura [17] if the surface $X$ is furthermore
rationally dominated by a product of curves (cf. [5] for many such examples).
Furthermore, for several other families of surfaces of general type with
$q=p_{g}=0$, it is known to hold either for the general member of the family
(eg. the Godeaux surfaces, cf. [28]), or for specific members of the family
(for example the Barlow surface [4]).

A slightly more general situation concerns surfaces equipped with the action
of a finite group $G$. This has been considered in the paper [27], where the
following theorem concerning group actions on complete intersection surfaces
is proved: Let $X$ be a smooth projective variety with trivial Chow groups
(i.e. the cycle class map $CH_{i}(X)_{\mathbb{Q}}\rightarrow
H^{2n-2i}(X,\mathbb{Q}),\,n={\rm dim}\,X$ is injective for all $i$). Let $G$
be a finite group acting on $X$ and let $E$ be a $G$-equivariant rank $n-2$
vector bundle on $X$ which has “enough” $G$-invariant sections (for example,
if the group action is trivial, one asks that $E$ is very ample). Let
$\pi\in\mathbb{Q}[G]$ be a projector. Then $\pi$ gives a self-correspondence
$\Gamma^{\pi}$ with $\mathbb{Q}$-coefficients (which is a projector) of the
$G$-invariant surfaces $S=V(\sigma),\,\sigma\in H^{0}(X,E)^{G}$ (cf. Section
1). We use the notation

$H^{2,0}(S)^{\pi}:={\rm Im}\,([\Gamma^{\pi}]^{*}:H^{2,0}(S)\rightarrow
H^{2,0}(S)),$ $CH_{0}(S)_{\mathbb{Q},hom}^{\pi}:={\rm
Im}\,([\Gamma^{\pi}]_{*}:CH_{0}(S)_{\mathbb{Q},hom}\rightarrow
CH_{0}(S)_{\mathbb{Q},hom})).$

######  Theorem 0.2.

Assume that the smooth surfaces $S=V(\sigma),\,\sigma\in H^{0}(E)^{G}$,
satisfy $H^{2,0}(S)^{\pi}=0$. Then we have
$CH_{0}(S)_{\mathbb{Q},hom}^{\pi}=0$.

Note that the Bloch conjecture for finite group actions on surfaces which do
not fit at all in the above geometric setting, namely finite order symplectic
automorphisms of $K3$ surfaces, has been recently proved in [15] and [30] by
completely different methods. For these symplectic automorphisms, one
considers the cycle $\Delta_{X}-\frac{1}{|G|}\sum_{g\in G}{\rm Graph}\,g$,
which acts as the identity minus the projector onto the $G$-invariant part,
and one proves that it acts as $0$ on $CH_{0}(X)_{hom}$ (in fact on the whole
of $CH_{0}$) according to Conjecture 0.1.

Theorem 0.2 is rather restrictive geometrically, due to the fact that not only
we consider $0$-sets of sections of a vector bundle, but also we impose this
very ampleness assumption on the vector bundle. Our first result in this paper
is a relaxed version of this theorem, which works in a much more general
geometric context and will be applicable in particular to the case of
Campedelli surfaces.

Let $\mathcal{S}\rightarrow B$ be a smooth projective morphism with two
dimensional connected fibers, where $B$ is quasi-projective. Let $\Gamma\in
CH^{2}(\mathcal{S}\times_{B}\mathcal{S})_{\mathbb{Q}}$ be a relative $0$-self
correspondence. Let
$\Gamma_{t}:=\Gamma_{\mid\mathcal{S}_{t}\times\mathcal{S}_{t}}$ be the
restricted cycle, with cohomology class $[\Gamma_{t}]\in
H^{4}(\mathcal{S}_{t}\times\mathcal{S}_{t},\mathbb{Q})$. We have the actions

$\Gamma_{t*}:CH_{0}(\mathcal{S}_{t})_{\mathbb{Q}}\rightarrow
CH_{0}(\mathcal{S}_{t})_{\mathbb{Q}},\,\,[\Gamma_{t}]^{*}:H^{i,0}(\mathcal{S}_{t})\rightarrow
H^{i,0}(\mathcal{S}_{t}).$

######  Theorem 0.3.

Assume the following:

(1) The fibers $\mathcal{S}_{t}$ satisfy $h^{1,0}(\mathcal{S}_{t})=0$ and
$[\Gamma_{t}]^{*}:H^{2,0}(\mathcal{S}_{t})\rightarrow
H^{2,0}(\mathcal{S}_{t})$ is equal to zero.

(2) A smooth projective (equivalently any smooth projective) completion
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ of the fibered self-product
$\mathcal{S}\times_{B}\mathcal{S}$ is rationally connected.

Then $\Gamma_{t*}:CH_{0}(\mathcal{S}_{t})_{hom}\rightarrow
CH_{0}(\mathcal{S}_{t})_{hom}$ is nilpotent for any $t\in B$.

This statement is both weaker and stronger than Theorem 0.2 since on the one
hand, the conclusion only states the nilpotence of $\Gamma_{t*}$, and not its
vanishing, while on the other hand the geometric context is much more flexible
and the assumption on the total space of the family is much weaker.

In fact the nilpotence property is sufficient to imply the vanishing in a
number of situations which we describe below. The first situation is the case
where we consider a family of surfaces with $h^{2,0}=h^{1,0}=0$. Then we get
the following consequence (the Bloch conjecture for surfaces with $q=p_{g}=0$
under assumption (2) below):

######  Corollary 0.4.

Let $\mathcal{S}\rightarrow B$ be a smooth projective morphism with two
dimensional connected fibers, where $B$ is quasi-projective. Assume the
following:

(1) The fibers $\mathcal{S}_{t}$ satisfy
$H^{1,0}(\mathcal{S}_{t})=H^{2,0}(\mathcal{S}_{t})=0$.

(2) A projective completion (or, equivalently, any projective completion)
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ of the fibered self-product
$\mathcal{S}\times_{B}\mathcal{S}$ is rationally connected.

Then $CH_{0}(\mathcal{S}_{t})_{hom}=0$ for any $t\in B$.

We refer to Section 1, Theorem 1.6 for a useful variant involving group
actions, which allows to consider many more situations (cf. [27] and Section 2
for examples).

######  Remark 0.5.

The proof will show as well that we can replace assumption (2) in these
statements by the following one:

(2’) A (equivalently any) smooth projective completion
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ of the fibered self-product
$\mathcal{S}\times_{B}\mathcal{S}$ has trivial $CH_{0}$ group.

However, it seems more natural to put a geometric assumption on the total
space since this is in practice much easier to check.

In the second section of this paper, we will apply these results to prove
Bloch’s conjecture 0.1 for Campedelli surfaces (cf. [11], [25], [10]).
Campedelli surfaces can be constructed starting from a $5\times 5$ symmetric
matrix $M(a),\,a\in\mathbb{P}^{11}$, of linear forms on $\mathbb{P}^{3}$
satisfying certain conditions (cf. (3)) making their discriminant invariant
under the Godeaux action (4) of $\mathbb{Z}/5\mathbb{Z}$ on $\mathbb{P}^{3}$.
The general quintic surface $V(a)$ defined by the determinant of $M(a)$ has
$20$ nodes corresponding to the points $x\in\mathbb{P}^{3}$ where the matrix
$M(a,x)$ has rank $3$, and it admits a double cover $S(a)$ which is étale away
from the nodes, and to which the $\mathbb{Z}/5\mathbb{Z}$-action lifts. Then
the Campedelli surface $\Sigma(a)$ is the quotient of $S(a)$ by this lifted
action.

Campedelli surfaces have a $4$-dimensional moduli space. For our purpose, the
geometry of the explicit $11$-dimensional parameter space described in [25] is
in fact more important than the structure of the moduli space.

######  Theorem 0.6.

Let $S$ be a Campedelli surface. Then $CH_{0}(S)=\mathbb{Z}$.

The starting point of this work was a question asked by the authors of [10]:
They needed to know that the Bloch conjecture holds for a simply connected
surface with $p_{g}=0$ (eg a Barlow surface [3]) and furthermore, they needed
it for a general deformation of this surface. The Bloch conjecture was proved
by Barlow [4] for some Barlow surfaces admitting an extra group action
allowing to play on group theoretic arguments as in [16], but it was not known
for the general Barlow surface.

Theorem 0.6 implies as well the Bloch conjecture for the Barlow surfaces since
the Barlow surfaces can be constructed as quotients of certain Campedelli
surfaces admitting an extra involution, namely the determinantal equation
defining $V(a)$ has to be invariant under the action of the dihedral group of
order $10$ (cf. [25]). The Campedelli surfaces appearing in this construction
of Barlow surfaces have only a $2$-dimensional moduli space. It is interesting
to note that we get the Bloch conjecture for the general Barlow surface via
the Bloch conjecture for the general Campedelli surface, but that our strategy
does not work directly for the Barlow surface, which has a too small parameter
space (cf. [11], [25]).

The third section of this paper applies Theorem 0.3 to prove a conditional
result on the Chow motive of $K3$ surfaces which can be realized as $0$-sets
of sections of a vector bundle on a rationally connected variety (cf. [20],
[21]). Recall that the Kuga-Satake construction (cf. [18], [13]) associates to
a polarized $K3$ surface $S$ an abelian variety $K(S)$ with the property that
the Hodge structure on $H^{2}(S,\mathbb{Q})$ is a direct summand of the Hodge
structure of $H^{2}(K(S),\mathbb{Q})$. The Hodge conjecture predicts that the
corresponding degree $4$ Hodge class on $S\times K(S)$ is algebraic. This is
not known in general, but this is established for $K3$ surfaces with large
Picard number (cf. [19], [23]). The next question concerns the Chow motive (as
opposed to the numerical motive) of these $K3$ surfaces. The Kuga-Satake
construction combined with the Bloch conjecture implies that the Chow motive
of a $K3$ surface is a direct summand of the Chow motive of its Kuga-Satake
variety. In this direction, we prove the following Theorem 0.7: Let $X$ be a
rationally connected variety of dimension $n$ and let $E\rightarrow X$ be a
rank $n-2$ globally generated vector bundle satisfying the following
properties:

(i) The restriction map $H^{0}(X,E)\rightarrow H^{0}(z,E_{\mid z})$ is
surjective for general $z=\\{x,\,y\\}\subset X$.

(ii) The general section $\sigma$ vanishing at two general points $x,\,y$
determines a smooth surface $V(\sigma)$.

We consider the case where the surfaces $S=V(\sigma)$ are algebraic $K3$
surfaces. For example, this is the case if ${\rm det}\,E=-K_{X}$, and the
surfaces $S=V(\sigma)$ for general $\sigma\in H^{0}(X,E)$ have irregularity
$0$. Almost all general algebraic $K3$ surfaces of genus $\leq 20$ have been
described this way by Mukai (cf. [20], [21]), where $X$ is a homogeneous
variety with Picard number $1$. Many more examples can be constructed starting
from an $X$ with Picard number $\geq 2$.

######  Theorem 0.7.

Assume the variational Hodge conjecture in degree $4$. Then the Chow motive of
a $K3$ surface $S$ as above is a direct summand of the Chow motive of an
abelian variety.

######  Remark 0.8.

The variational Hodge conjecture for degree $4$ Hodge classes is implied by
the Lefschetz standard conjecture in degrees $2$ and $4$. It is used here only
to conclude that the Kuga-Satake correspondence is algebraic for any $S$ as
above. Hence we could replace the variational Hodge conjecture by the
Lefschetz standard conjecture or by the assumption that the cohomological
motive of a general $K3$ surfaces $S$ in our family is a direct summand of the
cohomological motive of an abelian variety. The contents of the theorem is
that we then have the same result for the Chow motive.

As a consequence of this result, we get the following (conditional)
corollaries.

######  Corollary 0.9.

With the same assumptions as in Theorem 0.7, let $S$ be a member of the family
of $K3$ surfaces parameterized by $\mathbb{P}(H^{0}(X,E))$, and let $\Gamma\in
CH^{2}(S\times S)$ be a correspondence such that
$[\Gamma]^{*}:H^{2,0}(S)\rightarrow H^{2,0}(S)$ is zero. Then
$Z_{*}:CH_{0}(S)_{hom}\rightarrow CH_{0}(S)_{hom}$ is nilpotent.

######  Remark 0.10.

Note that there is a crucial difference between Theorem 0.3 and Corollary 0.9:
In Corollary 0.9, the cycle $\Gamma$ is not supposed to exist on the general
deformation $S_{t}$ of $S$. (Note also that the result in Corollary 0.9 is
only conditional since we need the Lefschetz standard conjecture, or at least
to know that the Kuga-Satake correspondence is algebraic for general $S_{t}$,
while Theorem 0.3 is unconditional!)

######  Corollary 0.11.

With the same assumptions as in Theorem 0.7, the transcendental part of the
Chow motive of any member of the family of $K3$ surfaces parameterized by
$\mathbb{P}(H^{0}(X,E))$ is indecomposable, that is, any submotive of it is
either the whole motive or the $0$-motive.

Thanks. I thank Christian Böhning, Hans-Christian Graf von Bothmer, Ludmil
Katzarkov and Pavel Sosna for asking me the question whether general Barlow
surfaces satisfy the Bloch conjecture, and for providing references on Barlow
versus Campedelli surfaces.

##  1 Proof of Theorem 0.3 and some consequences

This section is devoted to the proof of Theorem 0.3 and its consequences
(Corollary 0.4 or its more general form Theorem 1.6).

The proof will follow essentially the idea of [27]. The main novelty in the
proof lies in the use of Proposition 1.3. For completeness, we also outline
the arguments of [27], restricted to the surface case.

Consider the codimension $2$-cycle

$\Gamma\in CH^{2}(\mathcal{S}\times_{B}\mathcal{S})_{\mathbb{Q}}.$

Assumption (1) tells us that the restricted cycle
$\Gamma_{t}:=\Gamma_{\mid\mathcal{S}_{t}\times\mathcal{S}_{t}}$ is
cohomologous to the sum of a cycle supported on a product of (nonnecessarily
irreducible curves) in $\mathcal{S}_{t}$ and of cycles pulled-back from
$CH_{0}(\mathcal{S}_{t})$ via the two projections. We deduce from this (cf.
[27, Prop. 2.7]):

######  Lemma 1.1.

There exist a codimension $1$ closed algebraic subset
$\mathcal{C}\subset\mathcal{S}$, a codimension $2$ cycle $\mathcal{Z}$ on
$\mathcal{S}\times_{B}\mathcal{S}$ with $\mathbb{Q}$-coefficients supported on
$\mathcal{C}\times_{B}\mathcal{C}$, and two codimension $2$ cycles
$\mathcal{Z}_{1},\,\mathcal{Z}_{2}$ with $\mathbb{Q}$-coefficients on
$\mathcal{S}$, such that the cycle

$\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}_{1}-p_{2}^{*}\mathcal{Z}_{2}$

has its restriction to each fiber $\mathcal{S}_{t}\times\mathcal{S}_{t}$
cohomologous to $0$, where
$p_{1},\,p_{2}:\mathcal{S}\times_{B}\mathcal{S}\rightarrow\mathcal{S}$ are the
two projections.

This lemma is one of the key observations in [27]. The existence of the data
above is rather clear after a generically finite base change
$B^{\prime}\rightarrow B$ because it is true fiberwise. The key point is that,
working with cycles with $\mathbb{Q}$-coefficients, we can descend to $B$ and
hence, do not need in fact this base change which would ruin assumption (2).

The next step consists in passing from the fiberwise cohomological equality

$[\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}_{1}-p_{2}^{*}\mathcal{Z}_{2}]_{\mid\mathcal{S}_{t}\times\mathcal{S}_{t}}=0\,\,{\rm
in}\,\,H^{4}(\mathcal{S}_{t}\times\mathcal{S}_{t},\mathbb{Q})$

to the following global vanishing:

######  Lemma 1.2.

(cf. [27, Lemma 2.12]) There exist codimension $2$ algebraic cycles
$\mathcal{Z}^{\prime}_{1}$, $\mathcal{Z}^{\prime}_{2}$ with
$\mathbb{Q}$-coefficients on $\mathcal{S}$ such that

$[\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}^{\prime}_{1}-p_{2}^{*}\mathcal{Z}^{\prime}_{2}]=0\,\,{\rm
in}\,\,H^{4}(\mathcal{S}\times_{B}\mathcal{S},\mathbb{Q}).$

The proof of this lemma consists in the study of the Leray spectral sequence
of the fibration $p:\mathcal{S}\times_{B}\mathcal{S}\rightarrow B$. We know
that the class
$[\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}_{1}-p_{2}^{*}\mathcal{Z}_{2}]$
vanishes in the Leray quotient $H^{0}(B,R^{4}p_{*}\mathbb{Q})$ of
$H^{4}(\mathcal{S}\times_{B}\mathcal{S},\mathbb{Q})$. It follows that it is of
the form $p_{1}^{*}\alpha_{1}+p_{2}^{*}\alpha_{2}$, for some rational
cohomology classes $\alpha_{1},\,\alpha_{2}$ on $\mathcal{S}$. One then proves
that $\alpha_{i}$ can be chosen to be algebraic on $\mathcal{S}$.

The new part of the argument appears in the following proposition:

######  Proposition 1.3.

Under assumption (2), the following hold :

(i) The cycle
$\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}^{\prime}_{1}-p_{2}^{*}\mathcal{Z}^{\prime}_{2}$
is algebraically equivalent to $0$ on $\mathcal{S}\times_{B}\mathcal{S}$.

(ii) The restriction to the fibers $\mathcal{S}_{t}\times\mathcal{S}_{t}$ of
the codimension $2$ cycle
$\mathcal{Z}^{\prime}=\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}^{\prime}_{1}-p_{2}^{*}\mathcal{Z}^{\prime}_{2}$
is a nilpotent element (with respect to the composition of self-
correspondences) of
$CH^{2}(\mathcal{S}_{t}\times\mathcal{S}_{t})_{\mathbb{Q}}$.

Proof. We work now with a smooth projective completion
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$. Let
$D:=\overline{\mathcal{S}\times_{B}\mathcal{S}}\setminus{\mathcal{S}\times_{B}\mathcal{S}}$
be the divisor at infinity. Let $\widetilde{D}\stackrel{{\scriptstyle
j}}{{\rightarrow}}\overline{\mathcal{S}\times_{B}\mathcal{S}}$ be a
desingularization of $D$. The codimension $2$ cycle $\mathcal{Z}^{\prime}$
extends to a cycle $\overline{\mathcal{Z}^{\prime}}$ over
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$. We know from Lemma 1.2 that

$[\overline{\mathcal{Z}^{\prime}}]=0\,\,{\rm
in}\,\,H^{4}(\overline{\mathcal{S}\times_{B}\mathcal{S}},\mathbb{Q})$

and this implies by [31, Prop. 3] that there is a degree $2$ Hodge class
$\alpha$ on $\widetilde{D}$ such that

$j_{*}\alpha=[\overline{\mathcal{Z}^{\prime}}]\,\,{\rm
in}\,\,H^{4}(\overline{\mathcal{S}\times_{B}\mathcal{S}},\mathbb{Q}).$

By the Lefschetz theorem on $(1,1)$-classes, $\alpha$ is the class of a
codimension $1$ cycle $\mathcal{Z}^{\prime\prime}$ of $\widetilde{D}$ and we
conclude that

$[\overline{\mathcal{Z}^{\prime}}-j_{*}\mathcal{Z}^{\prime\prime}]=0\,\,{\rm
in}\,\,H^{4}(\overline{\mathcal{S}\times_{B}\mathcal{S}},\mathbb{Q}).$

We use now assumption (2) which says that the variety
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ is rationally connected. It has
then trivial $CH_{0}$, and so any codimension $2$ cycle homologous to $0$ on
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ is algebraically equivalent to
$0$ by the following result to to Bloch and Srinivas:

######  Theorem 1.4.

[8] On a smooth projective variety $X$ with $CH_{0}(X)$ supported on a
surface, homological equivalence and algebraic equivalence coincide for
codimension $2$ cycles.

We thus conclude that
$\overline{\mathcal{Z}^{\prime}}-j_{*}\mathcal{Z}^{\prime\prime}$ is
algebraically equivalent to $0$ on
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$, hence that
$\mathcal{Z}^{\prime}=(\overline{\mathcal{Z}^{\prime}}-j_{*}\mathcal{Z}^{\prime\prime})_{\mid\mathcal{S}\times_{B}\mathcal{S}}$
is algebraically equivalent to $0$ on ${\mathcal{S}\times_{B}\mathcal{S}}$.

(ii) This is a direct consequence of (i), using the following nilpotence
result proved independently in [26] and [29]:

######  Theorem 1.5.

On any smooth projective variety, self-correspondences algebraically
equivalent to $0$ are nilpotent for the composition of correspondences.

Proof of Theorem 0.3. Using the same notations as in the previous steps, we
know by Proposition 1.3 that under assumptions (1) and (2), the self-
correspondence

$\displaystyle\Gamma_{t}-\mathcal{Z}_{t}-p_{1}^{*}\mathcal{Z}^{\prime}_{1,t}-p_{2}^{*}\mathcal{Z}^{\prime}_{2,t}$
(1)

on $\mathcal{S}_{t}$ with $\mathbb{Q}$-coefficients is nilpotent. In
particular, the morphism it induces at the level of Chow groups is nilpotent.
On the other hand, recall that $\mathcal{Z}_{t}$ is supported on a product of
curves on $\mathcal{S}_{t}$, hence acts trivially on
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q}}$. Obviously, both cycles
$p_{1}^{*}\mathcal{Z}^{\prime}_{1,t},\,p_{2}^{*}\mathcal{Z}^{\prime}_{2,t}$
act trivially on $CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}$. Hence the self-
correspondence (1) acts as $\Gamma_{t*}$ on
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}$.

Let us now turn to our main application, namely Corollary 0.4, or a more
general form of it which involves a family of surfaces $S_{t}$ with an action
of a finite group $G$ and a projector $\pi\in\mathbb{Q}[G]$. Writing such a
projector as $\pi=\sum_{g\in G}a_{g}g$, $a_{g}\in\mathbb{Q}$, such a projector
provides a codimension $2$ cycle

$\displaystyle\Gamma^{\pi}_{t}=\sum_{g}a_{g}{\rm Graph}\,g\in
CH^{2}(S_{t}\times S_{t})_{\mathbb{Q}},$ (2)

with actions

${\Gamma^{\pi}_{t}}^{*}=\sum_{g}a_{g}g^{*}$

on the holomorphic forms of $S$, and

${\Gamma^{\pi}_{t}}_{*}=\sum_{g}a_{g}g_{*}$

on $CH_{0}(S_{t})_{\mathbb{Q}}$. Denote respectively
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}^{\pi}$ the image of the projector
${\Gamma^{\pi}_{t}}_{*}$ acting on $CH_{0}(S_{t})_{\mathbb{Q},hom}$ and
$H^{2,0}(S_{t})^{\pi}$ the image of the projector ${\Gamma^{\pi}_{t}}^{*}$
acting on $H^{2,0}(S_{t})$.

######  Theorem 1.6.

Let $\mathcal{S}\rightarrow B$ be a smooth projective morphism with two
dimensional connected fibers, where $B$ is quasi-projective. Let $G$ a finite
group acting in a fiberwise way on $\mathcal{S}$ and let $\pi\in\mathbb{Q}[G]$
be a projector. Assume the following:

(1) The fibers $\mathcal{S}_{t}$ satisfy $H^{1,0}(\mathcal{S}_{t})=0$ and
$H^{2,0}(\mathcal{S}_{t})^{\pi}=0$.

(2) A smooth projective completion (or, equivalently, any smooth projective
completion) $\overline{\mathcal{S}\times_{B}\mathcal{S}}$ of the fibered self-
product $\mathcal{S}\times_{B}\mathcal{S}$ is rationally connected.

Then $CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}^{\pi}=0$ for any $t\in B$.

Proof. The group $G$ acts fiberwise on $\mathcal{S}\rightarrow B$. Thus we
have the universal cycle

$\Gamma^{\pi}\in CH^{2}(\mathcal{S}\times_{B}\mathcal{S})_{\mathbb{Q}}$

defined as $\sum_{g\in G}a_{g}{\rm Graph}\,g$, where the graph is taken over
$B$. Since by assumption the action of $[\Gamma^{\pi}]^{*}$ on
$H^{2,0}(\mathcal{S}_{t})$ is $0$, we can apply Theorem 0.3, and conclude that
$\Gamma^{\pi}_{t*}$ is nilpotent on $CH_{0}(S_{t})_{\mathbb{Q,}hom}$. On the
other hand, $\Gamma^{\pi}_{t*}$ is a projector onto
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}^{\pi}$. The fact that it is
nilpotent implies thus that it is $0$, hence that
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}^{\pi}=0$.

##  2 Campedelli and Godeaux surfaces

Our main goal in this section is to check the main assumption of Theorem 1.6,
namely the rational connectedness of the fibered self-product of the universal
Campedelli surface, in order to prove Theorem 0.6.

We follow [11], [25]. Campedelli surfaces can be described as follows:
Consider the following symmetric $5\times 5$ matrix

$\displaystyle
M_{a}=\begin{pmatrix}a_{1}x_{1}&a_{2}x_{2}&a_{3}x_{3}&a_{4}x_{4}&0\\\
a_{2}x_{2}&a_{5}x_{3}&a_{6}x_{4}&0&a_{7}x_{1}\\\
a_{3}x_{3}&a_{6}x_{4}&0&a_{8}x_{1}&a_{9}x_{2}\\\
a_{4}x_{4}&0&a_{8}x_{1}&a_{10}x_{2}&a_{11}x_{3}\\\
0&a_{7}x_{1}&a_{9}x_{2}&a_{11}x_{3}&a_{12}x_{4}\end{pmatrix}$ (3)

depending on $12$ parameters $a_{1},\ldots,a_{12}$ and defining a symmetric
bilinear (or quadratic) form $q(a,x)$ on $\mathbb{C}^{5}$ depending on
$x\in\mathbb{P}^{3}$. This is a homogeneous degree $1$ matrix in the variables
$x_{1},\ldots,x_{4}$, and the vanishing of its determinant gives a degree $5$
surface $V(a)$ in $\mathbb{P}^{3}$ with nodes at those points
$x=(x_{1},\ldots,x_{4})$ where the matrix has rank only $3$. We will denote by
$T$ the vector space generated by $a_{1},\ldots,a_{12}$ and
$B\subset\mathbb{P}(T)$ the open set of parameters $a$ satisfying this last
condition.

The surface $V(a)$ is invariant under the Godeaux action of
$\mathbb{Z}/5\mathbb{Z}$ on $\mathbb{P}^{3}$, given by

$g^{*}x_{i}=\zeta^{i}x_{i},\,i=1,\ldots,\,4,$

where $g$ is a generator of $\mathbb{Z}/5\mathbb{Z}$ and $\zeta$ is a fifth
primitive root of unity. This follows either from the explicit development of
the determinant (see [25], where one monomial is incorrectly written:
$x_{1}x_{2}^{2}x_{4}^{2}$ should be $x_{1}x_{3}^{2}x_{4}^{2}$) or from the
following argument that we will need later on: Consider the following linear
action of $\mathbb{Z}/5\mathbb{Z}$ on $\mathbb{C}^{5}$:

$\displaystyle g(y_{1},\ldots,y_{5})=(\zeta
y_{1},\zeta^{2}y_{2},\ldots,\zeta^{5}y_{5}).$ (4)

Then one checks immediately that

$\displaystyle q(a,x)(gy,gy^{\prime})=\zeta q(a,gx)(y,y^{\prime}),$ (5)

so that the discriminant of $q(a,x)$, as a function of $x$, is invariant under
the action of $g$.

The Campedelli surface $\Sigma(a)$ is obtained as follows: there is a natural
double cover $S(a)$ of $V(a)$, which is étale away from the nodes, and
parameterizes the rulings in the rank four quadric $Q(a,x)$ defined by
$q(a,x)$ for $x\in V(a)$. This double cover is naturally equipped with a lift
of the $\mathbb{Z}/5\mathbb{Z}$-action on $V(a)$, which is explicitly
described as follows: Note first of all that the quadrics $Q(a,x)$ pass
through the point $y_{0}=(0,0,1,0,0)$ of $\mathbb{P}^{4}$.

######  Lemma 2.1.

For the general point $(a,x)\in\mathbb{P}(T)\times\mathbb{P}^{3}$ such that
$x\in V(a)$, the quadric $Q(a,x)$ is not singular at the point $y_{0}$.

Proof. It suffices to exhibit one pair $(a,x)$ satisfying this condition, and
such that the surface $V(a)$ is well-defined, that is the discriminant of
$q(a,x)$, seen as a function of $x$, is not identically $0$. Indeed, the
family of surfaces $V(a)$ is flat over the base near such a point, and the
result for the generic pair $(a,x)$ will then follow because the considered
property is open on the total space of this family.

We choose $(a,x)$ in such a way that the first column of the matrix (3) is
$0$, so that $x\in V(a)$. For example, we impose the conditions:

$a_{1}=0,\,a_{2}=0,\,x_{3}=0,\,x_{4}=0.$

Then the quadratic form $q(a,x)$ has for matrix

$\begin{pmatrix}0&0&0&0&0\\\ 0&0&0&0&a_{7}x_{1}\\\
0&0&0&a_{8}x_{1}&a_{9}x_{2}\\\ 0&0&a_{8}x_{1}&a_{10}x_{2}&0\\\
0&a_{7}x_{1}&a_{9}x_{2}&0&0\end{pmatrix}$

It is clear that $y_{0}$ is not a singular point of the corresponding quadric
if $a_{8}x_{1}\not=0$. On the other hand, for $a$ satisfying
$a_{1}=0,\,a_{2}=0$, and for a general point $x=(x_{1},\ldots,x_{4})$, the
matrix of $q(a,x)$ takes the form

$\begin{pmatrix}0&0&a_{3}x_{3}&a_{4}x_{4}&0\\\
0&a_{5}x_{3}&a_{6}x_{4}&0&a_{7}x_{1}\\\
a_{3}x_{3}&a_{6}x_{4}&0&a_{8}x_{1}&a_{9}x_{2}\\\
a_{4}x_{4}&0&a_{8}x_{1}&a_{10}x_{2}&a_{11}x_{3}\\\
0&a_{7}x_{1}&a_{9}x_{2}&a_{11}x_{3}&a_{12}x_{4}.\end{pmatrix}$

It is elementary to check that this matrix is generically of maximal rank.

It follows from this lemma that for generic $a$ and generic $x\in V(a)$, the
two rulings of $Q(a,x)$ correspond bijectively to the two planes through
$y_{0}$ contained in $Q(a,x)$. Hence for generic $a$, the surface $S(a)$ is
birationally equivalent to the surface $S^{\prime}(a)$ parameterizing planes
passing through $y_{0}$ and contained in $Q(a,x)$ for some
$x\in\mathbb{P}^{3}$ (which lies then necessarily in $V(a)$). It follows from
formula (5) that $S^{\prime}(a)$, which is a surface contained in the
Grassmannian $G_{0}$ of planes in $\mathbb{P}^{4}$ passing through $y_{0}$, is
invariant under the $\mathbb{Z}/5\mathbb{Z}$-action on $G_{0}$ induced by (4).
Hence there is a canonical $\mathbb{Z}/5\mathbb{Z}$-action on $S^{\prime}(a)$
which is compatible with the $\mathbb{Z}/5\mathbb{Z}$-action on $V(a)$, and
this immediately implies that the latter lifts to an action on the surface
$S(a)$ for any $a\in B$.

The Campedelli surface is defined as the quotient of $S(a)$ by
$\mathbb{Z}/5\mathbb{Z}$. In the following, we are going to apply Theorem 1.6,
and will thus work with the universal family $\mathcal{S}\rightarrow B$ of
double covers, with its $\mathbb{Z}/5\mathbb{Z}$-action defined above, where
$B\subset\mathbb{P}^{11}$ is the Zariski open set parameterizing smooth
surfaces $S(a)$.

We prove now:

######  Proposition 2.2.

The universal family $\mathcal{S}\rightarrow B$ has the property that the
fibered self-product $\mathcal{S}\times_{B}\mathcal{S}$ has a rationally
connected smooth projective compactification.

Proof. By the description given above, the family $\mathcal{S}\rightarrow B$
of surfaces $S(a),\,a\in B$, maps birationally to an irreducible component of
the following variety

$\displaystyle W=\\{(a,x,[P])\in\mathbb{P}(T)\times\mathbb{P}^{3}\times
G_{0},\,q(a,x)_{\mid P}=0\\},$ (6)

by the rational map which to a general point $(a,x),x\in V(a)$ and a choice of
ruling in the quadric $Q(a,x)$ associates the unique plane $P$ passing through
$y_{0}$, contained in $Q(a,x)$ and belonging to the chosen ruling. It follows
that $\mathcal{S}\times_{B}\mathcal{S}$ maps birationally by the same map
(which we will call $\Psi$) onto an irreducible component $W_{2}^{0}$ of the
following variety

$\displaystyle
W_{2}:=\\{(a,x,y,[P],[P^{\prime}])\in\mathbb{P}(T)\times\mathbb{P}^{3}\times\mathbb{P}^{3}\times
G_{0}\times G_{0},\,q(a,x)_{\mid P}=0,\,q(a,y)_{\mid P^{\prime}}=0\\}.$ (7)

Let $\mathcal{E}$ be the vector bundle of rank $5$ on $G_{0}$ whose fiber at a
point $[P]$ parameterizing a plane $P\subset\mathbb{P}^{4}$ passing through
$y_{0}$ is the space

$H^{0}(P,\mathcal{O}_{P}(2)\otimes\mathcal{I}_{y_{0}}).$

The family of quadrics $q(a,x)$ on $\mathbb{P}^{4}$ provides a $12$
dimensional linear space $T$ of sections of the bundle

$\mathcal{F}_{0}:=pr_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
pr_{3}^{*}\mathcal{E}\oplus pr_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
pr_{4}^{*}\mathcal{E}$

on

$Y_{0}:=\mathbb{P}^{3}\times\mathbb{P}^{3}\times G_{0}\times G_{0},$

where as usual the $pr_{i}$’s denote the various projections from $Y_{0}$ to
its factors. For a point $a\in T$, the corresponding section of
$\mathcal{F}_{0}$ is equal to $(q(a,x)_{\mid P},q(a,x^{\prime})_{\mid
P^{\prime}})$ at the point $(x,x^{\prime},[P],[P^{\prime}])$ of $Y_{0}$.
Formula (7) tells us that $W_{2}$ is the zero set of the corresponding
universal section of the bundle

${pr^{\prime}_{1}}^{*}\mathcal{O}_{\mathbb{P}(T)}(1)\otimes{pr^{\prime}_{2}}^{*}\mathcal{F}_{0}$

on $\mathbb{P}(T)\times Y_{0}$, where the $pr^{\prime}_{i}$ are now the two
natural projections from $\mathbb{P}(T)\times Y_{0}$ to its summands. Note
that $W_{2}^{0}$ has dimension $15$, hence has the expected codimension $10$,
since ${\rm dim}\,\mathbb{P}(T)\times Y_{0}=25$.

There is now a subtlety in our situation: It is not hard to see that $T$
generates generically the bundle $\mathcal{F}_{0}$ on $Y_{0}$. Hence there is
a “main” component of $W_{2}$ which is also of dimension $15$, and is
generically fibered into $\mathbb{P}^{1}$’s over $Y_{0}$. This component is
not equal to $W_{2}^{0}$ for the following reason: If one takes two general
planes $P,\,P^{\prime}$ through $y_{0}$, and two general points
$x,\,x^{\prime}\in\mathbb{P}^{3}$, the conditions that $q(a,x)$ vanishes
identically on $P$ and $q(a,x^{\prime})$ vanishes identically on $P^{\prime}$
implies that the third column of the matrix $M(a)$ is identically $0$, hence
that the point $y_{0}$ generates in fact the kernel of both matrices $M(a,x)$
and $M(a,x^{\prime})$. On the other hand, by construction of the map $\Psi$,
generically along ${\rm Im}\,\Psi\subset W_{2}^{0}$, the point $y_{0}$ is a
smooth point of the quadrics $Q(a,x)$ and $Q(a,x^{\prime})$.

The following lemma describes the component $W_{2}^{0}$.

######  Lemma 2.3.

Let $\Phi:W_{2}^{0}\rightarrow Y_{0}=\mathbb{P}^{3}\times\mathbb{P}^{3}\times
G_{0}\times G_{0}$ be the restriction to $W_{2}^{0}$ of the second projection
$\mathbb{P}(T)\times Y_{0}\rightarrow Y_{0}$.

(i) The image ${\rm Im}\,\Phi$ is a hypersurface $\mathcal{D}$ which admits a
rationally connected desingularization.

(ii) The generic rank of the evaluation map restricted to $\mathcal{D}$:

$T\otimes\mathcal{O}_{\mathcal{D}}\rightarrow{\mathcal{F}_{0}}_{\mid\mathcal{D}}$

is $9$. (Note that the generic rank of the evaluation map
$T\otimes\mathcal{O}_{Y_{0}}\rightarrow{\mathcal{F}_{0}}$ is $10$.)

Proof. We already mentioned that the map $\Phi\circ\Psi$ cannot be dominating.
Let us explain more precisely why, as this will provide the equation for
$\mathcal{D}$: Let $[P],\,[P^{\prime}]\in G_{0}$ and
$x,\,x^{\prime}\in\mathbb{P}^{3}$. The condition that
$(x,x^{\prime},P,P^{\prime})\in\Phi(W_{2}^{0})={\rm Im}\,(\Phi\circ\Psi)$ is
that for some $a\in\mathbb{P}(V)$,

$\displaystyle q(a,x)_{\mid P}=0,\,q(a,x^{\prime})_{\mid P^{\prime}}=0$ (8)

and furthermore that $y_{0}$ is a smooth point of both quadrics $Q(a,x)$ and
$Q(a,x^{\prime})$.

Let $e,\,f\in P$ be vectors such that $y_{0},e,f$ form a basis of $P$, and
choose similarly $e^{\prime},f^{\prime}\in P^{\prime}$ in order to get a basis
of $P^{\prime}$. Then among equations (8), we get

$\displaystyle
q(a,x)(y_{0},e)=0,\,q(a,x)(y_{0},f)=0,\,q(a,x^{\prime})(y_{0},x^{\prime})(y_{0},e^{\prime})=0,\,q(a,x^{\prime})(y_{0},f^{\prime})=0.$
(9)

For fixed $P,\,P^{\prime},\,x,\,x^{\prime}$, these equations are linear forms
in the variables $a_{3},\,a_{6},\,a_{8},\,a_{9}$ and it is not hard to see
that they are linearly independent for a generic choice of
$P,\,P^{\prime},\,x,\,x^{\prime}$, so that (9) implies that
$a_{3}=a_{6}=a_{8}=a_{9}=0$. But then, looking at the matrix $M(a)$ of (3) we
see that $y_{0}$ is in the kernel of $Q(a,x)$ and $Q(a,x^{\prime})$. As
already mentioned, the latter does not happen generically along ${\rm
Im}\,\Psi$, and we deduce that ${\rm Im}\,(\Phi\circ\Psi)$ is contained in the
hypersurface $\mathcal{D}$ where the four linear forms (9) in the four
variables $a_{3},\,a_{6},\,a_{8},\,a_{9}$ are not independent.

We claim that $\mathcal{D}$ is rationally connected. To see this we first
prove that it is irreducible, which is done by restricting the equation $f$ of
$\mathcal{D}$ (which is the determinant of the $(4\times 4)$ matrix whose
columns are the linear forms (9) written in the basis
$a_{3},\,a_{6},\,a_{8},\,a_{9}$), to a subvariety $Z$ of $Y_{0}$ of the form
$Z=\mathbb{P}^{3}\times\mathbb{P}^{3}\times C\times C^{\prime}\subset Y_{0}$,
where $C,\,C^{\prime}$ are curves in $G_{0}$.

Consider the following $1$ dimensional families $C\cong\mathbb{P}^{1}$,
$C^{\prime}\cong\mathbb{P}^{1}$ of planes passing through $y_{0}$:

$P_{t}=<e_{1},\lambda e_{2}+\mu
e_{4},e_{3}>,\,t=(\lambda,\mu)\in\mathbb{P}^{1}\cong C,$
$P^{\prime}_{t^{\prime}}=<e_{5},\lambda^{\prime}e_{2}+\mu^{\prime}e_{4},e_{3}>,\,t^{\prime}=(\lambda^{\prime},\mu^{\prime})\in\mathbb{P}^{1}\cong
C^{\prime}.$

The equations (9) restricted to the parameters $t,t^{\prime},x,x^{\prime}$
give the following four combinations of $a_{3},\,a_{6},\,a_{8},\,a_{9}$
depending on $\lambda,\,\mu,\,x,x^{\prime}$:

$a_{3}x_{3},\,\,\lambda a_{6}x_{4}+\mu
a_{8}x_{1},\,\,\,a_{9}x^{\prime}_{2},\,\,\lambda^{\prime}a_{6}x^{\prime}_{4}+\mu^{\prime}a_{8}x^{\prime}_{1}.$

Taking the determinant of this family gives

$f_{\mid
Z}=x_{3}x^{\prime}_{2}(\lambda\mu^{\prime}x_{4}x^{\prime}_{1}-\lambda^{\prime}\mu
x^{\prime}_{4}x_{1}).$

The hypersurface in $Z=\mathbb{P}^{3}\times\mathbb{P}^{3}\times C\times
C^{\prime}$ defined by $f_{\mid Z}$ has three irreducible components, which
belong respectively to the linear systems

$|pr_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,\,|pr_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,\,|pr_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
pr_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
pr_{3}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)\otimes
pr_{4}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,$

where the $pr_{i}$’s are now the projections from
$Z=\mathbb{P}^{3}\times\mathbb{P}^{3}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
to its factors.

As the restriction map from ${\rm Pic}\,Y_{0}$ to ${\rm Pic}\,Z$ is injective,
we see that if $\mathcal{D}$ was reducible, it would have an irreducible
component in one of the linear systems

$|p_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,\,|p_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,\,|p_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
p_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
p_{3}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)\otimes
p_{4}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,$

where the $p_{i}$’s are now the projections from
$Y_{0}=\mathbb{P}^{3}\times\mathbb{P}^{3}\times G_{0}\times G_{0}$ to its
factors. In particular, its restriction to either any
$\mathbb{P}^{3}\times\\{x^{\prime}\\}\times\\{[P]\\}\times\\{[P^{\prime}]\\}$
or any $\\{x\\}\times\mathbb{P}^{3}\times\\{[P]\\}\times\\{[P^{\prime}]\\}$
would have an irreducible component of degree $1$.

That this is not the case is proved by considering now the case where

$P=<e_{1}+e_{2},\,e_{4}+e_{5},\,e_{3}>,\,\,P^{\prime}=<e_{1}+e_{4},\,e_{2}+e_{5},\,e_{3}>.$

Then the four linear forms in (9) become

$a_{3}x_{3}+a_{6}x_{4},\,a_{8}x_{1}+a_{9}x_{2},\,a_{3}x^{\prime}_{3}+a_{8}x^{\prime}_{1},\,a_{6}x^{\prime}_{4}+a_{9}x^{\prime}_{2}.$

It is immediate to check that for generic choice of $x^{\prime}$, the
quadratic equation in $x$ given by this determinant has rank at least $4$, and
in particular is irreducible. Hence the restriction of $f$ to
$\mathbb{P}^{3}\times\\{x^{\prime}\\}\times\\{[P]\\}\times\\{[P^{\prime}]\\}$
has no irreducible component of degree $1$, and similarly for
$\\{x\\}\times\mathbb{P}^{3}\times\\{[P]\\}\times\\{[P^{\prime}]\\}$.

Rational connectedness of $\mathcal{D}$ now follows from the fact that the
projection on the last three summands of $Y_{0}$, restricted to $\mathcal{D}$,

$p:\mathcal{D}\hookrightarrow Y_{0}\rightarrow\mathbb{P}^{3}\times G_{0}\times
G_{0},$

has for general fibers smooth two-dimensional quadrics, as shows the
computation we just made. Hence $\mathcal{D}$ admits a rationally connected
smooth projective model, for example by [14].

We already proved that ${\rm Im}\,(\Phi\circ\Psi)\subset\mathcal{D}$. We will
now prove that ${\rm Im}\,(\Phi\circ\Psi)$ contains a Zariski open subset of
$\mathcal{D}$ and also statement ii). This is done by the following argument
which involves an explicit computation at a point $a_{0}\in\mathbb{P}(T)$
where the surface $V(a)$ becomes very singular but has enough smooth points to
conclude.

We start from the following specific matrix :

$\displaystyle M_{0}=\begin{pmatrix}x_{1}&0&x_{3}&0&0\\\ 0&x_{3}&x_{4}&0&0\\\
x_{3}&x_{4}&0&x_{1}&0\\\ 0&0&x_{1}&x_{2}&0\\\ 0&0&0&0&x_{4}\end{pmatrix}$ (10)

We compute ${\rm
det}\,M_{0}=-x_{4}x_{3}x_{1}^{3}-x_{1}x_{2}x_{4}^{3}-x_{4}x_{2}x_{3}^{3}$. The
surface $V(a_{0})$ defined by ${\rm det}\,M_{0}$ is rational and smooth at the
points

$\displaystyle x=(1,-\frac{1}{2},1,1),\,\,\,x^{\prime}=(1,-\frac{2}{9},2,-1).$
(11)

For each of the corresponding planes $P$ and $P^{\prime}$ contained in
$Q(a_{0},x)$, $Q(a_{0},x^{\prime})$ respectively and passing through $y_{0}$,
we have two choices. We choose the following:

$\displaystyle
P:=<y_{0},\,e,\,f>,\,e=e_{1}+e_{2}-2e_{4},\,f=e_{2}-e_{4}+\sqrt{-\frac{1}{2}}e_{5},$
(12) $\displaystyle
P^{\prime}=<y_{0},\,e^{\prime},\,f>,\,e^{\prime}=4e_{1}-e_{2}-9e_{4},\,f^{\prime}=e_{2}+e_{4}+\frac{4}{3}e_{5}.$

(We use here the standard basis $(e_{1},\ldots,e_{5})$ of $\mathbb{C}^{5}$, so
that $y_{0}=e_{3}$.) The vector $e-e_{3}$ (resp. $e^{\prime}-2e_{3}$)
generates the kernel of $q(a_{0},x)$ (resp. $q(a_{0},x^{\prime})$). As the
quadrics $q(a_{0},x)$ and $q(a_{0},x^{\prime})$ have rank $3$ and the surface
$V(a_{0})$ is smooth at $x$ and $x^{\prime}$, the points $(x,[P])$,
$(x^{\prime},[P^{\prime}])$ determine smooth points of the surface $S(a_{0})$.
Even if the surface $S(a_{0})$ is not smooth, so that $a_{0}\not\in B$, the
universal family $\mathcal{S}\rightarrow B$ extends to a smooth non proper map
$p:\mathcal{S}_{e}\rightarrow B_{e}\subset\mathbb{P}(T)$ near the points
$(a_{0},x,[P])$ and $(a_{0},x^{\prime},[P^{\prime}])$. Furthermore the
morphism $\Psi$ is well defined at the points $(a_{0},x,P)$,
$(a_{0},x^{\prime},P^{\prime})$ since the point $y_{0}$ is not a singular
point of the quadric $Q(a_{0},x)$ or $Q(a_{0},x^{\prime})$.

We claim that the map

$\Phi\circ\Psi:\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}\rightarrow
Y_{0}=\mathbb{P}^{3}\times\mathbb{P}^{3}\times G_{0}\times G_{0}$

has constant rank $13$ near the point

$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))\in\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}.$

This implies that the image of $\Phi\circ\Psi$ is of dimension $13$, which is
the dimension of $\mathcal{D}$, so that ${\rm Im}\,\Phi\circ\Psi$ has to
contain a Zariski open set of $\mathcal{D}$ since $\mathcal{D}$ is
irreducible. This implies that ${\rm Im}\,\Phi=\mathcal{D}$ as desired.

To prove the claim, we observe that it suffices to show the weaker claim that
the rank of

$\displaystyle(\Phi\circ\Psi)_{*}:T_{\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}}\rightarrow
T_{Y_{0}}$ (13)

is equal to $13$ at the given point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$. Indeed, as
$\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}$ is smooth at the point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$, the rank of the
differential in (13) can only increase in a neighborhood of
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$ and on the other hand, we
know that it is not of maximal rank at any point of
$\mathcal{S}^{\prime}\times_{B^{\prime}}\mathcal{S}^{\prime}$, since ${\rm
Im}\,(\Phi\circ\Psi)\subset\mathcal{D}$. Hence it must stay constant near
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$.

We now compute the rank of $(\Phi\circ\Psi)_{*}$ at the point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$. Note that the birational
map $\Psi:\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}\rightarrow W_{2}$ is a
local isomorphism near the point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$, because it is well defined
at this point and its inverse too. This argument proves that not only
$W_{2}^{0}$ but also $W_{2}$ is smooth of dimension $15$ near the point
$(a_{0},x,x^{\prime},[P],[P^{\prime}])$. Hence it suffices to compute the rank
of the differential

$\displaystyle\Phi_{*}:T_{W_{2},(a_{0},x,x^{\prime},[P],[P^{\prime}])}\rightarrow
T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])}$ (14)

of the map $\Phi:W_{2}\rightarrow Y_{0}$ at the point
$(a_{0},x,x^{\prime},[P],[P^{\prime}])$.

Recalling that $W_{2}$ is defined as the zero set of the universal section
$\sigma_{univ}$ of the bundle

${pr^{\prime}_{1}}^{*}\mathcal{O}_{\mathbb{P}(T)}(1)\otimes{pr^{\prime}_{2}}^{*}\mathcal{F}_{0}$

on $\mathbb{P}(T)\times Y_{0}$, the tangent space of $W_{2}$ at the point
$(a_{0},x,x^{\prime},[P],[P^{\prime}])$ is equal to

${\rm Ker}\,(d\sigma_{univ}:T_{\mathbb{P}(T),a_{0}}\times
T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])}\rightarrow\mathcal{F}_{0,(x,x^{\prime},[P],[P^{\prime}])}).$

Clearly the differential $d\sigma_{univ}$ restricted to
$T_{\mathbb{P}(T),a_{0}}$ is induced by the evaluation map

$\displaystyle{\rm
ev}_{(x,x^{\prime},[P],[P^{\prime}])}:T\rightarrow\mathcal{F}_{0,(x,x^{\prime},[P],[P^{\prime}])}$
(15)

at the point $(x,x^{\prime},[P],[P^{\prime}])$ of $Y_{0}$. On the other hand,
the differential $d\sigma_{univ}$ restricted to the tangent space
$T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])}$ is surjective because the variety
$S(a_{0})\times S(a_{0})$ is smooth of codimension $4$ and isomorphic via
$\Phi$ to $V(\sigma_{univ}(a_{0}))$ near $(x,x^{\prime},[P],[P^{\prime}])$. It
follows from this that the corank of the second projection

$\Phi_{*}:{\rm
Ker}\,d\sigma_{univ}=T_{W_{2},(a_{0},x,x^{\prime},[P],[P^{\prime}])}\subset
T_{\mathbb{P}(T),a_{0}}\oplus
T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])}\rightarrow
T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])},$

is equal to the corank of the map
$d\sigma_{univ}:T_{\mathbb{P}(T),a_{0}}\rightarrow\mathcal{F}_{0,(x,x^{\prime},[P],[P^{\prime}])})$,
that is to the corank of the evaluation map ${\rm
ev}_{(x,x^{\prime},[P],[P^{\prime}])}$ of (15).

In particular, the rank of $\Phi_{*}$ in (14) is equal to $13$ if and only if
the rank of the evaluation map ${\rm ev}_{x,x^{\prime},[P],[P^{\prime}]}$ is
equal to $9$, which is our statement (ii).

In conclusion, we proved that (i) is implied by (ii) and that (ii) itself
implied by (ii) at the given point $(x,x^{\prime},[P],[P^{\prime}])$ of (11),
(12). It just remains to prove that the rank of ${\rm
ev}_{(x,x^{\prime},[P],[P^{\prime}])}$ is equal to $9$ at this point, which is
done by the explicit computation of the rank of the family of linear forms in
the $a_{i}$ given by

$q(a,x)(y_{0},e),\,\,\,q(a,x)(y_{0},f),\,\,\,q(a,x)(e,e),\,\,\,q(a,x)(f,f),\,\,\,q(a,x)(e,f),$
$q(a,x^{\prime})(y_{0},e^{\prime}),\,\,\,q(a,x^{\prime})(y_{0},f^{\prime}),\,\,\,q(a,x^{\prime})(e^{\prime},e^{\prime}),\,\,\,q(a,x^{\prime})(f^{\prime},f^{\prime}),\,\,\,q(a,x^{\prime})(e^{\prime},f^{\prime}).$

These forms are the following:

$a_{3}+a_{6}-2a_{8},\,\,\,\,\,\,a_{6}-a_{8}-\frac{1}{2}\sqrt{-\frac{1}{2}}a_{9},\,\,\,\,\,\,a_{1}+a_{5}-2a_{10}-a_{2}-4a_{4},$
$a_{5}-\frac{a_{10}}{2}-\frac{a_{12}}{2}+2\sqrt{-\frac{1}{2}}a_{7}-2\sqrt{-\frac{1}{2}}a_{11},\,\,\,\,\,\,-\frac{a_{2}}{2}-a_{4}+a_{5}+\sqrt{-\frac{1}{2}}a_{7}-a_{10}-2\sqrt{-\frac{1}{2}}a_{11}$
$8a_{3}+a_{6}-9a_{8},\,\,\,\,\,\,-a_{6}+a_{8}-\frac{8}{27}a_{9},\,\,\,\,\,\,16a_{1}+2a_{5}-18a_{10}+\frac{16}{9}a_{2}+72a_{4},$
$2a_{5}-\frac{2}{9}a_{10}-\frac{16}{9}a_{12}+\frac{8}{3}a_{7}+\frac{16}{3}a_{11},\,\,\,-\frac{8}{9}a_{2}-4a_{4}-2a_{5}-\frac{4}{3}a_{7}+2a_{10}-24a_{11}.$

It is immediate to check that the rank of this family is $9$.

It follows from Lemma 2.3 that $W_{2}^{0}$ (or rather a smooth projective
birational model of $W_{2}^{0}$) is rationally connected, since a Zariski open
set of $W_{2}^{0}$ is a $\mathbb{P}^{2}$-bundle over a Zariski open set
$\mathcal{D}^{0}$ of $\mathcal{D}$ which is smooth and admits a rationally
connected completion. Hence $\mathcal{S}\times_{B}\mathcal{S}$ is also
rationally connected, as it is birationally equivalent to $W_{2}^{0}$.

We get by application of Theorem 1.6 the following statement (cf. Theorem
0.6):

######  Corollary 2.4.

The Campedelli surface $\Sigma(a)$ has $CH_{0}$ equal to $\mathbb{Z}$.

Proof. Indeed, the surface $\Sigma(a)$ has $h^{1,0}=h^{2,0}=0$ (cf. [11])
which means equivalently that the surface $S(a)$ introduced above has
$h^{2,0}_{inv}=0$ where “inv” means invariant under the
$\mathbb{Z}/5\mathbb{Z}$-action, which by the above description is defined on
$\mathcal{S}$. The result is then a consequence of Theorem 1.6 applied to the
projector $\pi_{inv}$ onto the $\mathbb{Z}/5\mathbb{Z}$-invariant part, which
shows that

$CH_{0}(\Sigma(a))_{\mathbb{Q},hom}=CH_{0}(S(a))^{\pi_{inv}}_{\mathbb{Q},hom}=0$

and from Roitman’s theorem [24] which says that $CH_{0}(\Sigma(a))$ has no
torsion.

######  Corollary 2.5.

The Barlow surface $\Sigma^{\prime}(b)$ has $CH_{0}$ equal to $\mathbb{Z}$.

Proof. Indeed, the Barlow surface $\Sigma^{\prime}(b)$ is a quotient of the
Campedelli surface $\Sigma(b)$ by an involution, hence
$CH_{0}(\Sigma^{\prime}(b))\hookrightarrow CH_{0}(\Sigma(b))$ since
$CH_{0}(\Sigma^{\prime}(b))$ has no torsion by [24].

##  3 On the Chow motive of complete intersections $K3$ surfaces

Let $S$ be a $K3$ surface. The Hodge structure on $H^{2}(S,\mathbb{Z})$ is a
weight $2$ polarized Hodge structure with $h^{2,0}=1$. In [18], Kuga and
Satake construct an abelian variety $K(S)$ associated to this Hodge structure.
Its main property is the fact that there is a natural injective morphism of
weight $2$ Hodge structures :

$H^{2}(S,\mathbb{Z})\rightarrow H^{2}(K(S),\mathbb{Z}).$

Such a morphism of Hodge structures in turn provides, using Künneth
decomposition and Poincaré duality a Hodge class

$\alpha_{S}\in{\rm Hdg}^{4}(S\times K(S))$

where ${\rm Hdg}^{4}(X)$ denotes the space of rational Hodge classes on $X$.

This class is not known in general to satisfy the Hodge conjecture, that is,
to be the class of an algebraic cycle. This is known to hold for Kummer
surfaces (see [19]), and for some $K3$ surfaces with Picard number $16$ (see
[23]). The deformation theory of $K3$ surfaces, and more particularly the fact
that any projective $K3$ surface deforms to a Kummer surface, combined with
the global invariant cycle theorem of Deligne [12] imply the following result,
(cf. [13], [2]):

######  Theorem 3.1.

Let $S$ be a projective $K3$ surface. There exist a connected quasi-projective
variety $B$, a family of projective $K3$ surfaces $\mathcal{S}\rightarrow B$,
a family of abelian varieties $\mathcal{K}\rightarrow B$, (where all varieties
are quasi-projective and all morphisms are smooth projective), and a Hodge
class

$\eta\in{\rm Hdg}^{4}(\mathcal{S}\times_{B}\mathcal{K}),$

such that :

  1. 1.

for some point $t_{0}\in B$, $\mathcal{S}_{t_{0}}\cong S$;

  2. 2.

For any point $t\in B$, $\mathcal{K}_{t}\cong K(S_{t})$ and
$\eta_{t}=\alpha_{S_{t}}$.

  3. 3.

For some point $t_{1}\in B$, the class $\eta_{t_{1}}$ is algebraic.

Here by “Hodge class $\eta$”, we mean that the class $\eta$ comes from a Hodge
class on some smooth projective compactification of
$\mathcal{S}\times_{B}\mathcal{K}$.

The existence of the family is an algebraicity statement for the Kuga-Satake
construction (see [13]), which can then be done in family for $K3$ surfaces
with given polarization, while the last item follows from the fact that the
locally complete such families always contain Kummer fibers $S^{\prime}$ for
which the class $\alpha_{S^{\prime}}$ is known to be algebraic.

######  Corollary 3.2.

[2] The Hodge class $\alpha_{S}$ is “motivated”.

This means (cf. [2]) that this Hodge class can be constructed via algebraic
correspondences from Hodge classes on auxiliary varieties, which are either
algebraic or obtained by inverting Lefschetz operators. In particular the
class $\alpha_{S}$ is algebraic if the standard Lefschetz conjecture holds.

######  Corollary 3.3.

Assume the variational form of the Hodge conjecture holds, or assume the
Lefschetz standard conjecture in degrees $2$ and $4$. Then the class
$\alpha_{S}$ is algebraic for any projective $K3$ surface $S$.

Indeed, the variational Hodge conjecture states that in the situation of
Theorem 3.1, if a Hodge class $\eta$ on the total space has an algebraic
restriction on one fiber, then its restriction to any fiber is algebraic. In
our situation, it will be implied by the Lefschetz conjecture for degree $2$
and degree $4$ cohomology on a smooth projective compactification of
$\mathcal{S}\times_{B}\mathcal{K}$.

From now on, we assume that the Kuga-Satake correspondence $\alpha_{S}$ is
algebraic for a general projective $K3$ surface $S$ of genus $g$ (which means
by definition that $S$ comes equipped with an ample line bundle of self-
intersection $2g-2$). We view the class $\alpha_{S}$ as an injective morphism
of Hodge structures

$\displaystyle\alpha_{S}:H^{2}(S,\mathbb{Q})\rightarrow
H^{2}(K(S),\mathbb{Q})$ (16)

and use a polarization $h$ of $K(S)$ to construct an inverse of $\alpha_{S}$
by the following lemma (which can be proved as well by the explicit
description $K(S)$ as a complex torus and its polarization, cf. [18]). In the
following, $H^{2}(S,\mathbb{Q})_{tr}$ denotes the transcendental cohomology of
$S$, which is defined as the orthogonal complement of the Néron-Severi group
of $S$.

######  Lemma 3.4.

Let $h=c_{1}(H)\in H^{2}(K(S),\mathbb{Q})$ be the class of an ample line
bundle, where $S$ is a very general $K3$ surface of genus $g$. Then there
exists a nonzero rational number $\lambda_{g}$ such that the endomorphism

${}^{t}\alpha_{S}\circ(h^{N-2}\cup)\circ\alpha_{S}:H^{2}(S,\mathbb{Q})\rightarrow
H^{2}(S,\mathbb{Q})$

restricts to $\lambda_{g}Id$ on $H^{2}(S,\mathbb{Q})_{tr}$, where $N={\rm
dim}\,K(S)$ and

${}^{t}\alpha_{S}:H^{2N-2}(K(S),\mathbb{Q})\rightarrow H^{2}(S,\mathbb{Q})$

is the transpose of the map $\alpha_{S}$ of (16) with respect to Poincaré
duality.

Proof. The composite ${}^{t}\alpha_{S}\circ h^{N-2}\circ\alpha_{S}$ is an
endomorphism of the Hodge structure on $H^{2}(S,\mathbb{Q})$. As $S$ is very
general, this Hodge structure has only a one dimensional $\mathbb{Q}$-vector
space of algebraic classes, generated by the polarization, and its orthogonal
is a simple Hodge structure with only the homotheties as endomorphisms. We
conclude that ${}^{t}\alpha_{S}\circ h^{N-2}\circ\alpha_{S}$ preserves
$H^{2}(S,\mathbb{Q})_{tr}$ and acts on it as an homothety with rational
coefficient (which is thus independent of $S$). It just remains to show that
it does not act as zero on $H^{2}(S,\mathbb{Q})_{tr}$. This follows from the
second Hodge-Riemmann bilinear relations which say that for $\omega\in
H^{2,0}(S),\,\omega\not=0$, we have

$\langle\omega,\,{{}^{t}\alpha_{S}}(h^{N-2}\cup\alpha_{S}(\overline{\omega}))\rangle_{S}=\langle\alpha_{S}(\omega),h^{N-2}\cup\alpha_{S}(\overline{\omega})\rangle_{K(S)}$

which is $>0$ because $\alpha_{S}(\omega)\not=0$ in $H^{2,0}(K(S))$. Hence
${}^{t}\alpha_{S}(h^{N-2}\cup\alpha_{S}(\overline{\omega}))\not=0$.

We now start from a rationally connected variety $X$ of dimension $n$, with a
vector bundle $E$ of rank $n-2$ on $X$, such that $-K_{X}={\rm det}\,E$ and
the following properties hold:

(*) For general $x,\,y\in X$, and for general $\sigma\in
H^{0}(X,E\otimes\mathcal{I}_{x}\otimes\mathcal{I}_{y})$, the zero locus
$V(\sigma)$ is a smooth connected surface with $0$ irregularity (hence a
smooth $K3$ surface).

Let $L$ be an ample line bundle on $X$, inducing a polarization of genus $g$
on the $K3$ surface $S_{\sigma}:=V(\sigma)$ for $\sigma\in
B\subset\mathbb{P}(H^{0}(X,E))$.

We now prove:

######  Theorem 3.5.

Assume the Kuga-Satake correspondence $\alpha_{S}$ is algebraic, for the
general $K3$ surface $S$ with such a polarization. Then, for any $\sigma\in
B$, the Chow motive of $S_{\sigma}$ is a direct summand of the motive of an
abelian variety.

Let us explain the precise meaning of this statement. The algebraicity of the
Kuga-Satake correspondence combined with Lemma 3.4 implies that there is a
codimension $2$ algebraic cycle

$Z_{S}\in CH^{2}(S\times K(S))_{\mathbb{Q}}$

with the property that the cycle $\Gamma_{S}$ defined by

$\Gamma_{S}={{}^{t}Z_{S}}\circ h^{N-2}\circ Z_{S}\in CH^{2}(S\times
S)_{\mathbb{Q}}$

has the property that its cohomology class $[\Gamma]\in H^{4}(S\times
S,\mathbb{Q})$ induces a nonzero homothety

$\displaystyle[\Gamma]_{*}=\lambda Id:H^{2}(S,\mathbb{Q})_{tr}\rightarrow
H^{2}(S,\mathbb{Q})_{tr},$ (17)

which can be equivalently formulated as follows: Let us introduce the cycle
$\Delta_{S,tr}$, which in the case of a $K3$ surface is canonically defined,
by the formula

$\displaystyle\Delta_{S,tr}=\Delta_{S}-o_{S}\times S-S\times
o_{S}-\sum_{ij}\alpha_{ij}C_{i}\times C_{j},$ (18)

where $\Delta_{S}$ is the diagonal of $S$, $o_{S}$ is the canonical $0$-cycle
of degree $1$ on $S$ introduced in [6], the $C_{i}$ form a basis of $({\rm
Pic}\,S)\otimes\mathbb{Q}={\rm NS}(S)_{\mathbb{Q}}$ and the $\alpha_{ij}$ are
the coefficients of the inverse of the matrix of the intersection form of $S$
restricted to $NS(S)$. This corrected diagonal cycle is a projector and it has
the property that its action on cohomology is the orthogonal projector
$H^{*}(S,\mathbb{Q})\rightarrow H^{2}(S,\mathbb{Q})_{tr}$. Formula (17) says
that we have the cohomological equality

$\displaystyle[\Gamma\circ\Delta_{S,tr}]=\lambda[\Delta_{S,tr}]\,\,{\rm
in}\,\,H^{4}(S\times S,\mathbb{Q}).$ (19)

A more precise form of Theorem 3.5 says that we can get in fact such an
equality at the level of Chow groups:

######  Theorem 3.6.

Assume the Kuga-Satake correspondence $\alpha_{S}$ is algebraic, for a general
$K3$ surface with such a polarization. Then, for any $\sigma\in B$, there is
an abelian variety $A_{\sigma}$, and cycles $Z\in CH^{2}(S_{\sigma}\times
A_{\sigma})_{\mathbb{Q}}$, $Z^{\prime}\in CH^{N^{\prime}}(A_{\sigma}\times
S_{\sigma})_{\mathbb{Q}}$, $N^{\prime}={\rm dim}\,A_{\sigma}$, with the
property that

$\displaystyle Z^{\prime}\circ Z\circ\Delta_{S,tr}=\lambda\Delta_{S,tr}$ (20)

for a nonzero rational number $\lambda$.

The proof will use two preparatory lemmas. Let
$B\subset\mathbb{P}(H^{0}(X,E))$ be the open set parameterizing smooth
surfaces $S_{\sigma}=V(\sigma)\subset X$. Let $\pi:\mathcal{S}\rightarrow B$
be the universal family, that is:

$\displaystyle\mathcal{S}=\\{(\sigma,x)\in B\times X,\,\sigma(x)=0\\}.$ (21)

The first observation is the following:

######  Lemma 3.7.

Under assumption (*), the fibered self-product
$\mathcal{S}\times_{B}\mathcal{S}$ is rationally connected (or rather, admits
a smooth projective rationally connected completion).

Proof. From (21), we deduce that

$\displaystyle\mathcal{S}\times_{B}\mathcal{S}=\\{(\sigma,x,y)\in B\times
X\times X,\,\sigma(x)=\sigma(y)=0\\}.$ (22)

Hence $\mathcal{S}\times_{B}\mathcal{S}$ is Zariski open in the following
variety:

$W:=\\{(\sigma,x,y)\in\mathbb{P}(H^{0}(X,E))\times X\times
X,\,\sigma(x)=\sigma(y)=0\\}.$

In particular, as it is irreducible, it is Zariski open in one irreducible
component $W^{0}$ of $W$.

Consider the projection on the two last factors:

$(p_{2},p_{3}):W\rightarrow X\times X.$

Its fibers are projective spaces, so that there is only one “main” irreducible
component $W^{1}$ of $W$ dominating $X\times X$ and it admits a smooth
rationally connected completion since $X\times X$ is rationally connected.

Assumption (*) now tells us that at a general point of $W^{1}$, the first
projection $p_{1}:W\rightarrow B$ is smooth of relative dimension $4$. It
follows that $W$ is smooth at this point which belongs to both components
$W^{0}$ and $W^{1}$. Thus $W^{0}=W^{1}$ and
$\mathcal{S}\times_{B}\mathcal{S}\cong_{birat}W^{0}$ admits a smooth
rationally connected completion.

The next step is the following lemma:

######  Lemma 3.8.

Assume the Kuga-Satake correspondence $\alpha_{S}$ is algebraic for the
general polarized $K3$ surface of genus $g$. Then there exist a rational
number $\lambda\not=0$, a family $\mathcal{A}\rightarrow B$ of polarized
$N^{\prime}$-dimensional abelian varieties, with relative polarization
$\mathcal{L}$ and a codimension $2$ cycle

$\mathcal{Z}\in CH^{2}(\mathcal{S}\times_{B}\mathcal{A})_{\mathbb{Q}}$

such that for very general $t\in B$, the cycle

$\Gamma_{t}:={{}^{t}\mathcal{Z}_{t}}\circ
c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}\circ\mathcal{Z}_{t}\circ\Delta_{tr,t}$

satisfies:

$\displaystyle[\Gamma_{t}]=\lambda[\Delta_{tr,t}]\in H^{4}(S_{t}\times
S_{t},\mathbb{Q}).$ (23)

In this formula, the term $c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}$ is defined
as the self-correspondence of $\mathcal{A}_{t}$ which consists of the cycle
$c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}$ supported on the diagonal of
$\mathcal{A}_{t}$. We also recall that the codimension $2$-cycle
$\Delta_{tr,t}\in CH^{2}(S_{t}\times S_{t})_{\mathbb{Q}}$ is the projector
onto the transcendental part of the motive of $S_{t}$. The reason why the
result is stated only for the very general point $t$ is the fact that due to
the possible jump of the Picard group of $S_{t}$, the generic cycle
$\Delta_{tr,\overline{\eta}}$ does not specialize to the cycle $\Delta_{tr,t}$
at any closed point $t\in B$, but only at the very general one. In fact, the
statement is true at any point, but the cycle $\Delta_{tr,t}$ has to be
modified when the Picard group jumps.

Proof of Lemma 3.8. By our assumption, using the countability of relative
Hilbert schemes and the existence of universal objects parameterized by them,
there exist a generically finite cover $r:B^{\prime}\rightarrow B$, a
universal family of polarized abelian varieties

$\mathcal{K}\rightarrow B^{\prime},\,\mathcal{L}_{K}\in{\rm Pic}\,\mathcal{K}$

and a codimension $2$ cycle $\mathcal{Z}^{\prime}\in
CH^{2}(\mathcal{S}^{\prime}\times_{B^{\prime}}\mathcal{K})_{\mathbb{Q}}$,
where $\mathcal{S}^{\prime}:=\mathcal{S}\times_{B}B^{\prime}$, with the
property that

$\displaystyle[\mathcal{Z}_{t}]=\alpha_{\mathcal{S}_{t}}\,\,{\rm
in}\,\,H^{4}(\mathcal{S}_{t}\times K(\mathcal{S}_{t}),\mathbb{Q})$ (24)

for any $t\in B^{\prime}$. Furthermore, by Lemma 3.4, we know that there
exists a nonzero rational number $\lambda_{g}$ such that for any $t\in
B^{\prime}$, we have

${}^{t}\alpha_{\mathcal{S}_{t}}\circ(h^{N-2}\cup)\circ\alpha_{\mathcal{S}_{t}}=\lambda_{g}Id:H^{2}(\mathcal{S}_{t},\mathbb{Q})_{tr}\rightarrow
H^{2}(\mathcal{S}_{t},\mathbb{Q})_{tr},$ (25)

where as before $N$ is the dimension of $K(\mathcal{S}_{t})$, and
$h_{t}=c_{1}(\mathcal{L}_{K\mid\mathcal{K}_{t}})$. We now construct the
following family of abelian varieties on $B$ (or a Zariski open set of it)

$\mathcal{A}_{t}=\prod_{t^{\prime}\in r^{-1}(t)}\mathcal{K}_{t^{\prime}},$

with polarization given by

$\displaystyle\mathcal{L}_{t}=\sum_{t^{\prime}\in
r^{-1}(t)}pr_{t^{\prime}}^{*}(\mathcal{L}_{K,t^{\prime}}),$ (26)

where $pr_{t^{\prime}}$ is the obvious projection from
$\mathcal{A}_{t}=\prod_{t^{\prime}\in r^{-1}(t)}K_{t^{\prime}}$ to its factor
$\mathcal{K}_{t^{\prime}}$, and the following cycle $\mathcal{Z}\in
CH^{2}(\mathcal{S}\times_{B}\mathcal{K})_{\mathbb{Q}}$, with fiber at $t\in B$
given by

$\displaystyle\mathcal{Z}_{t}=\sum_{t^{\prime}\in
r^{-1}(t)}(Id_{\mathcal{S}_{t}},pr_{t^{\prime}})^{*}\mathcal{Z}^{\prime}_{t^{\prime}}.$
(27)

In the last formula, we use of course the identification

$\mathcal{S}^{\prime}_{t^{\prime}}=\mathcal{S}_{t},\,r(t^{\prime})=t.$

It just remains to prove formula (23). Combining (26) and (27), we get, using
again the notation $h_{t}=c_{1}(\mathcal{L}_{t})\in H^{2}(A_{t},\mathbb{Q})$,
$h_{t^{\prime}}=c_{1}(\mathcal{L}_{K,t^{\prime}})\in
H^{2}(K_{t^{\prime}},\mathbb{Q})$:

$[\Gamma_{t}]^{*}=$ $(\sum_{t^{\prime}\in
r^{-1}(t)}[(pr_{t^{\prime}},Id_{\mathcal{S}_{t}})^{*}(^{t}\mathcal{Z}^{\prime}_{t^{\prime}})]^{*})\circ(\sum_{t^{\prime\prime}\in
r^{-1}(t)}pr_{t^{\prime\prime}}^{*}(h_{t^{\prime\prime}}))^{N^{\prime}-2}\cup\circ(\sum_{t^{\prime\prime\prime}\in
r^{-1}(t)}(Id_{\mathcal{S}_{t}},pr_{t^{\prime\prime\prime}})^{*}[\mathcal{Z}^{\prime}_{t^{\prime\prime\prime}}]^{*})\circ\pi_{tr,t}:$
$H^{*}(\mathcal{S}_{t},\mathbb{Q})\rightarrow
H^{*}(\mathcal{S}_{t},\mathbb{Q}).$

Note that $N^{\prime}={\rm dim}\,\mathcal{A}_{t}=\natural(r^{-1}(t))\,N={\rm
deg}\,(B^{\prime}/B)\,\,N$. We develop the product above, and observe that the
only nonzero terms appearing in this development come from by taking
$t^{\prime}=t^{\prime\prime\prime}$ and putting the monomial

$pr_{t^{\prime}}^{*}(h_{t^{\prime}}))^{N-2}\cup_{t^{\prime\prime}\not=t^{\prime}}pr_{t^{\prime\prime}}^{*}(h_{t^{\prime\prime}}))^{N}$

in the middle term. The other terms are $0$ due to the projection formula. Let
us explain this in the case of only two summands $K_{1},\,K_{2}$ of dimension
$r$ with polarizations $l_{1},\,l_{2}$ and two cycles $Z_{i}\in CH^{2}(S\times
K_{i})$ giving rise to a cycle of the form

$(Id_{S},pr_{1})^{*}Z_{1}+(Id_{S},pr_{2})^{*}Z_{2}\in CH^{2}(S\times
K_{1}\times K_{2}),$

where the $pr_{i}$’s are the projections from $S\times K_{1}\times K_{2}$ to
$K_{i}$: Then we have

$[{(pr_{1},Id_{S})^{*}}^{t}Z_{1}]^{*}\circ(l_{1}+l_{2})^{2r-2}\cup\circ[{(Id_{S},pr_{2})}^{*}Z_{2}]^{*}=0:H^{2}(S,\mathbb{Q})_{tr}\rightarrow
H^{2}(S,\mathbb{Q})_{tr}$

by the projection formula and for the same reason

$[{(pr_{1},Id_{S})^{*}}^{t}Z_{1}]^{*}\circ(l_{1}+l_{2})^{2r-2}\cup\circ[{(Id_{S},pr_{1})}^{*}Z_{1}]^{*}$
$=[{(pr_{1},Id_{S})^{*}}^{t}Z_{1}]^{*}\circ\sum_{0\leq k\leq
r}\binom{2r-2}{k}(l_{1}^{k}l_{2}^{2r-2-k})\cup\circ[{(Id_{S},pr_{1})}^{*}Z_{1}]^{*}$
$\underset{proj.\,formula}{=}\binom{2r-2}{r}{\rm
deg}\,(l_{2}^{r})\,\,[Z_{1}]^{*}\circ
l_{1}^{r-2}\cup\circ[^{t}Z_{1}]^{*}:H^{2}(S,\mathbb{Q})_{tr}\rightarrow
H^{2}(S,\mathbb{Q})_{tr}.$

We thus get (as the degrees ${\rm deg}\,(h_{t^{\prime}}^{N})$ of the
polarizations $h_{t^{\prime}}$ on the abelian varieties
$\mathcal{K}_{t^{\prime}}$ are all equal):

$[\Gamma_{t}]^{*}=M({\rm deg}\,(h_{t^{\prime}}^{N}))^{{\rm
deg}\,(B^{\prime}/B)-1}(\sum_{t^{\prime}\in
r^{-1}(t)}[^{t}\mathcal{Z}^{\prime}_{t^{\prime}})]^{*})\circ
h_{t^{\prime}}^{N-2}\cup\circ[\mathcal{Z}^{\prime}_{t^{\prime}}]^{*})\circ\pi_{tr,t}:$
$H^{*}(\mathcal{S}_{t},\mathbb{Q})\rightarrow
H^{*}(\mathcal{S}_{t},\mathbb{Q}),$

where $M$ is the multinomial coefficient appearing in front of the monomial
$x_{1}^{N-2}x_{2}^{N}\ldots x_{{\rm deg}\,(B^{\prime}/B)}^{N}$ in the
development of $(x_{1}+\ldots+x_{{\rm deg}\,(B^{\prime}/B)})^{N{\rm
deg}\,(B^{\prime}/B)-2}$.

By (24) and (25), we conclude that

$[\Gamma_{t}]^{*}=M({\rm deg}\,(h_{t^{\prime}}^{N}))^{{\rm
deg}\,(B^{\prime}/B)-1}{{\rm
deg}\,(B^{\prime}/B)}\lambda_{g}\pi_{tr,t}:H^{*}(\mathcal{S}_{t},\mathbb{Q})\rightarrow
H^{*}(\mathcal{S}_{t},\mathbb{Q}),$

which proves formula (23) with $\lambda=M({\rm
deg}\,(h_{t^{\prime}}^{N}))^{{\rm deg}\,(B^{\prime}/B)-1}{{\rm
deg}\,(B^{\prime}/B)}\lambda_{g}$.

Proof of Theorem 3.6. Consider the cycle $\mathcal{Z}\in
CH^{2}(\mathcal{S}\times_{B}\mathcal{A})_{\mathbb{Q}}$ of Lemma 3.8, and the
cycle $\Gamma\in CH^{2}(\mathcal{S}\times_{B}\mathcal{S})_{\mathbb{Q}}$

$\displaystyle\Gamma:=\Delta_{tr}\circ{{}^{t}\mathcal{Z}}\circ
c_{1}(\mathcal{L})^{N^{\prime}-2}\circ\mathcal{Z}\circ\Delta_{tr}$ (28)

where now $c_{1}(\mathcal{L})$ is the class of $\mathcal{L}$ in
$CH^{1}(\mathcal{A})$ and we denote by $c_{1}(\mathcal{L})^{N-2}$ the relative
self-correspondence of $\mathcal{A}$ given by the cycle
$c_{1}(\mathcal{L})^{N-2}$ supported on the relative diagonal of $\mathcal{A}$
over $B$ (it thus induces the intersection product with
$c_{1}(\mathcal{L})^{N-2}$ on Chow groups). Furthermore the composition of
correspondences is the relative composition over $B$, and $\Delta_{tr}$ is the
generic transcendental motive, (which is canonically defined in our case, at
least after restricting to a Zariski open set of $B$) obtained as follows:
Choose a $0$-cycle $o_{\overline{\eta}}$ of degree $1$ on the generic
geometric fiber $\mathcal{S}_{\overline{\eta}}$ and choose a basis
$L_{1},\ldots,L_{k}$ of ${\rm Pic}\,\mathcal{S}_{\overline{\eta}}$. We have
then the projector $\Delta_{alg,\overline{\eta}}\in
CH^{2}(\mathcal{S}_{\overline{\eta}})_{\mathbb{Q}}$ defined as in (18) using
the fact that the intersection pairing on the group of cycles
$<o_{\overline{\eta}},\,\mathcal{S}_{\overline{\eta}},\,L_{i}>$ is
nondegenerate. The transcendental projector $\Delta_{tr,\overline{\eta}}$ is
defined as
$\Delta_{\mathcal{S}_{\overline{\eta}}}-\Delta_{alg,\overline{\eta}}$. This is
a codimension $2$ cycle on
$\mathcal{S}_{\overline{\eta}}\times_{\overline{\mathbb{C}(B)}}\mathcal{S}_{\overline{\eta}}$
but it comes from a cycle on
$\mathcal{S}^{\prime\prime}\times_{B^{\prime\prime}}\mathcal{S}^{\prime\prime}$
for some generically finite covers $B^{\prime\prime}\rightarrow
B,\,\mathcal{S}^{\prime\prime}=\mathcal{S}\times_{B}B^{\prime\prime}$, and the
latter descends finally to a codimension $2$-cycle on
$\mathcal{S}\times_{B}\mathcal{S}$, which one checks to be a multiple of a
projector, at least over a Zariski open set of $B$.

The cycle $\Gamma$ satisfies (23), which we rewrite as

$\displaystyle[\Gamma^{\prime}_{t}]=0\,\,{\rm
in}\,\,H^{4}(\mathcal{S}_{t}\times\mathcal{S}_{t},\mathbb{Q}),$ (29)

where $\Gamma^{\prime}:=\Gamma-\lambda\Delta_{tr}$.

Lemma 3.7 tells us that the fibered self-product
$\mathcal{S}\times\mathcal{S}$ is rationally connected. We can thus apply
Theorem 0.3 and conclude that $\Gamma^{\prime}_{t}\in
CH^{2}(\mathcal{S}_{t}\times\mathcal{S}_{t})_{\mathbb{Q}}$ is nilpotent. It
follows that $\Gamma_{t}=\lambda\Delta_{tr,t}+N_{t}$, where $\lambda\not=0$
and $N_{t}$ is a nilpotent cycle in $\mathcal{S}_{t}\times\mathcal{S}_{t}$
having the property that

$\Delta_{tr,t}\circ N_{t}=N_{t}\circ\Delta_{tr,t}=N_{t}.$

We conclude immediately from the standard inversion formula for
$\lambda\,\mathbb{I}+N$, with $N$ nilpotent and $\lambda\not=0$, that there
exists a correspondence $\Phi_{t}\in
CH^{2}(\mathcal{S}_{t}\times\mathcal{S}_{t})_{\mathbb{Q}}$ such that

$\Phi_{t}\circ\Gamma_{t}=\Delta_{tr,t}\,\,{\rm in}\,\
CH^{2}(\mathcal{S}_{t}\times\mathcal{S}_{t})_{\mathbb{Q}}.$

Recalling now formula (28):

$\Gamma_{t}:=\Delta_{tr,t}\circ{{}^{t}\mathcal{Z}_{t}}\circ
c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}\circ\mathcal{Z}_{t}\circ\Delta_{tr,t},$

we proved (20) with

$Z^{\prime}=\Phi_{t}\circ\Delta_{tr,t}\circ{{}^{t}\mathcal{Z}_{t}}\circ
c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}.$

Let us finish by explaining the following corollaries. They all follow from
Kimura’s theory of finite dimensionality and are a strong motivation to
establish the Kuga-Satake correspondence at a Chow theoretic level rather than
cohomological one.

######  Corollary 3.9.

With the same assumptions as in Theorem 3.5, for any $\sigma\in B$, any self-
correspondence of $S_{\sigma}$ which is homologous to $0$ is nilpotent. In
particular, for any finite group action $G$ on $S_{\sigma}$ and any projector
$\pi\in\mathbb{Q}[G]$, if $H^{2,0}(S_{\sigma})^{\pi}=0$, then
$CH_{0}(S_{\sigma})_{\mathbb{Q},hom}^{\pi}=0$.

Proof. The first statement follows from Theorem 3.5 Kimura’s work (cf. [17])
which says that abelian varieties have a finite dimensional motive and that
for any finite dimensional motive, self-correspondences homologous to $0$ are
nilpotent.

In the case of a finite group action, we consider as in the previous section
the self-correspondence $\Gamma^{\pi}$. It is nonnecessarily homologous to
$0$, but as it acts as $0$ on $H^{2,0}(S)$, its class can be written as

$[\Gamma^{\pi}]=\sum_{i}\alpha_{i}[C_{i}]\times[C^{\prime}_{i}]\,\,{\rm
in}\,\,H^{4}(S_{\sigma}\times S_{\sigma},\mathbb{Q})$

for some rational numbers $\alpha_{i}$ and curves $C_{i},\,C^{\prime}_{j}$ on
$S_{\sigma}$. Then we have

$[\Gamma^{\pi}-\sum_{i}\alpha_{i}C_{i}\times C^{\prime}_{i}]=0\,\,{\rm
in}\,\,H^{4}(S_{\sigma}\times S_{\sigma},\mathbb{Q}),$

from which we conclude by Kimura’s theorem that the self-correspondence

$Z:=\Gamma^{\pi}-\sum_{i}\alpha_{i}C_{i}\times C^{\prime}_{i}\in
CH^{2}(S_{\sigma}\times S_{\sigma})_{\mathbb{Q}}$

is nilpotent. It follows that
$Z_{*}:CH_{0}(S_{\sigma})_{\mathbb{Q},hom}\rightarrow
CH_{0}(S_{\sigma})_{\mathbb{Q},hom}$ is nilpotent. As it is equal to

$\Gamma^{\pi}_{*}:CH_{0}(S_{\sigma})_{\mathbb{Q},hom}\rightarrow
CH_{0}(S_{\sigma})_{\mathbb{Q},hom},$

which is the projector on $CH_{0}(S_{\sigma})_{\mathbb{Q},hom}^{\pi}$, we
conclude that $CH_{0}(S_{\sigma})_{\mathbb{Q},hom}^{\pi}=0$.

######  Corollary 3.10.

With the same assumptions as in Theorem 3.5, the transcendental part of the
Chow motive of any member of the family of $K3$ surfaces parameterized by
$\mathbb{P}(H^{0}(X,E))$ is indecomposable, that is, any submotive of it is
either the whole motive or the $0$-motive.

Proof. Recall that the transcendental motive of $S_{\sigma}$ is $S_{\sigma}$
equipped with the projector $\pi_{tr}$ defined in (18). Let now $\pi\in
CH^{2}(S_{\sigma}\times S_{\Sigma})_{\mathbb{Q}}$ be a projector of the
transcendental motive of $S_{\sigma}$, that is
$\pi\circ\pi_{tr}=\pi_{tr}\circ\pi=\pi$. Since $h^{2,0}(S_{\sigma})=1$,
$\pi_{*}$ acts either as $0$ or as $Id$ on $H^{2,0}(S_{\sigma})$. In the first
case, ${\rm Ker}\,(\pi_{*})_{\mid H^{2}(S,\mathbb{Q})_{tr}}$ is a sub-Hodge
structure with $(2,0)$-component equal to $H^{2,0}(S)$. Its orthogonal
complement is then contained in $NS(S_{\sigma})_{\mathbb{Q}}$ which implies
that $\pi_{*}=0$ on $H^{2}(S,\mathbb{Q})_{tr}$. In the second case, we find
similarly that $\pi_{*}=Id$ on $H^{2}(S,\mathbb{Q})_{tr}$. Since
$\pi=\pi_{tr}\circ\pi=\pi\circ\pi_{tr}$, it follows that $\pi_{*}$ acts either
by $0$ or as $\pi_{tr}$ on $H^{*}(S,\mathbb{Q})$. Hence the cohomology class
of either $\pi$ or $\pi_{tr}-\pi$ is equal to $0$, from which we conclude by
theorem 3.5 that $\pi$ or $\pi_{tr}-\pi$ is nilpotent. As both are projectors,
we find that $\pi=0$ or $\pi_{tr}=0$ in $CH^{2}(S_{\sigma}\times
S_{\sigma})_{\mathbb{Q}}$.

## References

  * [1] Y. André. Pour une théorie inconditionnelle des motifs, Publications Mathématiques de l’IHÉS, 83 (1996), p. 5-49.
  * [2] Y. André. On the Shafarevich and Tate conjectures for hyper-Kähler varieties. Math. Ann. 305 (1996), no. 2, 205-248.
  * [3] R. Barlow, A simply connected surface of general type with $p_{g}=0$, Invent. Math. 79 (1985), no. 2, 293-301.
  * [4] R. Barlow, Rational equivalence of zero cycles for some more surfaces with $p_{g}=0$, Invent. Math. 79 (1985), no. 2, 303-308.
  * [5] I. Bauer, F. Catanese, F. Grünewald, R. Pignatelli. Quotients of products of curves, new surfaces with $p_{g}=0$ and their fundamental groups, Amer. J. of Math., Volume 134 (2012), pp. 993-1049.
  * [6] A. Beauville, C. Voisin. On the Chow ring of a K3 surface, J. Algebraic Geom. 13 (2004), no. 3, 417–426.
  * [7] S. Bloch. Lectures on algebraic cycles, Duke University Mathematics Series. IV (1980). Durham, North Carolina: Duke University, Mathematics Department.
  * [8] S. Bloch, V. Srinivas. Remarks on correspondences and algebraic cycles, Amer. J. of Math. 105 (1983) 1235-1253.
  * [9] S. Bloch, A. Kas, D. Lieberman. Zero cycles on surfaces with $p_{g}=0$. Compositio Math. 33 (1976), no. 2, 135-145.
  * [10] C. Böhning, H.-C. Graf von Bothmer, L. Katzarkov, P. Sosna. Determinantal Barlow surfaces and phantom categories, preprint arXiv:1210.0343.
  * [11] F. Catanese. Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math. 63 (1981), no. 3, 433-465.
  * [12] P. Deligne. Théorie de Hodge II, Publ. Math. IHES 40 (1971), 5-57.
  * [13] P. Deligne. La conjecture de Weil pour les surfaces $K3$, Inventiones Math. 15, 206-226 (1972).
  * [14] T. Graber, J. Harris et J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc., 16 (1), 57-67 (2003).
  * [15] D. Huybrechts. Symplectic automorphisms of K3 surfaces of arbitrary order, preprint arXiv:1205.3433.
  * [16] H. Inose, M. Mizukami. Rational equivalence of $0$-cycles on some surfaces of general type with $p_{g}=0$. Math. Ann. 244 (1979), no. 3, 205-217.
  * [17] S.-I. Kimura. Chow groups are finite dimensional, in some sense. Math. Ann. 331 (2005), no. 1, 173-201.
  * [18] M. Kuga, I. Satake. Abelian varieties attached to polarized $K3$ surfaces, Math. Annalen 169, 239-242 (1967).
  * [19] D. Morrison. The Kuga-Satake variety of an abelian surface. J. Algebra 92 (1985), no. 2, 454-476.
  * [20] S. Mukai. Polarized K3 surfaces of genus 18 and 20, in Complex projective geometry, Cambridge Univ. Press, 1992, pp. 264–276.
  * [21] S. Mukai. Polarized K3 surfaces of genus thirteen, in Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math. 45, Math. Soc. Japan, Tokyo, (2006), pp. 315-326.
  * [22] D. Mumford. Rational equivalence of $0$-cycles on surfaces. J. Math. Kyoto Univ. 9 (1968) 195-204.
  * [23] K. Paranjape. Abelian varieties associated to certain $K3$ surfaces. Compositio Math. 68 (1988), no. 1, 11-22.
  * [24] A. Roitman. The torsion of the group of zero-cycles modulo rational equivalence, Ann. of Math. 111 (1980), 553-569.
  * [25] P. Supino. A note on Campedelli surfaces. Geom. Dedicata 71 (1998), no. 1, 19-31.
  * [26] V. Voevodsky. A nilpotence theorem for cycles algebraically equivalent to zero. Internat. Math. Res. Notices, no. 4, 187-198, (1995).
  * [27] C. Voisin. The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, preprint arXiv:1107.2600, to appear in Ann. Scient. Norm. Sup..
  * [28] C. Voisin. Sur les zéro-cycles de certaines hypersurfaces munies d’un automorphisme. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 4, 473-492.
  * [29] C. Voisin. Remarks on zero-cycles of self-products of varieties, dans Moduli of vector bundles (Proceedings of the Taniguchi congress on vector bundles), Maruyama Ed., Decker (1994) 265-285.
  * [30] C. Voisin. Symplectic involutions of $K3$ surfaces act trivially on $CH_{0}$, preprint arXiv:1204.6684, to appear in Documenta Math..
  * [31] C. Voisin. Lectures on the Hodge and Grothendieck-Hodge conjectures, Rendiconti del Seminario Matematico, Vol. 69, n0 2, (2011).

Centre de mathématiques Laurent Schwartz

91128 Palaiseau Cédex, France

<EMAIL_ADDRESS>