Introduction - Liverpoolpcguletski/papers/INV6.pdf · Let W be the resolution of singularities of the quotient surface S/ι. If A0(W) = 0, ... 0-cycles on algebraic surfaces. A new

BLOCH’S CONJECTURE FOR SURFACES WITHINVOLUTIONS AND OF GEOMETRIC GENUS ZERO

VLADIMIR GULETSKII

Abstract. Let S be a smooth projective surface with pg = 0, let ι be aregular involution acting on S, and let W be the resolution of singularitiesof the quotient surface S/ι. In the paper we prove that Bloch’s conjectureholds for the surface S if and only if it holds for the surface W . This yieldsBloch’s conjecture for all surfaces S whenever the same conjecture is true forthe desingularized quotient W . In particular, Bloch’s conjecture holds true forall numerical Godeaux surfaces with involutions, a “half” of Campedelli sur-faces with involutions, the surface of Craighero and Gattazzo, some Catanesesurfaces and other examples. Applying the same method to K3-surfaces, weprove that if a K3-surface S admits a regular involution whose quotient is ofEnriques type, then the motive M(S) is finite-dimensional.

1. Introduction

Let k be a field, and let V be an algebraic variety over k. An r-dimensional

algebraic cycle Z on V is a linear combination of integral r-dimensional closed

subschemes in V . Two r-cycles Z and Z ′ are rationally equivalent on V if there

exists a positive r + 1-cycle Z on V × P1, dominant over P1, and an r-cycle

W on V , such that the cycle-theoretical specializations Z (0) and Z (∞) at

two fundamental points on P1 are Z +W and Z ′ +W respectively. This is the

minimal adequate equivalence relation on algebraic cycles in the sense of Samuel,

[42], and the Chow group CHr(V ) of r-cycles modulo rational equivalence on V

is an important object of study in algebraic geometry.

If C is a smooth projective curve over k having a k-rational point on it, the

classical Abel-Jacobi theorem asserts that the subgroup A0(C) generated by

degree zero cycle classes in CH0(C) is isomorphic to the Jacobian variety of

the curve C. More generally, algebraically trivial divisors (i.e. algebraic cycles

of codimension one) modulo rational equivalence on V are parametrized by the

Picard variety of the variety V , which makes the study of divisors accomplishable

and efficient in applications. But already in codimension 2 the behaviour of

rational equivalence is very different from what could be expected based on the

intuition taken from the codimension one case.

Indeed, the first instance of codimension 2 algebraic cycles is when we study

rational equivalence of 0-cycles on a smooth projective surface S over k. Assume

for simplicity that the ground field k is C. Let A0(S) be the subgroup of degree

Date: 27 April 2017.2010 Mathematics Subject Classification. 14C15, 14C25, 14J29.Key words and phrases. 0-cycles, rational equivalence, Bloch’s conjecture, motives, sur-

faces of general type, projective duality, vanishing cycles, monodromy, involutions, numericalGodeaux surfaces, Campedelli and Catanese surfaces, K3 and Enriques surfaces.

1

2 VLADIMIR GULETSKII

0 zero-cycle classes in CH0(S), and let H2tr(S) be the transcendental part of the

second (Weil) cohomology group of S. Let also T (S) be the kernel of the Albanese

homomorphism from A0(S) onto the Albanese variety of the surface S. In [37]

Mumford proved that if the group H2tr(S) is not trivial, then T (S) is nonzero and,

moreover, A0(S) cannot be parametrized by an abelian variety, in any reasonable

sense. If, for example, S is a K3-surface over C, all the intersections of curves

on S are all proportional modulo rational equivalence to the second Chern class

of S, [8], whereas the whole group A0(S) is huge by Mumford’s result.

On the other hand, some deep Hodge-theoretical intuition led Bloch to a

conjecture saying that T (S) is 0 provided H2tr(S) is trivial, see [9] and [10].

Soon after, it was generalized to what is nowadays known as the Bloch-Beilinson

conjectures, whose comprehensive exposition can be found in [28]. The Bloch-

Beilinson conjectures imply that all Chow motives with coefficients in Q must

be finite-dimensional in the sense of Kimura and O’Sullivan, see [1]. If M(S) is

the motive of S and H2tr(S) = 0, then M(S) is finite-dimensional if and only if

Bloch’s conjecture holds for S, see [25] and [30]. In other words, Bloch’s con-

jecture is a particular case of the expected finite-dimensionality of all motives of

smooth projective varieties over a field. Notice that if a motive M is of abelian

type, i.e. it lies in the subcategory additively and tensorily generated by curves,

then it is finite-dimensional, see [30].

The status of Bloch’s conjecture is as follows. If S the Kodaira dimension of

S is strictly smaller than 2, it was proven in [11] by ad hoc methods. If S is

of general type, vanishing of H2tr(S) implies that S is regular, so that there are

neither analytic nor abelian invariants living on S, and the conjecture simply

asserts that any two points on S are rationally equivalent to each other. This is

the hard case of Bloch’s conjecture, unknown for the most part, except for several

cases: (i) the surfaces studied by Barlow in [3] and [4], (ii) the classical Godeaux

surfaces studied by Voisin in [50] and [51], (iii) the Catanese surfaces and general

Barlow surfaces solved by Voisin in [53], and finite quotients of products of curves

proven by Kimura in [30] (see also [25] and [5]).

Recently we also proved the Bloch conjecture for the numerical Godeaux sur-

face with ample canonical class, constructed by Craighero and Gattazzo in [16],

and predicted by Stagnaro in [47], see [24]. Notice that this surface is simply

connected by the main result in [40].

The purpose of the present paper is to generalize and strengthen the method

presented in [24], and then apply the generalized method to surfaces of general

type with involutions and pg = 0, and also to K3-surfaces with involutions of

Enriques type.

THEOREM A. Let k be an algebraically closed field of characteristic 0,

let S be a smooth projective surface with a regular involution ι acting on

S, and assume that q(S) = 0. Let W be the resolution of singularities

of the quotient surface S/ι. If A0(W ) = 0, then the motive M(S) is of

abelian type, and hence finite-dimensional.

BLOCH’S CONJECTURE FOR SURFACES WITH INVOLUTIONS 3

If pg(S) = 0 and S is of general type, the vanishing of A0(W ) is equivalent to

finite-dimensionality of the motive of the surface W . The following modification

of Theorem A shows that Bloch’s conjecture for the surface S not only implies

but rather is equivalent to the same conjecture for the surface W .

THEOREM B. Let S be a smooth projective surface over k with a regular

involution ι and pg(S) = 0. As above, let W be the resolution of singular-

ities of the quotient surface S/ι. Then Bloch’s conjecture holds true for

the surface S if and only if it holds true for the surface W .

These two theorems, closely related to each other, can be applied to many

concrete types of surfaces over algebraically closed fields of characteristic zero.

We collect a few such applications in the following two corollaries.

COROLLARY C. Let S be a smooth projective surface over k with a

regular involution ι and pg(S) = 0. Bloch’s conjecture holds true for S if

the desingularized quotient W is not of general type. In particular, it is

true for all numerical Godeaux surfaces with involutions, each numerical

Campedelli surface whenever the bicanonical map is composed with an

involution on it, the surface of Craighero and Gattazzo, some Catanese

surfaces and other examples.

COROLLARY D. Let S be an algebraic K3-surface over k with a regular

involution ι acting with no fixed points on S, so that the quotient surface

S/ι is smooth and of Enriques type. Then the motive M(S) is abelian,

and hence finite-dimensional.

The paper is organized as follows. In Section 2 we fix notation and prove

a few motivic lemmas convenient for what follows. In Section 3 we recall how

the irreducibility of the monodromy action on vanishing cycles helps to study

0-cycles on algebraic surfaces. A new result is, however, Theorem 6 which shows

how the monodromy argument helps to prove finite-dimensionality of the motive

of a surface embedded into a projective space. The proofs of Theorems A and B

are presented in Section 4, where we develop all the necessary projective duality

arguments, and then compose them with the monodromy argument and motivic

finite-dimensionality lemmas in order to prove the results claimed by Theorems

A and B. In the last Section 5 we show how to deduce Corollaries C and D

from the main theorems. In particular, we prove Bloch’s conjecture for new

families of surfaces of general type with pg = 0, including numerical Godeaux

and Campedelli surfaces with involutions and other examples.

Acknowledgements. The author is grateful to Kalyan Banerjee, Ivan

Cheltsov and Alexander Tikhomirov for helpful discussions on the internet.


2. Notation and auxiliary results

Let k be an algebraically closed field of characteristic 0. If V is an algebraic

variety over k, let CHr(V ) be the Chow group of r-cycles modulo rational equiv-

alence on V . If V is equidimensional of dimension n, we may write CHn−r(V )

instead of CHr(V ). Let also Ar(V ) be the subgroup generated by algebraically

trivial cycle classes in CHr(V ), and the same for upper indices.

For any abelian group A let AQ be the tensor product of A and Q over Z. If Wand V are two smooth projective varieties over k, and W = ∪jWj the connected

components of W ,

CHm(W,V )Q = ⊕jCHej+m(Wj × V )Q ,

is the group of correspondences of degree m from W to V , where ej = dim(Wj).

For any two composable correspondences f ∈ CHn(W,V )Q and g ∈ CHm(V, U)Qthe composition g ◦ f is an element in CHm+n(W,V )Q. The category of smooth

projective varieties and correspondences between them is additive, with the di-

agonals ∆V playing the role of identity morphisms in it. The category of Chow

motives CHM(k)Q over k is then the pseudoabelian envelope of the category of

correspondences. The objects in CHM(k)Q are triples (V,Σ, n), where Σ is an

idempotent in the associative algebra CH0(V, V )Q, and n ∈ Z. For two motives

M = (W,Ξ,m) and N = (V,Σ, n) ,

the group of morphisms from M to N is given by the formula

Hom(M,N) = Σ ◦ CHn−m(W,V )Q ◦ Ξ .

The tensor product in CHM is given by the formula

(W,Ξ,m)⊗ (V,Σ, n) = (W × V,Ξ⊗ Σ,m+ n) ,

where Ξ⊗Σ is the fibred product of the two correspondences with appropriately

permuted factors. The tensor unity 1 is the motive of Spec(k), and the Lefschetz

motive L is defined to have

M(P1) = 1⊕ L .

The motive L is invertible, and Lm is the m-fold tensor power L⊗m for any

integer m. The tensor category CHM(k) is rigid with the duality given by the

formula

(V,Σ, n)∨ = (V,Σt, d− n) ,

where X is a smooth projective variety of pure dimension d over k. The motive

M(V ) of a smooth projective V is a triple (X,∆X , 0), and for any morphism

f : W → V the morphism M(f) : M(V ) → M(W ) is the transposed graph

Γtf . This gives a monoidal contravariant functor from the category of all smooth

projective varieties to the category of Chow motives CHM(k)Q. Further details

on the language of Chow motives can be found in [43].

Since the category CHM(k) is tensor, for any Chow motive M and any non-

negative n one can easily define its wedge and symmetric powers ∧nM and


SymnM . A Chow motive M is said to be finite-dimensional, if it decomposes

into two direct summands,

M = M+ ⊕M− ,

such that

∧mM+ = 0 and SymnM− = 0 ,

for somem and n, see [30]. Then we also say thatM+ is evenly finite-dimensional

and M− is oddly finite-dimensional. The dimension of M+ is then the maximal

m, such that ∧mM+ = 0, and the dimension of M− is the maximal n, such that

SymnM+ = 0. The total dimension of a finite-dimensional motive is the sum of

the dimensional of the even and odd parts.

If f : M → N is a morphism in CHM(k)Q, we will be saying that f is

an embedding of M into N , or that M is a submotive in N , if there exists

a morphism g : N → M , such that g ◦ f = idM . Then, of course, f ◦ g is

an idempotent in End(N) which cuts out M from N . If M is a submotive in

N and N is evenly (oddly) finite-dimensional, then M is evenly (respectively,

oddly) finite-dimensional. Moreover, all other expected formal properties of

finite-dimensionality hold true in CHM(k)Q.

An essential result along this line says that if M is finite-dimensional and

f : M →M a numerically trivial endomorphism of M , then f is nilpotent in the

associative algebra End(M), see Proposition 7.5 in [30].

Let now S be a smooth projective surface over k. LetKS be a canonical divisor

on S and OS(KS) the canonical sheaf on S. Then

pg(S) = dimH0(S,OS(KS))

is the geometrical genus of S.

For any positive integer n we have the linear system |nKS| which determines

the rational map

ϕ|nKS | : S 99K PN ,

where N = dim |nKS|. The maximum of dimensions of the images of the rational

maps ϕ|nKS |, for all n ≥ 1, is called the Kodaira dimension of S and denoted by

κ(S). By definition, κ(S) = −1 if the linear system |nKS| is empty for all n ≥ 1.

Obviously κ(S) ≤ 2. Another definition of this important invariant is that

κ(S) = −1 + tr.deg(R/k) ,

where

R = ⊕+∞n=0 H

0(S,OS(nKS))

is the pluricanonical ring of the surface S. If κ(S) < 2 then we say that S is of

special type, and S is said to be of general type if κ(S) = 2.

Consider the Albanese variety Alb(S) of S, i.e. the dual to the Picard variety

of S. The irregularity q(S) of the surface S is, by definition, the dimension of the

abelian variety Alb(S). If CH0(S) is the Chow group of 0-cycles modulo rational

equivalence on S, and if A0(S) is a subgroup generated by degree 0 cycles classes


in CH0(S), the Albanese variety arises together with the surjective Albanese

homomorphism

aS : A0(S)→ Alb(S)

whose kernel will be denoted by T (S). By Roitman’s theorem the Albanese

homomorphism induces an isomorphism on the torsion subgroups, so that the

Albanese kernel T (S) is torsion free. If k is C, the Albanese homomorphism can

be also understood as an Abel-Jacobi map, and the Albaneser kernel T (S) as

the corresponding Abel-Jacobi kernel for S.

Let

χ(OS) =∑i

dim(−1)iH i(S,OS)

is the Euler characteristic of the structural sheaf OS. Then we have the Noether

formula

χ(OS) = 1− q + pg ,

see, for example, p. 63 in [41] or any other book on algebraic surfaces.

If S is of general type,

χ(OS) > 0 ,

see, for example, Theorem X.4 on page 114 in [6]. The Noether formula then

gives us the inequality

q(S) ≤ pg(S) .

Therefore, if pg(S) = 0 then q(S) = 0 and the Albanese variety vanishes. In such

a case, T (S) coincides with A0(S).

Notice also that A0(S) is a divisible group, see the first lemma in [7]. If

q(S) = 0, then A0(S) is torsion free, and therefore it is uniquely divisible, i.e. a

vector space over Q.

As we mentioned already in Introduction, Bloch’s conjecture asserts that if

pg(S) = 0, then T (S) = 0. If S is of general type, pg(S) = 0 implies q(S) = 0,

and therefore the conjecture simply states that for any two closed points P and

Q on the surface S the point P is rationally equivalent to the point Q on S.

Fix a closed point P0 on the surface S and set

π0 = [P0 × S] and π4 = [S × P0] .

These two correspondences are idempotents and define the motives M0(S) = 1

and M4(S) = L2 of the surface S. Using a smooth hyperplane section C of S,

and the Poincare divisor on the product of C and the Jacobian of C, one can

construct the Picard and its dual Albanese projectors, π1 and π3 respectively,

both with coefficients in Q, and the corresponding motives M1(S) and M3(S)

have the expected behaviour, see [38]. Subtracting, π0, π4, the Picard and Al-

banese projectors π1 and π3 from the diagonal ∆S we get the middle projector

π2. Respectively, we obtain the decomposition of M(S) into the direct sum of

five motives M i(S), i = 0, . . . , 4, in the category CHM(k)Q. The latter decom-

position can be refined further by splitting the algebraic part from M2(S), see

[29]. Namely, let ρ be the Picard number of S and choose ρ divisors

D1, . . . , Dρ


whose cohomology classes generate the second Weil cohomology group H2(S).

Choose the Poincare dual divisors D′1, . . . , D

′ρ, so that the intersection number

⟨Di ·D′j⟩ is the Kronecker symbol. For each index i let

π2,i = [Di ×D′i]

be class of the product of Di and D′i on S × S. Then π2 decomposes into the

algebraic idempotent

πalg2 =

ρ∑i=1

π2,i

and the transcendental projector

π2,tr = π2 − π2,alg .

The resulting decomposition is

M(S) = 1⊕M1(S)⊕M2alg(S)⊕M2

tr(S)⊕M3(S)⊕ L2

in CHM(k)Q.

Lemma 1. Let S be a smooth projective surface over k with pg(S) = 0. Bloch’s

conjecture holds true for S if and only if

∧nM2tr(S) = 0

for some n.

Proof. By Theorem 7.10 in [30], and by Theorem 7 in [25], we have that the Bloch

conjecture holds for S if and only if the motive M(S) is finite-dimensional. The

motives M0(S), M2alg(S) and M4(S) = L

2 are evenly finite-dimensional. The

motives M1(S) and M3(S) are oddly finite-dimensional. Then the motive M(S)

is finite-dimensional if and only if the transcendental motive M2tr(S) is evenly

finite-dimensional.

Lemma 2. Let L/k be an extension of algebraically closed fields of characteristic

0, let S be a smooth projective surface over k with pg(S) = 0, and let

SL = S ×Spec(k) Spec(L) .

Then Bloch’s conjecture holds for the surface S if and only if it holds for the

surface SL.

Proof. Indeed, the scalar extension functor from CHM(k)Q to CHM(L)Q is ad-

ditive and monoidal. It follows that this functor preserves finite-dimensional ob-

jects. If Bloch’s conjecture holds for S, i.e. the motiveM(S) is finite-dimensional,

so is the motive M(SL), and hence Bloch’s conjecture holds also for SL. Vice

versa, if Bloch’s conjecture holds for M(SL), i.e. ∧nM2tr(S) = 0 for some n,

the projector Σ defining the wedge power ∧nM2tr(S) vanishes under the field

extension from k to L. Then Σ = 0 by the main theorem in [21].

Let now k0 be the algebraic closure of the minimal subfield of definition of the

surface S in k, let S0 be a smooth projective model of the surface S over k0, so

that

S = S0 ×Spec(k0) Spec(k) .


Fix an embedding

σ : k0 → C

of the field k0 into C. Let S0,σ be the surface over σ(k0) induced by S0 and the

embedding σ in the obvious way, and let

S0,σC = S0,σ ×Spec(σ(k0)) Spec(C) .

The following lemma shows us that in proving Bloch’s conjecture for S over an

arbitrary algebraically closed field k of characteristic 0, one can assume, without

loss of generality, that k = C.

Lemma 3. The following items are equivalent:

• Bloch’s conjecture holds for S;

• Bloch’s conjecture holds for S0;

• Bloch’s conjecture holds for S0,σ;

• Bloch’s conjecture holds for S0,σC.

Proof. Apply Lemma 2.

3. The monodromy argument for surfaces over C

The purpose of this section is dual. First we recall, for the convenience of the

reader, the monodromy argument considered in the 2-dimensional case only (see

[2] and [52]). For simplicity, and thanks to Lemma 3, we may assume that the

ground field k is C, in which case the results can be made stronger. Secondly,

we will combine the monodromy argument with the idea of the transcendence

degree of a 0-cycle class and its relation to the structure of the motive, as in [22].

So, let S be a smooth projective surface given over C, and let Σ be the linear

system of a very ample divisor on S. Let d be the dimension of Σ, and let

S ⊂ Pd∗

be the projective embedding of S induced by Σ, where Pd∗ is the dual projective

space parametrizing hyperplanes in Pd. For any closed point t ∈ Σ let Ct be the

corresponding member of Σ, and let

rt : Ct → S

be the embedding of Ct into S. Let also ∆ be the discriminant locus of the linear

system Σ, i.e. a subset, such that

t ∈ Σr∆ ⇔ Ct is smooth .

Notice that if the section Ct is smooth, then it is also connected by the theorem of

Fulton and Hansen, see [20], or the modern treatment in [36]. The discriminant

locus ∆ can be also identified with the dual hypersurface S∨ to S in Pd∗. Recall

that S∨ parametrizes hyperplanes in Pd which are tangent to S.

Now, let A0(S) be the Chow group of degree zero 0-cycles modulo rational

equivalence on S. For any t ∈ Σ let A0(Ct) be the Chow group of degree zero


0-cycles modulo rational equivalence on Ct. If Ct is smooth and connected, we

obtain the Gysin push-forward homomorphism

rt∗ : Jt = A0(Ct)→ A0(S)

where the Chow group A0(Ct) is identified with the Jacobian

Jt = J(Ct)

of the curve Ct. We will be saying that rt∗ is a Gysin homomorphism on Chow

groups induced by rt, and the kernel

Gt = ker(rt∗)

will be called the Gysin kernel associated with the hyperplane section Ct.

We also have the Gysin homomorphism

rt∗ : H1(Ct,Q)→ H3(S,Q)

on cohomology groups, whose kernel, when Ct is smooth and connected, is noth-

ing else but the group of vanishing cycles

H1(Ct,Q)van ,

see, for example, [52]. Let Bt be the abelian subvariety in Jt corresponding to

the Q-vector subspace H1(Ct,Q)van in the space

H1(Ct,Q) ≃ H1(Jt,Q) .

Since the Gysin homomorphism on cohomolgy groups corresponds to the regular

homomorphism on Albanese varieties,

Jt ≃ Alb(Ct)→ Alb(S) ,

one can show that, up to torsion, Bt is the kernel of the latter.

For short, let

U = Σr∆ .

The following theorem was stated as an exercise on pages 304 - 305 in [52], and

is the case p = 1 of Theorem 19 in [2].

Theorem 4. For every closed point t in U there exists an abelian subvariety At

in Bt, such that the Gysin kernel Gt is the union of a countable collection of

shifts of At inside Jt. Moreover, for every t in U either At = Bt, and then Gt is

the union of a countable collection of shifts of Bt, or At = 0, in which case the

kernel Gt is countable.

Proof. This result is a consequence of the irreducibility of the monodromy action

on vanishing cycles, see [44], [17], [18] and [34], and Mumford’s countability

Lemma 3 in [37]. Here we give only an outline of how to deduce Theorem 4 from

these results, merely for the convenience of the reader. The details can be found

in [2].

Let p1 and p2 be the projections of the universal hyperplane H on Pd and Pd∗

respectively, let C → S be the pull-back of p1 with respect to S ⊂ Pd, and let

f : C → Pd∗


be the composition of the closed embedding C ⊂H with p2. For any morphism

of schemes

D → Pd∗

let HD → D be the pull-back of p2 w.r.t. D → Pd∗, let CD be the fibred product

of C and HD over H , and let

fD : CD → D

be the induced projection. Let also

SD = S ×D → D

be the pull-back of trivial family S × Pd∗ → Pd∗ w.r.t. the morphism D → Pd∗.

For any t ∈ U , choose D to be a line passing through t in Pd∗, and such that

fD is a Lefschetz pencil for S. It means, in particular, that if Ct is the effective

divisor on S, corresponding to the point t in Σ, then Ct = F ∩Ht, and Ht, as a

point of Pd∗, sits on D. Moreover, the surface CD can be obtain as the blow up

of S at the base locus Ct ∩ Ct′ of the Lefschetz pencil fD.

Let

V = (Σr∆) ∩D ,

and let

π1(V, t)

be the fundamental group of V pointed at t. The group H1(Ct,Q)van is generated

by vanishing cycles. The standard Noether-Lefschetz formula gives us that the

local monodromy representation of π1(V, t) on H1(Ct,Q)van is irreducible, see,

for example, [34] or [52]. Zarisky’s theorem allows us to deduce that the global

monodromy representation

π1(U, t)→ H1(Ct,Q)van

is irreducible too, see 3.2.2 in [52].

Now, for any natural n, let

Symn(S)

be the n-th symmetric power of the surface S. By Mumford’s Lemma 3 in [37],

the preimage of 0, under the natural map

Symn(S)× Symn(S)→ CH0(S)

sending (A,B) to A−B, is the union of a countable collection of Zariski closed

subsets in Symn(S)× Symn(S). Fix a point P0 on S, and construct the infinite

symmetric power Sym∞(S) with the aid of P0. Then rational equivalence of

0-cycles on S can be encoded via rational connectivity on the group completion

(Sym∞(S))+

of the monoid, see Theorem 6.8 in [48]. Together with Mumford’s result and

the uncountability of the ground field, this allows us to show that there exists

an abelian subvariety At in Jt, such that ker(rt∗) is the union of a countable

collection of shifts of At inside Jt.


Let Lt be the Q-vector subspace in

H1(Ct,Q) ≃ H1(Jt,Q) .

corresponding to the abelian subvariety At in Jt. As

Lt ⊂ H1(Ct,Q)van ,

we have that

At ⊂ Bt ,

for each

t ∈ Σr∆ .

Moreover, Lt is a π1(U, t)-submodule in H1(Ct,Q)van. Since the monodromy

action on H1(Ct,Q)van is irreducible, either

Lt = 0 ,

and then

At = 0 ,

or

Lt = H1(Ct,Q)van ,

and then

At = Bt .

This finishes the proof of the theorem.

Corollary 5. If S is regular, i.e. q(S) = 0, then, for any t ∈ U , either Gt = Jt,

i.e. rt∗ = 0, or Gt is countable.

Proof. If q = 0, then H3(S,Q) = 0, so that H1(Ct,Q)van coincides with

H1(Ct,Q), and hence Bt coincides with Jt, for each closed point t in U .

Let us now connect the monodromy argument to the motivic point of view

presented in the previous section. Let CHM(k)abQ be the minimal possible tensor

and pseudoabelin subcategory of the category CHM(k)Q generated1 by motives

of smooth projective curves. Then, of course, the motives of abelian varieties are

there, so that one may call objects in CHM(k)abQ abelian motives over k, with

coefficients in Q. It is well-known that all abelian motives are finite-dimensional,

see [30].

The group A0(S) is said to be weakly representable if one of the equivalent

conditions of Proposition 1.6 in [28] is satisfied. If pg(S) = 0, the standard

properties of weak representability give us that Bloch’s conjecture holds for S,

i.e. M(S) is finite-dimensional, if and only if the Chow group A0(S) is weakly

representable. If, moreover, q(S) = 0, finite-dimensionality of M(S) is the same

as vanishing of A0(S). If pg(S) = 0, we do not know in general whether the tran-

scendental motive M2tr(S) is finite-dimensional or of abelian type (see, however,

the results in [39]).

1additively and tensorially


Theorem 6. Let q(S) = 0 and assume that there exists a curve T in Pd∗, such

that Gt = Jt for any t in some nonempty Zariski open subset in T ∩ U . Then

the motive M(S) is of abelian type, and hence finite-dimensional.

Proof. Let k0 be the algebraic closure in k = C of the minimal field of definition

(see [54]) of the surface S, and let S0 be the model of S over k0, so that S is the

surface S0× Spec(C). Let P be a closed point on S whose transcendence degree

over k0, in the sense of [22], is equal to 2. In other words, if P0 is the image of

the unique point under the composition of the morphism

P : Spec(C)→ S

with the canonical morphism

S → S0 ,

the transcendence degree of the residue field of the point P0 over k0 is 2.

Since the union of all sections Ct, for all closed points t ∈ T , is the whole

surface S, we can choose a closed point t in T ∩ U , such that Ct passes through

P on S. On the other hand, we can also choose an integral curve D0 on S0 and

look at the corresponding curve

D = D0 × Spec(C)

on S. Since the divisor C = Ct is very ample, by the Nakai-Moishezon criterion

we have that

C ·D > 0 .

It means, in particular, that the set-theoretic intersection of C andD is nonempty.

Choose a point Q in C ∩D. Since, by assumption, Gt = Jt for any t in T ∩U ,

we have that the point P is rationally equivalent to the point Q on S.

Now, since Q sits on a curve having a model over k0, the transcendence degree

of the rational cycle class [P ] is less or equal to 1, see Lemma 3.2 in [22]. Then

the motive M(S) is a submotive in the direct sum of motives of curves and points

by Theorem 3.6 on page 564 in loc.cit. It follows that the motive M(S) is of

abelian type, and hence finite-dimensional.

Corollary 7. Under the assumptions of Theorem 6, if, moreover, pg(S) = 0, the

Chow group A0(S) is trivial.

Proof. Obvious.

4. The proofs of Theorems A and B

Let k be an algebraically closed field of characteristic 0, and let S be a smooth

projective surface over k. Assume there exists a regular involution

ι : S → S

acting on S, i.e. a regular automorphism of order 2 of the surface S. Without

loss of generality, we may also assume that the surface S is minimal, i.e. not

containing (−1)-curves. This is our basic input in this paper.


Since S is minimal, the locus of fixed points of the involution ι is either empty

or the union of a smooth possibly reducible curve R and a finite collection of

points

P1, . . . , Ps

on S. Consider the quotient surface

V = S/ι

and the quotient regular map

τ : S → V .

If the fixed locus is nonempty, the surface V is obviously singular with quotient

singularities concentrated at the points

Qi = τ(Pi) , i = 1, . . . , s ,

on V . Since quotient singularities are rational, see Proposition 5.15 on page 158

in [33], the surface V is normal. Moreover, as ι is an involution, the singularities

of V are are ordinary double points. Let

µ : F → S

be the blowing up of the surface S at the points P1, . . . , Ps, and denote by the

same symbol

ι : F → F

the induced involution on F . The fixed locus of the action of ι on F consists

of the preimage of the curve R under the map µ, which we denote by the same

symbol R, and the exceptional curves

E1, . . . , Es

over the points P1, . . . , Ps. The the quotient surface

W = F/ι ,

however, is smooth. We obtain the commutative square

F

π

��

µ // S

τ

��W

ν // V

of surfaces and regular morphisms over k. The surface W is the resolution of

singularities on the normal surface V . One could also write

W = WS, ι

in order to stress that the surface W is uniquely defined by the surface S and

the involution ι acting on S.

If the fixed locus of the involution ι is empty on S, then V is smooth, and we

set W = V and F = S.


Let D be a divisor on SF . The Riemann-Roch formula gives us that

χ(OF (D)) =1

2D · (D −KF ) + pa(F ) + 1 ,

see for example page 433 in [26].

Lemma 8. For any positive integer h there exists a divisor D on F , such that

D is very ample, ι-invariant and h0(F,OF (D)) > h.

Proof. Take any ample divisor D0 on F . As D0 is ample, i.e. the sheaf OF (D0) is

generated by its global sections, so is the divisor ι∗(D0). Since the tensor product

of two OF -modules generated by global sections possesses the same property (see,

for example, Corollary 4.5.7 in [23]) it follows that the sum of two ample divisors

is ample. In particular, the divisor

D1 = D0 + ι∗D0

is ample on F . Moreover, it is ι-invariant on F . Then, for a big enough m ∈ Zthe divisor

D2 = mD1

is very ample and ι-invariant on F . Find a positive integer c0, such that

H i(F,OF (cD2)) = 0

for all c > c0 and each i > 0 by Serre’s vanishing theorem, see Theorem 5.2 on

page 228 in [26]. Then h0(F,OF (cD2)) is the same as χ(OF (cD2)), and hence,

by Riemann-Roch,

h0(F,OF (cD2)) =1

2· cD2 · (cD2 −KF ) + pa(F ) + 1 .

By Nakai-Moishezon criterion, the first summand from the right hand side is

positive and big, when c is big. Set

D = cD2 .

Increasing c we see that D possesses all the properties required.

Fix once and for all a very ample ι-invariant divisor D on F with big enough

odd number

h0(F,OF (D)) ,

which exists by Lemma 8. Let

Σ = |D|be the linear system of the divisor m and let

d = dim(Σ) = h0(F,OF (D))− 1

be the dimension of it.

The linear system Σ can be identified with the dual projective space Pd∗. Since

the divisor D is very ample, the regular morphism

ϕΣ : F → Pd


is a closed embedding of the surface F into the d-dimensional projective space

by means of the linear system Σ. Recall that for each closed point P on F the

collection of all divisors in Σ passing through P is a hyperplane HP in

Pd = P(H0(F,OF (D))) .

The hyperplane HP gives the corresponding point ϕΣ(P ) in Pd.

For any closed point t in Σ let Ct be the corresponding effective divisor on F ,

and let

∆ = {t ∈ Σ | Ct is not smooth}be the discriminant locus of the linear system Σ. The discriminant locus ∆ is

nothing else but the projectively dual variety F∨ to F in Pd∗, i.e.

F∨ = ∆ .

The defect

def(Σ) = d− 1− dim(∆)

of the linear system Σ is 0, i.e. ∆ is a hypersurface in Σ, see Example 7.5 in [49].

For any closed point P on F let TP be the embedded tangent space to F at

P . Let then

C = {(P,H) ∈ F × Pd∗ | TP ⊂ H}be the conormal variety of the surface F , see Definition 1.9 in [49] or any other

source on projective duality. Since F is smooth, the condition TP ⊂ H is equiv-

alent to the condition H ∩ F is singular at P , see, for example, the top of page

in 360 in [31] or page 173 in [32]. Therefore, we also have that

C = {(P,H) ∈ F × Pd∗ | H ∩ F is singular at P} .

The dual hypersurface F∨ = ∆ is the image of the conormal variety C under

the second projection from F × Pd∗ onto Pd∗. Let

Fα←− C

β−→ ∆

be the obvious projections of the conormal variety C onto F and ∆.

For any point P on F , the fibre α−1(P ) consists of the hyperplanes in Pd

containing the embedded tangent space TP to F at P , and therefore it is a

d−3-dimensional linear subspace in Pd∗. Therefore, the variety C is a projective

bundle over F with fibres of dimension d − 3. It follows that C is irreducible,

and therefore the dual variety ∆ is also irreducible, see, for example, page 7 in

[49].

The variety C is smooth, and the morphism β is a resolution of singularities

on ∆, see Theorem 1.15 in [49]. For any hyperplane H in ∆ = F∨ let

CH = H ∩ F

be the curve cut out on F by the hyperplane H. Then β−1(H) is the set of closed

points P on the surface F , such that the curve CH is singular at P .

Next, since the divisor D is ι-invariant, the order two group

G = {id, ι}


acts on the projective space Σ = Pd∗. The latter involution ι on Pd∗ induces the

involution on Pd, such that the embedding F ⊂ Pd is ι-equivariant.

In what follows, for any linear subspace Π of dimension m in Pd let Π⊥ be

the corresponding orthogonal linear subspace of dimension d − 1 − m in Pd∗

parametrizing all hypersurfaces in Pd which contain Π. The linear space Π⊥ can

be also considered as the variety dual to the linear space Π, so that we may

also write Π⊥ = Π∨. Let, furthermore, Π∗ be the projective space of the same

dimension m dual to Π = Pm.

The locus of fixed points in Pd under the action of G is the disjoint union of

two linear subspaces,

Fix G(Pd) = Λ ∪ Ξ ,

and, respectively, the locus of fixed points of the action of G in Σ is the disjoint

union,

Fix G(Σ) = Λ⊥ ∪ Ξ⊥ ,

of two polar linear subspaces in Σ. Clearly,

dim(Λ) + dim(Ξ) = d− 1 ,

and the same for the orthogonal linear spaces Λ⊥ and Ξ⊥.

Now we are ready to prove Theorems A and B stated in Introduction.

Theorem 9. Let S, ι and W = WS, ι be as above. If A0(W ) = 0, then the motive

M(S) is of abelian type, and hence finite-dimensional.

Proof. Since F is birational to S, Bloch’s conjecture for S is equivalent to the

same conjecture for S. Apart for the standard monodromy argument, i.e. Corol-

lary 5, and the cycle-theoretic Corollary 7, the essential geometrical component

of the proof consists of showing that the union Λ⊥ ∪ Ξ⊥ is not contained in ∆.

To prove the latter we assume the opposite, i.e. that both linear spaces Λ⊥ and

Ξ⊥ are contained in the dual hypersurface ∆ = F∨ in Pd∗. Since

dim(Λ) + dim(Ξ) = d− 1

and

d > 5

by Lemma 8, it follows that the dimension of either Λ or Ξ is strictly less than

d− 3, say

dim(Λ) < d− 3 .

Consider the projection

πΛ : Pd 99K Ξ

sending any point

P ∈ Pd r Λ

to the unique point of intersection of the span PΛ and Ξ. Let

FΛ = πΛ(F )

be the Zariski closure of the image of the quasi-projective surface F r Λ under

the projection πΛ. The dual variety F∨Λ to the variety FΛ is a closed irreducible


subvariety in the dual projective space Ξ∗. For any hyperplane Π in Ξ the span

of Λ and Π is a hyperplane in Pd containing Λ. Vice versa, any hyperplane

containing Λ in Pd intersects Ξ along a hyperplane in Ξ. This allows us to

identify Ξ∗ with the orthogonal space Λ⊥ in Pd∗ and look at the dual variety F∨Λ

as a subvariety in Pd∗. Since dim(Λ) < d − 3, the variety F∨Λ is an irreducible

component of the set-theoretical intersection Λ⊥ ∩F∨ by Proposition 1.1 in [14].

But we assumed that Λ⊥ is a subset in F∨, whence F∨Λ = Λ⊥ = Ξ∗. This is

contradiction, as the dual variety F∨Λ has a nonzero codimension in Ξ∗.

Thus, Λ⊥ is not a subset in ∆, just because the dimension of Λ is strictly

smaller than d− 3, and then we have that

UΛ = (Σr∆) ∩ Λ⊥ = ∅ .

Now, let C be a member of UΛ ⊂ Σ, i.e. C = Ct is a curve for some closed

point t in UΛ. Then C is smooth and ι-invariant member of Σ. Moreover, C is

connected by Corollary 7.9 on page 244 in [26] (see also [20] or [36]).

Notice also that since the fixed locus of ι on F is 1-dimensional, the fact that

Λ⊥ ⊂ ∆ implies that Λ∩F = ∅, and hence the fixed locus on F is cut out by the

linear space Ξ. That would be not the case if we could work straightforwardly

with the surface S, in which case the end of the proof (see below) would be as

in [24].

Now, if F is rational or ruled, then the motive M(F ), and hence the motive

M(S), is finite-dimensional for trivial reasons. So, we may assume, without loss

of generality, that F is neither rational nor ruled. In such a case, the genus g(C)

of the curve C cannot be 0, and it cannot be 1 by Theorem 5.10 in [45]. Thus,

g(C) > 1 .

Let N be the image of the curve C under the quotient map π from F onto the

surface W , and let g(N) be the genus of the curve N . Let

θ : C → N

be the restriction of the regular map π on C. Notice that the fixed locus of the

action of the involution ι on C is the intersection of C with the set Λ∩ F . And,

certainly, the fixed locus is empty, if the intersection Λ ∩ F is empty.

Let N be the normalization of the curve N , and let

C → N

be the unique morphism from C to N induced by the morphism θ. Assume that

the genus of N is 0. In such a case we obtain that the curve C is hyperelliptic.

But since F is neither rational nor ruled, the hyperellipticity of C contradicts to

Theorem 5.10 in [45] again. Therefore, the genus of N is positive,

g(N) > 0 .


Consider the commutative square

C

θ

��

r // F

π

��N

s // W

in which the horizontal morphisms r and s are obvious closed embeddings. Since

C is ι-invariant, this square is Cartesian. And as s is a proper morphism and π

is a flat morphism of algebraic schemes, applying Proposition 1.7 from [19], we

obtain the commutative square

(1)

A0(C)r∗ // A0(F )

A0(N)

θ∗

OO

s∗ // A0(W )

π∗

OO

Next, Lemma 3 allows us to assume that k = C. The group A0(N) is isomor-

phic to the Jacobian JN of the smooth projective curve N . Since g(N) > 0, the

Jacobian

JN ≃ A0(N)

is an uncountable abelian group. By localization, A0(N) is also an uncountable

abelian group. Moreover, by the assumption of the theorem,

A0(W ) = 0 .

The latter commutative diagram gives us that the uncountable group A0(N)

modulo torsion is injectively embedded into the kernel of the Gysin homomor-

phism r∗ from JC ≃ A0(C) to A0(F ). Therefore, r∗ = 0 by Corollary 5.

Draw an appropriate curve T in the linear space Λ⊥ passing through its Zariski

open subset UΛ. As the genus of the curve N = Nt is positive for every closed

point t in T ∩UΛ, the previous argument gives us that the kernel ker(rt∗) cannot

be countable, and so rt∗ = 0 for each t in T ∩ UΛ.

By Theorem 6, the motive M(F ), and hence the motive M(S) is abelian. It

follows that both motives are finite-dimensional.

Theorem 10. In terms above, if pg(S) = 0, then Bloch’s conjecture holds for S

if and only if it holds for W .

Proof. Since W is the quotient surface of F under the free action of a finite

group, the equalities pg = q = 0 for the surface F imply the same equalities for

the surface W . Moreover, the morphism π induces the embedding

Γtπ : M(W )→M(F )


in the category CHM(k)Q. Therefore, if Bloch’s conjecture holds for the sur-

face F , i.e. the motive M(F ) is finite-dimensional, then, of course, the same

conjecture holds for the surface W .

To prove the converse, assume that Bloch’s conjecture holds for the surface W .

Since q(W ) = 0, it is equivalent to saying that A0(W ) = 0. Then, by Theorem

10, the motive M(F ) is finite-dimensional, i.e. Bloch’s conjecture holds for the

surface F .

5. Numerical Godeaux surfaces and other examples

Now we can also deduce a few corollaries from the main Theorems 9 and 10,

which show how to apply them to concrete families of surfaces with pg = 0, in

the context of Bloch’s conjecture, and also to K3-surfaces.

Corollary 11. Let S be a numerical Godeaux surface with a regular involution

on it. Then Bloch’s conjecture holds true for S.

Proof. Let ι be the regular involution of S, and let W be the resolution of

singularities of the normal quotient S/ι. Let also F be the blow up of S at the

isolated fixed points of the involution ι. By Corollary 3.5 and Proposition 4.5 in

[12] we have that W is either rational or the minimal model of W is an Enriques

surface. The motive of the Enriques surface is finite-dimensional by [15], and

hence Bloch’s conjecture is true for W by Lemma 1. Moreover,

K2F ≤ K2

W

and hence W cannot be birational to a hypersurface in P3. Theorem 9 completes

the proof.

By definition, numerical Campedelli surfaces are smooth projective minimal

surfaces S of general type with pg(S) = 0 and K2S = 2.

Corollary 12. Let S be a numerical Campedelli surface endowed with an in-

volution ι regularly acting on S. Assume that the bicanonical map of S factors

through the quotient map from S to quotient surface S/ι. Then Bloch’s conjecture

holds for S.

Proof. If the bicanonical map of S factors through the quotient map from S to

S/ι, the surface W is either rational or Enriques. Then Bloch’s conjecture holds

for S for the same reason as in Corollary 11.

Remark 13. If the bicanonical map of a numerical Campedelli surface S does

not factors through the quotient map, then W can be of general type, see [13].

If W is of general type, Theorem 9 only says that Bloch’s conjecture is true for

S if and only if it is true for W .

Applying Theorem 9 and the same argument as in Corollary 11, one can also

show that Bloch’s conjecture holds true for some Catanese surfaces as in [53], all

the surfaces listed in the table on page 430 in [46], the K2W = −2 and −4 cases in

the table on page 123 in [35], and other new examples, whenever we know that

W is not of general type.


Corollary 14. Let S be an algebraic K3-surface with a regular involution ι acting

without fixed points on S, so that the quotient V = S/ι is a smooth Enriques

surface. Then the motive M(S) is of abelian type, and hence finite-dimensional.

Proof. Since V is an Enriques surface, we have that pg(V ) = q(V ) = 0 and

the motive M(V ) is finite-dimensional. It follows that A0(V ) = 0. Then apply

Theorem 9.

References

[1] Y. Andre. Une introduction aux motifs (motifs purs, motifs mixtes, periodes). Panora-mas et Syntheses 17. Societe Mathematique de France 2004

[2] K. Banerjee. V. Guletskiı. Rational equivalence for line configurations on cubic hyper-surfaces in P5. Preprint arXiv:1405.6430v2

[3] R. Barlow. A simply connected surface of general type with pg = 0. Inventiones math-ematicae 79 (1985) 293 - 301

[4] R. Barlow. Rational equivalence of zero cycles for some more surfaces with pg = 0.Inventiones mathematicae 79 (1985) 303 - 308

[5] I. Bauer, F. Catanese, R. Pignatelli. Surfaces of general type with geometric genus zero:a survey. Complex and Differential Geometry. Springer Proceedings in Mathematics 8(2011) 1 - 48

[6] A. Beauville. Complex Algebraic Surfaces. London Mathematical Society Student Texts34. Cambridge University Press 1996

[7] A. Beauville. Varietes de Prym et jacobiennes intermediaires. Ann. Sci. Ec. Norm. Sup.10, 309 - 391 (1977)

[8] A. Beauville, C. Voisin. On the Chow ring of a K3 surface. Journal of Algebraic Geom-etry, Volume 13 (2004) 417 - 426

[9] S. Bloch. K2 of Artinian Q-algebras, with application to algebraic cycles. Comm. Alge-bra 3 (1975) 405 - 428

[10] S. Bloch. Lectures on Algebraic Cycles. Duke University Math. Series IV. Durham, NC.Duke University 1980

[11] S. Bloch, A. Kas, D. Lieberman. Zero cycles on surfaces with pg = 0. Compositio Math.33 (1976) 135 - 145

[12] A. Calabri, C. Ciliberto. M. Mendes Lopes. Numerical Godeaux surfaces with an invo-lution. Transactions of the American Mathematical Society. Vol. 359, No. 4 (2007) 1605- 1632

[13] A. Calabri, M. Mendes Lopes, R. Pardini. Involutions on numerical Campedelli surfaces.Tohoku Math. J. 60 (2008) 1 - 22

[14] C. Ciliberto, F. Russo, A. Simis. Homaloidal hypersurfaces and hypersurfaces withvanishing Hessian. Advances in Mathematics 218 (2008) 1759 - 1805

[15] K. Coombes. The K-cohomology of Enriques surfaces. Contemporary Mathematics 126(1992) 47 - 57

[16] P. Craighero, R. Gattazzo. Quintic surfaces of P3 having a nonsingular model withq = pg = 0, P2 = 0. Rend. Sem. Mat. Univ. Padova 91 (1994) 187 - 198

[17] P. Deligne. La conjecture de Weil I. Publications Mathematiques de l’IHES 43 (1974)273 - 307

[18] P. Deligne. La conjecture de Weil II. Publications Mathematiques de l’IHES 52 (1980)137 - 252

[19] W. Fulton. Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3Folge. Band 2. Springer-Verlag 1984

[20] W. Fulton, J. Hansen. A Connectedness Theorem for Projective Varieties, with Appli-cations to Intersections andSingularities of Mappings. Annals of Mathematics. Vol. 110,No. 1 (1979) 159 - 166

[21] S. Gille. On Chow motives of surfaces. J. reine angew. Math. 686 (2014) 149 - 166


[22] S. Gorchinskiy, V. Guletskiı. Transcendence degree of zero-cycles and the structure ofChow motives. Central European Journal of Mathematics. Volume 10, No. 2 (2012) 559- 568

[23] A. Grothedieck (avec la collaboration de Jean Dieudonne). Elements de geometrie

algebrique II. Etude globale elementaire de quelques classes de morphismes. PublicationsMathmatiques de l’IHES 8 (1961) 5 - 222

[24] V. Guletskiı. Bloch’s conjecture for the surface of Craighero and Gattazzo. PreprintarXiv:1609.04074v4

[25] V. Guletskiı, C. Pedrini. Finite-dimensional motives and the conjectures of Beilinsonand Murre. K-Theory. Vol. 30, No. 3 (2003) 243 - 263

[26] R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics. Springer-Verlag,1977

[27] A. Huckleberry, M. Sauer. On the order of the automorphism group of a surface ofgeneral type. Math. Z. Vol. 205 (1990) 321 - 329

[28] U. Jannsen. Motivic Sheaves and Filtratins on Chow Groups. In ”Motives”, Proc. Sym-posia in Pure Math. Vol. 55, Part 1 (1994) 245 - 302

[29] B. Kahn, J. Murre, C. Pedrini. On the transcendental part of the motive of a sur-face. Algebraic cycles and motives, Vol. 2. London Math. Soc. Lecture Note Series 344.Cambridge Univ. Press. Cambridge (2007) 143 - 202

[30] S.-I. Kimura. Chow groups are finite dimensional, in some sense. Math. Ann. Vol 331,No. 1 (2005) 173 - 201

[31] S. Kleiman. The enumerative theory of singularities. In Real and complex singularities.Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976. Sijthoff andNoordhoff, Alphen aan den Rijn (1977) pp. 297 - 396

[32] S. Kleiman. Tangency and duality. In Proceedings of the 1984 Vancouver Conferencein Algebraic Geometry. CMS Conference Proceedings. The American MathematicalSociety, Providence (1986) 163 - 226

[33] J. Kollar, S. Mori (with the collaboration of H. Clemens and A. Corti). BirationalGeometry of Algebraic Varieties. Cambridge Tracts in Mathematics 134. CambridgeUniversity Press (1998)

[34] K. Lamotke. The topology of complex projective varieties after S. Lefschetz. Topology,Volume 20 (1981) 15 - 51

[35] Y. Lee, Y. Shin. Involutions on a surface of general type with pg = q = 0, K2 = 7.Osaka J. Math. 51 (2014) 121 - 139

[36] D. Martinelli, J. C. Naranjo, G. P. Pirola. Connectedness Bertini Theorem via numericalequivalence. Preprint arXiv:1412.1978

[37] D. Mumford. Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ. Volume9 (1969) pp 195 - 204

[38] J. Murre. On the motive of an algebraic surface. J. Reine Angew. Math. 409 (1990) 190- 204

[39] C. Pedrini. On the finite dimensionality of a K3 surface. Manuscripta math. 138 (2012)59 - 72

[40] J. Rana, J. Tevelev, G. Urzua. The Craighero-Gattazzo surface is simply-connected.arXiv:1506.03529v2

[41] M. Reid. Chapters on Algebraic Surfaces. In Complex algebraic varieties. J. Kollar (Ed.)IAS/Park City Mathematics Series 3, AMS (1997) 1 - 154

[42] P. Samuel. Relations dequivalence en geometrie algebrique. Proceedings of ICM in Ed-inburgh (1958) 470 - 487

[43] A. Scholl. Classical motives. In ”Motives”, Proc. Symp. Pure Math. Vol.55, Part 1(1994) 163 - 187

[44] Seminaire de Geometrie Algebrique 7, II. Groupes de Monodromie en geometriealgebrique par P. Deligne and N. Katz. Lecture Notes in Mathematics, Volume 340,Springer-Verlag, Berlin (1973)

[45] F. Serrano. The adjunction mapping and hyperelliptic divisors on a surface. J. reineangew. Math. 281 (1987) 90 - 109


[46] Y. Shin. Involutions on surfaces of general type with pg = 0 I. The composed case.Commun. Korean Math. Soc. Vol. 28 No. 3 (2013) 425 - 432

[47] E. Stagnaro. Canonical and pluricanonical adjoints to an algebraic surface I. Rapportitecnici, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate 1991

[48] A. Suslin, V. Voevodsky. Singular homology of abstract algebraic varieties. Inventionesmathematicae. Volume 123, Issue 1 (1996) pp 61 - 94

[49] E. Tevelev. Projective duality and homogeneous spaces. Springer-Verlag 2005[50] C. Voisin. Sur les zero-cycles de certaine hypersurfaces munies d’un automorphisme.

Ann. Scuola Norm. Sup. Pisa Ck. Si. (4) 19 (1992) 473 - 492[51] C. Voisin. Variations de structure de Hodge et zero-cycles sur les surfaces generales.

Math. Ann. 299 (1994) 77 - 103[52] C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge

studies in advanced mathematics 76 (2002)[53] C. Voisin. Bloch’s conjecture for Catanese and Barlow surfaces. J. Differential Geom.

Volume 97, Number 1 (2014) 149 - 175[54] A. Weil. Foundations of algebraic geometry. AMS 1962

Department of Mathematical Sciences, University of Liverpool, PeachStreet, Liverpool L69 7ZL, England, UK

E-mail address: [email protected]

Documents

Introduction - Liverpoolpcguletski/papers/INV6.pdf · Let W be the resolution of singularities of the quotient surface S/ι. If A0(W) = 0, ... 0-cycles on algebraic surfaces. A new