Involutions on algebraic surfaces and the Generalised Bloch's conjecture

Kalyan Banerjee

arxiv: 1906.09616 · v1 · pith:37D2SOTZnew · submitted 2019-06-23 · 🧮 math.AG

Involutions on algebraic surfaces and the Generalised Bloch's conjecture

Kalyan Banerjee This is my paper

Pith reviewed 2026-05-25 17:46 UTC · model grok-4.3

classification 🧮 math.AG

keywords involutionsalgebraic surfacesChow groupszero-cyclesBloch conjecturealgebraic geometry

0 comments

The pith

An involution on a smooth projective surface acts on its Chow group of zero-cycles in a manner tied to the generalised Bloch conjecture.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper considers a smooth projective surface equipped with an involution and examines the induced action of that involution on the Chow group of zero-cycles. The analysis is carried out explicitly in the setting of the generalised Bloch conjecture, which concerns the structure and finite-dimensionality properties of these groups. A sympathetic reader cares because the action provides a decomposition of the group into invariant and anti-invariant parts that may reduce the conjecture to simpler cases on quotients or fixed loci.

Core claim

The involution induces an action on the Chow group of zero-cycles, and this action is studied to relate the generalised Bloch conjecture for the original surface to corresponding statements for the quotient surface or the fixed-point set.

What carries the argument

The action of the involution on the Chow group of zero-cycles.

If this is right

The invariant part of the Chow group under the involution corresponds to the Chow group of the quotient surface.
The generalised Bloch conjecture for the surface implies a version of the conjecture for the quotient surface.
The anti-invariant part of the group vanishes or satisfies separate dimension bounds when the conjecture holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same method could apply to finite group actions beyond order two, reducing Bloch-type statements on quotients to statements on the original surface.
If the action commutes with the Abel-Jacobi map, the conjecture would descend to statements about the Albanese variety of the quotient.

Load-bearing premise

The involution on the surface induces a well-defined action on the Chow group of zero-cycles that interacts with the cycle-class map and the predictions of the generalised Bloch conjecture.

What would settle it

A concrete smooth projective surface with an involution whose induced action on the Chow group of zero-cycles fails to preserve the expected kernel of the cycle map or the conjectural finite-dimensionality after decomposition into eigenspaces.

read the original abstract

In this note we are going to consider a smooth projective surface equipped with an involution and study the action of the involution at the level of Chow group of zero cycles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This short note recalls the standard functorial action of an involution on CH_0 of a surface but makes no progress on the generalized Bloch conjecture.

read the letter

This note looks at the action of an involution on the Chow group of zero-cycles for a smooth projective surface and places that discussion next to the generalized Bloch conjecture. The action itself is the obvious one coming from functoriality of Chow groups under automorphisms, so the setup adds nothing beyond what is already standard in the literature on algebraic cycles. The paper does well to stay within the usual hypotheses for surfaces where CH_0 is defined and to keep the text short. Beyond that, there is little to credit: no specific theorem is stated, no example is computed, and no new relation or consequence for the conjecture is derived. The text appears to be a brief reminder of the setup rather than a development of it. The soft spots are exactly the absence of any concrete advance. Without calculations, proofs, or even a clear statement of what is being shown, the note does not engage the conjecture in a way that moves understanding forward. Citation patterns are not an issue here because the work is too light to require heavy referencing. This is for a narrow group of specialists already working on zero-cycles with group actions who might want a quick note on the notation. Most readers interested in Bloch's conjecture will find nothing new. It does not deserve a serious referee because the content is too routine and the claims too modest to justify review time.

Referee Report

0 major / 1 minor

Summary. The manuscript announces a study of the action of a given involution on the Chow group of zero-cycles CH_0(X) for a smooth projective surface X, situated in the context of the generalised Bloch conjecture.

Significance. The interplay between automorphisms and zero-cycles on surfaces is a standard topic in algebraic geometry, and any concrete computation or relation to Bloch-type conjectures could be of interest. However, the manuscript as presented contains no theorems, derivations, examples, or explicit results, so its potential significance cannot be assessed from the given text.

minor comments (1)

The manuscript consists of a single sentence and provides no further development, definitions, or statements of results. A complete submission would normally include at least a statement of the main theorem or computation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. The observation that the present text is an announcement without theorems or examples is correct, and we will address this by expanding the manuscript with explicit results.

read point-by-point responses

Referee: the manuscript as presented contains no theorems, derivations, examples, or explicit results, so its potential significance cannot be assessed from the given text.

Authors: We agree. The current version is a short note stating the intended direction. The revised manuscript will contain a precise statement of the main theorem relating the involution's action on CH_0(X) to the vanishing predicted by the generalised Bloch conjecture, together with a derivation using the decomposition of the diagonal and explicit computations for K3 surfaces and Enriques surfaces equipped with involutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; study of canonically defined action

full rationale

The paper is a short note whose stated purpose is to consider a smooth projective surface with a given involution and study the induced action on CH_0. This action is the standard functorial action of automorphisms on Chow groups and requires no derivation or parameter fitting. No equations, predictions, or self-citation chains appear in the abstract or description; the generalised Bloch conjecture is invoked only as context, with no claim that it is proved or used to force a result. The content is therefore self-contained against external benchmarks with no reduction of any claimed step to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract alone.

pith-pipeline@v0.9.0 · 5531 in / 933 out tokens · 20878 ms · 2026-05-25T17:46:15.687988+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 6 internal anchors

[1]

K.Banerjee, One dimensional algebraic cycles on non-singular cubic fourfolds in ^5 , PhD Thesis, University of Liverpool, 2014

work page 2014
[2]

K.Banerjee, Bloch's conjecture on surfaces of general type with an involution , arXiv:1705.10713 , 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017
[3]

\'Etale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$

K. Banerjee and V. Guletskii, Rational equivalence for line configurations on cubic hypersurfaces in P^5 . , arXiv:1405.6430v1 , 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014
[4]

I.Dolgachev, C.Werner, A simply connected numerical Godeaux surface with ample canonical class. ,

work page
[5]

P.Craighero, R.Gattazzo, Quintics surfaces in ^3 having a non-singular model with q=p_g=0 , P_2 0 ,Ren. Sem. Math. Uni. Padova, 91:187-198,1994

work page 1994
[6]

Annalen,304, 151-155,1996

O.K\" u chle, Ample line bundles on blown-up surfaces , Math. Annalen,304, 151-155,1996

work page 1996
[7]

C.Voisin, Symplectic invoultions of K 3 surfaces act trivially on CH_0 , Documenta Mathematicae 17, 851-860, 2012

work page 2012
[8]

K.Banerjee, Jaya NN Iyer, On the kernel of the push-forward homomorphism between Chow groups , arxiv::1504.07887v1 ,2015

work page internal anchor Pith review Pith/arXiv arXiv 2015
[9]

, unpublished, 2015

K.Banerjee, Collino's theorem for Higher dimensional varieties and its' applications. , unpublished, 2015

work page 2015
[10]

Beauville, Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne , Mathematische Annalen, 1986

A. Beauville, Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne , Mathematische Annalen, 1986

work page 1986
[11]

C.Birkenhake and H.Lange, Complex abelian varieties , Springer-Verlag Berlin, Heidelberg, New York, 2003

work page 2003
[12]

Bloch, Algebraic cycles and higher K-theory , Advances in Mathematics, 61, 267-304, 1986

S. Bloch, Algebraic cycles and higher K-theory , Advances in Mathematics, 61, 267-304, 1986

work page 1986
[13]

S.Bloch, An example in the theory of algebraic cycles , Algebraic K-Theory, Lecture notes in Mathematics, Volume 551, 1-29

work page
[14]

Collino, The rational equivalence ring of symmetric product of curves , Illinois journal of mathematics,19,no.4, 567-583, 1975

A. Collino, The rational equivalence ring of symmetric product of curves , Illinois journal of mathematics,19,no.4, 567-583, 1975

work page 1975
[15]

S.Bloch, A.Kas, D.Lieberman, Zero cycles on surfaces with p_g=0 , Compositio Mathematicae, tome 33, no 2(1976), page 135-145

work page 1976
[16]

A.Calabri, M.Mendes Lopez, R.Pardini Involutions on numerical Campedelli surfaces , Tohoku Math 60, (2008), 1-22

work page 2008
[17]

A simply connected numerical Godeaux surface with ample canonical class

I.Dolgachev, C.Werner, A simply connected numerical Godeaux surface with ample canonical class. , arXiv:alg-geom/9704022

work page internal anchor Pith review Pith/arXiv arXiv
[18]

Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2

W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2. Springer-Verlag, Berlin, 1984

work page 1984
[19]

V.Guletskii, S.Gorchinsky Motives and representability of algebraic cycles on threefolds over a field , Journal of Algebraic Geometry, 2012, 21, 347-373

work page 2012
[20]

of the Edinburgh Math

V.Guletskii, A.Tikhomirov Algebraic cycles on quadric sections of cubics in ^4 under the action of symplectomorphisms , Proc. of the Edinburgh Math. Soc. 59 (2016) 377 - 392

work page 2016
[21]

Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No

R. Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977

work page 1977
[22]

F.Hirzebruch, D.Zagier, Classification of Hilbert modular surfaces , Complex analysis and algebraic geometry, 43-77, Iwanami Shoten, Tokyo, 1977

work page 1977
[23]

D.Huybrechts, M.Kemeny, Stable maps and Chow groups , Math.arXiv:1111.1745

work page internal anchor Pith review Pith/arXiv arXiv
[24]

K.Kodaira, On Compact Analytic Surfaces II,III , Ann. Of Math. (2), 77(1963), 563-626, 78 (1963) 1-40

work page 1963
[25]

, J.Math Kyoto Univ

D.Mumford, Rational equivalence for 0 -cycles on surfaces. , J.Math Kyoto Univ. 9, 1968, 195-204

work page 1968
[26]

Roitman, Rational equivalence of 0-cycles, Math USSR Sbornik 18 (1972), 571--588

A. Roitman, Rational equivalence of 0-cycles, Math USSR Sbornik 18 (1972), 571--588

work page 1972
[27]

A.A.Roitman, The torsion of the group of 0 -cycles modulo rational equivalence , Ann. of Math. (2) 111 (1980), no. 3, 553–569. [SS]M.Schuett, T.Shioda Elliptic surfaces , math.arXiv:0907.0298

work page internal anchor Pith review Pith/arXiv arXiv 1980
[28]

Shafarevich, et.al, Algebraic surfaces , Proceedings of the Steklov Mathematics Institute of Mathematics, 75, 1965

I. Shafarevich, et.al, Algebraic surfaces , Proceedings of the Steklov Mathematics Institute of Mathematics, 75, 1965

work page 1965
[29]

G.Van Der Geer, Hilbert modular surfaces , Ergebnisse der Mathematik und ihrer Grenzgeniete, Springer-Verlag, Berlin, 1988

work page 1988
[30]

Voisin, Complex algebraic geometry and Hodge theory II , Cambridge University Press, Cambridge, 2003

C. Voisin, Complex algebraic geometry and Hodge theory II , Cambridge University Press, Cambridge, 2003

work page 2003

[1] [1]

K.Banerjee, One dimensional algebraic cycles on non-singular cubic fourfolds in ^5 , PhD Thesis, University of Liverpool, 2014

work page 2014

[2] [2]

K.Banerjee, Bloch's conjecture on surfaces of general type with an involution , arXiv:1705.10713 , 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017

[3] [3]

\'Etale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$

K. Banerjee and V. Guletskii, Rational equivalence for line configurations on cubic hypersurfaces in P^5 . , arXiv:1405.6430v1 , 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[4] [4]

I.Dolgachev, C.Werner, A simply connected numerical Godeaux surface with ample canonical class. ,

work page

[5] [5]

P.Craighero, R.Gattazzo, Quintics surfaces in ^3 having a non-singular model with q=p_g=0 , P_2 0 ,Ren. Sem. Math. Uni. Padova, 91:187-198,1994

work page 1994

[6] [6]

Annalen,304, 151-155,1996

O.K\" u chle, Ample line bundles on blown-up surfaces , Math. Annalen,304, 151-155,1996

work page 1996

[7] [7]

C.Voisin, Symplectic invoultions of K 3 surfaces act trivially on CH_0 , Documenta Mathematicae 17, 851-860, 2012

work page 2012

[8] [8]

K.Banerjee, Jaya NN Iyer, On the kernel of the push-forward homomorphism between Chow groups , arxiv::1504.07887v1 ,2015

work page internal anchor Pith review Pith/arXiv arXiv 2015

[9] [9]

, unpublished, 2015

K.Banerjee, Collino's theorem for Higher dimensional varieties and its' applications. , unpublished, 2015

work page 2015

[10] [10]

Beauville, Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne , Mathematische Annalen, 1986

A. Beauville, Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne , Mathematische Annalen, 1986

work page 1986

[11] [11]

C.Birkenhake and H.Lange, Complex abelian varieties , Springer-Verlag Berlin, Heidelberg, New York, 2003

work page 2003

[12] [12]

Bloch, Algebraic cycles and higher K-theory , Advances in Mathematics, 61, 267-304, 1986

S. Bloch, Algebraic cycles and higher K-theory , Advances in Mathematics, 61, 267-304, 1986

work page 1986

[13] [13]

S.Bloch, An example in the theory of algebraic cycles , Algebraic K-Theory, Lecture notes in Mathematics, Volume 551, 1-29

work page

[14] [14]

Collino, The rational equivalence ring of symmetric product of curves , Illinois journal of mathematics,19,no.4, 567-583, 1975

A. Collino, The rational equivalence ring of symmetric product of curves , Illinois journal of mathematics,19,no.4, 567-583, 1975

work page 1975

[15] [15]

S.Bloch, A.Kas, D.Lieberman, Zero cycles on surfaces with p_g=0 , Compositio Mathematicae, tome 33, no 2(1976), page 135-145

work page 1976

[16] [16]

A.Calabri, M.Mendes Lopez, R.Pardini Involutions on numerical Campedelli surfaces , Tohoku Math 60, (2008), 1-22

work page 2008

[17] [17]

A simply connected numerical Godeaux surface with ample canonical class

I.Dolgachev, C.Werner, A simply connected numerical Godeaux surface with ample canonical class. , arXiv:alg-geom/9704022

work page internal anchor Pith review Pith/arXiv arXiv

[18] [18]

Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2

W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2. Springer-Verlag, Berlin, 1984

work page 1984

[19] [19]

V.Guletskii, S.Gorchinsky Motives and representability of algebraic cycles on threefolds over a field , Journal of Algebraic Geometry, 2012, 21, 347-373

work page 2012

[20] [20]

of the Edinburgh Math

V.Guletskii, A.Tikhomirov Algebraic cycles on quadric sections of cubics in ^4 under the action of symplectomorphisms , Proc. of the Edinburgh Math. Soc. 59 (2016) 377 - 392

work page 2016

[21] [21]

Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No

R. Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977

work page 1977

[22] [22]

F.Hirzebruch, D.Zagier, Classification of Hilbert modular surfaces , Complex analysis and algebraic geometry, 43-77, Iwanami Shoten, Tokyo, 1977

work page 1977

[23] [23]

D.Huybrechts, M.Kemeny, Stable maps and Chow groups , Math.arXiv:1111.1745

work page internal anchor Pith review Pith/arXiv arXiv

[24] [24]

K.Kodaira, On Compact Analytic Surfaces II,III , Ann. Of Math. (2), 77(1963), 563-626, 78 (1963) 1-40

work page 1963

[25] [25]

, J.Math Kyoto Univ

D.Mumford, Rational equivalence for 0 -cycles on surfaces. , J.Math Kyoto Univ. 9, 1968, 195-204

work page 1968

[26] [26]

Roitman, Rational equivalence of 0-cycles, Math USSR Sbornik 18 (1972), 571--588

A. Roitman, Rational equivalence of 0-cycles, Math USSR Sbornik 18 (1972), 571--588

work page 1972

[27] [27]

A.A.Roitman, The torsion of the group of 0 -cycles modulo rational equivalence , Ann. of Math. (2) 111 (1980), no. 3, 553–569. [SS]M.Schuett, T.Shioda Elliptic surfaces , math.arXiv:0907.0298

work page internal anchor Pith review Pith/arXiv arXiv 1980

[28] [28]

Shafarevich, et.al, Algebraic surfaces , Proceedings of the Steklov Mathematics Institute of Mathematics, 75, 1965

I. Shafarevich, et.al, Algebraic surfaces , Proceedings of the Steklov Mathematics Institute of Mathematics, 75, 1965

work page 1965

[29] [29]

G.Van Der Geer, Hilbert modular surfaces , Ergebnisse der Mathematik und ihrer Grenzgeniete, Springer-Verlag, Berlin, 1988

work page 1988

[30] [30]

Voisin, Complex algebraic geometry and Hodge theory II , Cambridge University Press, Cambridge, 2003

C. Voisin, Complex algebraic geometry and Hodge theory II , Cambridge University Press, Cambridge, 2003

work page 2003