Representability of codimension three cycles

Kalyan Banerjee

arxiv: 2604.20223 · v1 · submitted 2026-04-22 · 🧮 math.AG

Representability of codimension three cycles

Kalyan Banerjee This is my paper

Pith reviewed 2026-05-09 23:52 UTC · model grok-4.3

classification 🧮 math.AG

keywords representabilitycodimension three cyclesfourfoldszero-cyclesrational equivalencealgebraic cyclesChow groups

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The pith

Codimension three cycles on fourfolds can be represented using zero-cycles on surfaces modulo rational equivalence.

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

The paper introduces a definition of representability for codimension three cycles on a fourfold. This definition expresses such cycles through zero-cycles modulo rational equivalence on surfaces. A sympathetic reader would care because it reduces questions about cycles in higher dimension to data on lower-dimensional varieties. The approach relies on the existing machinery of rational equivalence to organize the geometry of these cycles.

Core claim

The author develops the notion of representability of codimension three cycles on a fourfold by associating them to zero-cycles modulo rational equivalence on surfaces.

What carries the argument

The proposed representability notion, which reduces codimension three cycles on fourfolds to zero-cycles on surfaces via rational equivalence.

Load-bearing premise

That this definition of representability via zero-cycles on surfaces is the right or useful one for capturing the geometry of codimension three cycles.

What would settle it

A specific fourfold and codimension three cycle where the proposed representability condition either holds for a geometrically nontrivial cycle or fails for a trivial one, contrary to independent calculations of the cycle class.

read the original abstract

In this paper, we develop the notion of representability of co-dimension three cycles on a fourfold in terms of zero cycles modulo rational equivalence on surfaces.

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 paper defines representability for codim-3 cycles on fourfolds by linking them to zero-cycles mod rational equivalence on surfaces, but offers little beyond the definition itself.

read the letter

The paper develops a notion of representability for codimension three cycles on a fourfold, expressing it through zero cycles modulo rational equivalence on surfaces. This is the central new element. It does a reasonable job of laying out the definition in a direct way. The construction translates between cycle classes of different dimensions without introducing circular reasoning or loose terms, which is a plus for a definitional paper. Credit is due for keeping the setup clean and focused on the specific case of fourfolds. Where it falls short is in providing reasons to care about this particular definition. The abstract gives no indication of how it relates to or improves upon earlier ideas in the theory of algebraic cycles, such as those involving Abel-Jacobi maps or other representability conditions in lower codimensions. There are also no concrete examples or consequences mentioned that would demonstrate its effectiveness. This makes the contribution feel preliminary. The paper is really for people working on algebraic cycles and Chow rings who might be interested in extending representability concepts to codimension three. A reader already familiar with the literature on fourfolds and surfaces could extract value if the full text includes more details on the maps involved. For a general audience or someone looking for major new theorems, it offers little. I think it deserves a serious referee. The formal part is solid enough to warrant review, even if the authors need to strengthen the motivation section in revisions. It is not groundbreaking, but it is the kind of incremental definitional work that can be useful if it catches on.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a formal definition of representability for codimension-three cycles on fourfolds, expressed in terms of zero-cycles modulo rational equivalence on associated surfaces. It presents this as a direct construction within the Chow ring and the category of algebraic cycles, with no accompanying theorems, examples, or applications supplied beyond the definition itself.

Significance. As a purely definitional contribution, the work supplies a precise translation between cycle classes of differing codimensions. If the notion proves effective in applications, it could serve as a useful framework for analyzing the geometry of higher-codimension cycles on fourfolds and their relation to zero-cycles on surfaces. The paper's internal construction is consistent once the maps and equivalence relations are stated.

minor comments (2)

The abstract and introduction would benefit from a short comparison to existing notions of representability for cycles (e.g., those appearing in the literature on motives or on surfaces) to clarify the novelty of the proposed definition.
Notation for the surfaces associated to the fourfold and for the induced maps on Chow groups should be introduced explicitly in the first section where the definition appears, to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for acknowledging the internal consistency of the construction. We respond to the observations raised in the report below.

read point-by-point responses

Referee: The manuscript develops a formal definition of representability for codimension-three cycles on fourfolds, expressed in terms of zero-cycles modulo rational equivalence on associated surfaces. It presents this as a direct construction within the Chow ring and the category of algebraic cycles, with no accompanying theorems, examples, or applications supplied beyond the definition itself.

Authors: The paper's purpose is precisely to introduce and formalize this notion of representability, providing a direct translation between codimension-three cycles on fourfolds and zero-cycles on surfaces within the Chow ring. As a definitional contribution, it does not claim or include additional theorems, examples, or applications; this is by design, as the focus is on establishing the framework itself. The referee correctly notes the absence of such elements, but we maintain that the definition stands as a self-contained contribution that can support future work on higher-codimension cycles. revision: no

Circularity Check

0 steps flagged

No circularity: purely definitional contribution

full rationale

The paper introduces a formal definition of representability for codimension-three cycles on fourfolds, expressed directly in terms of zero-cycles modulo rational equivalence on associated surfaces. No derivation chain, equations, fitted parameters, or predictive claims exist that could reduce to self-referential inputs. The central claim is the construction of this notion itself within the Chow ring, presented as a direct translation between cycle classes without any self-citation load-bearing steps or ansatz smuggling. As a self-contained definitional paper with no internal reductions to its own outputs, the work exhibits no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.0 · 5290 in / 983 out tokens · 21973 ms · 2026-05-09T23:52:21.426070+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

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

work page 2014
[2]

Banerjee and V

K. Banerjee and V. Guletskii, Etale monodromy and rational equivalence of 1 -cycles on cubic hypersurfaces in P^5 , Sbornik Math. Volume 211, no. 2, February 2020, 161-200

work page 2020
[3]

R.Barlow, Rational equivalence of zero cycles for some surfaces with p_g=0 , Invent. Math. 1985, no. 2, 303-308

work page 1985
[4]

2 (1976), page 135-145

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

work page 1976
[5]

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
[6]

V.Guletskii, S.Gorchinsky, Non-trivial elements in the Abel-Jacobi kernels of higher dimensional varieties , Advances in Mathematics, 241, 2013

work page 2013
[7]

V.Guletskii, C.Pedrini, Chow motive of a Goeadux surface , Algebraic Geometry: A Volume in Memory of Paolo Francia, 179-195, de Gruyter, Berlin - New York, 2002

work page 2002
[8]

V.Guletskii, Motivic obstruction to rationality , arXiv:1605.09434

work page arXiv
[9]

V.Guletskii, Bloch's conjecture for surfaces with involutions and of geometric genus zero , arXiv:1704.04187

work page arXiv
[10]

V.Guletskii, Bloch's conjecture for the surface of Craighero and Gattazzo , arXiv:1609.04074

work page arXiv
[11]

Annalen, 244, 1979, no

H.Inose, M.Mizukami, Rational equivalence of 0-cycles on some surfaces with p_g=0 , Math. Annalen, 244, 1979, no. 3, 205-217

work page 1979
[12]

B.Kahn, On the universal regular homomorphism in codimension 2 , arxiv.org: 2007.07592

work page arXiv 2007
[13]

, J.Math Kyoto Univ

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

work page 1968
[14]

J.Murre, Applications of algebraic K-theory to the theory of algebraic cycles , Algebraic Geometry, Sitges(Barcelona), 1983, 216-261, Lecture Notes in Mathematics, 1124, Springer, Berlin, 1985

work page 1983
[15]

Paranjape, Cohomological and cycle-theoretic connectivity , Ann

K. Paranjape, Cohomological and cycle-theoretic connectivity , Ann. of Math. (2) 139 (1994), no. 3, 641--660

work page 1994
[16]

C.Pedrini, C.Weibel, Some examples of surfaces of general type for which Bloch's conjecture holds , Recent advances in Hodge theory, Period Domains, Algebraic Cycles and Arithmetic, Edited by Matt Kerr and Gregory Pearlstein, Cambridge University Press, February 2016

work page 2016
[17]

A.Roitman, -equivalence of zero dimensional cycles (Russian) , Math. Sbornik. 86(128), 1971, 557-570

work page 1971
[18]

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

work page 1972
[19]

A.Roitman, The torsion of the group of 0-cycles modulo rational equivalence , Ann. of Math. (2), 111, 1980, no. 3, 553-569

work page 1980
[20]

M.Shen, On relations among 1 -cycles on cubic hypersurfaces , Journal of Algebraic Geometry, 23, 2014, 539-569

work page 2014
[21]

A.Suslin, V.Voevodsky, Relative cycles and Chow sheaves , Cycles, transfers, motivic homology theories, 10-86, Annals of Math studies

work page
[22]

88, no.3, 1993, 285-316

C.Schoen, On Hodge structures and non-representability of Chow groups , Composition Math. 88, no.3, 1993, 285-316

work page 1993
[23]

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

work page 2012
[24]

Tian, Hong R

Z. Tian, Hong R. Zong, One cycles on rationally connected varieties , Compositio Math. Volume 150, issue 3, March 2014, 396-408

work page 2014
[25]

C.Voisin, Complex algebraic geometry and Hodge theory II , Cambridge studies of Mathematics, 2002

work page 2002
[26]

1, 149-175

C.Voisin, Bloch's conjecture for Catanese and Barlow surfaces , Journal of Differential Geometry, 2014, no. 1, 149-175

work page 2014
[27]

Scuola Norm

C.Voisin, Sur les zero cycles de certaines hypersurfaces munies d'un automorphisme , Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 19, 1992, no.4, 473-492

work page 1992

[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]

Banerjee and V

K. Banerjee and V. Guletskii, Etale monodromy and rational equivalence of 1 -cycles on cubic hypersurfaces in P^5 , Sbornik Math. Volume 211, no. 2, February 2020, 161-200

work page 2020

[3] [3]

R.Barlow, Rational equivalence of zero cycles for some surfaces with p_g=0 , Invent. Math. 1985, no. 2, 303-308

work page 1985

[4] [4]

2 (1976), page 135-145

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

work page 1976

[5] [5]

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

[6] [6]

V.Guletskii, S.Gorchinsky, Non-trivial elements in the Abel-Jacobi kernels of higher dimensional varieties , Advances in Mathematics, 241, 2013

work page 2013

[7] [7]

V.Guletskii, C.Pedrini, Chow motive of a Goeadux surface , Algebraic Geometry: A Volume in Memory of Paolo Francia, 179-195, de Gruyter, Berlin - New York, 2002

work page 2002

[8] [8]

V.Guletskii, Motivic obstruction to rationality , arXiv:1605.09434

work page arXiv

[9] [9]

V.Guletskii, Bloch's conjecture for surfaces with involutions and of geometric genus zero , arXiv:1704.04187

work page arXiv

[10] [10]

V.Guletskii, Bloch's conjecture for the surface of Craighero and Gattazzo , arXiv:1609.04074

work page arXiv

[11] [11]

Annalen, 244, 1979, no

H.Inose, M.Mizukami, Rational equivalence of 0-cycles on some surfaces with p_g=0 , Math. Annalen, 244, 1979, no. 3, 205-217

work page 1979

[12] [12]

B.Kahn, On the universal regular homomorphism in codimension 2 , arxiv.org: 2007.07592

work page arXiv 2007

[13] [13]

, J.Math Kyoto Univ

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

work page 1968

[14] [14]

J.Murre, Applications of algebraic K-theory to the theory of algebraic cycles , Algebraic Geometry, Sitges(Barcelona), 1983, 216-261, Lecture Notes in Mathematics, 1124, Springer, Berlin, 1985

work page 1983

[15] [15]

Paranjape, Cohomological and cycle-theoretic connectivity , Ann

K. Paranjape, Cohomological and cycle-theoretic connectivity , Ann. of Math. (2) 139 (1994), no. 3, 641--660

work page 1994

[16] [16]

C.Pedrini, C.Weibel, Some examples of surfaces of general type for which Bloch's conjecture holds , Recent advances in Hodge theory, Period Domains, Algebraic Cycles and Arithmetic, Edited by Matt Kerr and Gregory Pearlstein, Cambridge University Press, February 2016

work page 2016

[17] [17]

A.Roitman, -equivalence of zero dimensional cycles (Russian) , Math. Sbornik. 86(128), 1971, 557-570

work page 1971

[18] [18]

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

work page 1972

[19] [19]

A.Roitman, The torsion of the group of 0-cycles modulo rational equivalence , Ann. of Math. (2), 111, 1980, no. 3, 553-569

work page 1980

[20] [20]

M.Shen, On relations among 1 -cycles on cubic hypersurfaces , Journal of Algebraic Geometry, 23, 2014, 539-569

work page 2014

[21] [21]

A.Suslin, V.Voevodsky, Relative cycles and Chow sheaves , Cycles, transfers, motivic homology theories, 10-86, Annals of Math studies

work page

[22] [22]

88, no.3, 1993, 285-316

C.Schoen, On Hodge structures and non-representability of Chow groups , Composition Math. 88, no.3, 1993, 285-316

work page 1993

[23] [23]

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

work page 2012

[24] [24]

Tian, Hong R

Z. Tian, Hong R. Zong, One cycles on rationally connected varieties , Compositio Math. Volume 150, issue 3, March 2014, 396-408

work page 2014

[25] [25]

C.Voisin, Complex algebraic geometry and Hodge theory II , Cambridge studies of Mathematics, 2002

work page 2002

[26] [26]

1, 149-175

C.Voisin, Bloch's conjecture for Catanese and Barlow surfaces , Journal of Differential Geometry, 2014, no. 1, 149-175

work page 2014

[27] [27]

Scuola Norm

C.Voisin, Sur les zero cycles de certaines hypersurfaces munies d'un automorphisme , Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 19, 1992, no.4, 473-492

work page 1992