Skip to main content
Log in


In this exposition we understand when the natural map from the two-fold self product of the Chow variety parametrizing codimension p cycles on a smooth projective variety X to the Chow group \(\mathrm{{CH}}^p(X)_0\) of degree zero cycles is surjective. We derive some consequences when the map is surjective.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Banerjee, K.: One dimensional algebraic cycles on non-singular cubic fourfolds in \({\mathbb{P}}^5\), PhD Thesis, University of Liverpool (2014)

  • Banerjee, K., Guletskii, V.: Etale monodromy and rational equivalence of \(1\)-cycles on cubic hypersurfaces in \({\mathbb{P}}^5\). Sbornik Math. 211(2), 161–200 (2020)

    Article  Google Scholar 

  • Barlow, R.: Rational equivalence of zero cycles for some surfaces with \(p_g=0\). Invent. Math. 2, 303–308 (1985)

    Article  MathSciNet  Google Scholar 

  • Bloch, S., Kas, A., Lieberman, D.: Zero cycles on surfaces with \(p_g=0\). Compos. Math. tome 33(2), 135–145 (1976)

    MATH  Google Scholar 

  • Fulton, W.: Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2. Springer, Berlin (1984)

    Google Scholar 

  • Guletskii, V., Gorchinsky, S.: Non-trivial elements in the Abel–Jacobi kernels of higher dimensional varieties. Adv. Math. 241 (2013)

  • Guletskii, V., Pedrini, C.: Chow motive of a Goeadux surface, Algebraic Geometry: A Volume in Memory of Paolo Francia, pp. 179–195. de Gruyter, Berlin (2002)

  • Inose, H., Mizukami, M.: Rational equivalence of 0-cycles on some surfaces with \(p_g=0\). Math. Ann. 244(3), 205–217 (1979)

    Article  MathSciNet  Google Scholar 

  • Kahn, B.: On the universal regular homomorphism in codimension \(2\). arxiv:2007.07592 (2020)

  • Mumford, D.: Rational equivalence for \(0\)-cycles on surfaces. J. Math Kyoto Univ. 9, 195–204 (1968)

    MathSciNet  MATH  Google Scholar 

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

  • Paranjape, K.: Cohomological and cycle-theoretic connectivity. Ann. Math. (2) 139(3), 641–660 (1994)

    Article  MathSciNet  Google Scholar 

  • Pedrini, C., Weibel, C.: Some examples of surfaces of general type for which Bloch’s conjecture holds. In: Kerr, M., Pearlstein, G. (eds.) Recent Advances in Hodge Theory, Period Domains, Algebraic Cycles and Arithmetic. Cambridge University Press, Cambridge (2016)

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  • Roitman, A.: The torsion of the group of 0-cycles modulo rational equivalence. Ann. Math. (2) 111(3), 553–569 (1980)

    Article  MathSciNet  Google Scholar 

  • Schoen, C.: On Hodge structures and non-representability of Chow groups. Compos. Math. 88(3), 285–316 (1993)

    MATH  Google Scholar 

  • Shen, M.: On relations among \(1\)-cycles on cubic hypersurfaces. J. Algebr. Geom. 23, 539–569 (2014)

    Article  MathSciNet  Google Scholar 

  • Suslin, A., Voevodsky, V.: Relative cycles and Chow sheaves, Cycles, transfers, motivic homology theories, pp. 10–86. Annals of Math studies (2011)

  • Voisin, C.: Sur les zero cycles de certaines hypersurfaces munies d’un automorphisme. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19(4), 473–492 (1992)

    MathSciNet  MATH  Google Scholar 

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

  • Voisin, C.: Bloch’s conjecture for Catanese and Barlow surfaces. J. Differ. Geom. 1, 149–175 (2014)

    MathSciNet  MATH  Google Scholar 

Download references


The author would like to thank the hospitality of Tata Institute Mumbai and Harish Chandra Research Institute, India, for hosting this project. The author also thanks the anonymous referee for careful reading of the manuscript and for advising the author about the improvements of certain technical points in the paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Kalyan Banerjee.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Banerjee, K. On finite dimensionality of Chow groups. Beitr Algebra Geom 63, 189–207 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification