Comptes Rendus
Volume articles
Complex analysis and geometry
On the GAGA principle for algebraic affine hypersurfaces
Vo Van Tan1 
1 Department of Mathematics, Suffolk University, Boston, Ma. 02114, USA
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 103-110.

For any complete ℂ-algebraic variety Y and its underlying compact ℂ-analytic space 𝒴, it follows from the well known GAGA principle that the algebraic Picard group Pic(Y) and the analytic Picard group ℙic(𝒴) are isomorphic. Our main purpose here is to provide a simple proof of an analogous situation for non complete ℂ-algebraic varieties, namely ℂ-algebraic affine hypersurfaces with at most isolated singularities.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.254
Classification: 32E10, 32Q28, 14B07, 14C22, 14J30, 57P10

Vo Van Tan 1

1 Department of Mathematics, Suffolk University, Boston, Ma. 02114, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{CRMATH_2022__360_G2_103_0,
     author = {Vo Van Tan},
     title = {On the {GAGA} principle for algebraic affine hypersurfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {103--110},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.254},
     language = {en},
}
TY  - JOUR
AU  - Vo Van Tan
TI  - On the GAGA principle for algebraic affine hypersurfaces
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 103
EP  - 110
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.254
LA  - en
ID  - CRMATH_2022__360_G2_103_0
ER  - 
%0 Journal Article
%A Vo Van Tan
%T On the GAGA principle for algebraic affine hypersurfaces
%J Comptes Rendus. Mathématique
%D 2022
%P 103-110
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.254
%G en
%F CRMATH_2022__360_G2_103_0
Vo Van Tan. On the GAGA principle for algebraic affine hypersurfaces. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 103-110. doi : 10.5802/crmath.254. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.254/

[1] Markus Banagl; Laurentiu Maxim Deformations of singularities and the homology of intersection spaces, J. Topol. Anal., Volume 4 (2012) no. 4, p. 413--448 | DOI | MR | Zbl

[2] Alexandru Dimca On the homology and cohomology of complete intersection with isolated singularities, Compos. Math., Volume 58 (1986), pp. 321-339 | Numdam | MR | Zbl

[3] Alexandru Dimca Singularities and topology of hypersurfaces, Universitext, Springer, 1992 | DOI | Zbl

[4] Robert M. Fossum; Birger Iverson On Picard groups of algebraic fibre spaces, J. Pure Appl. Algebra, Volume 3 (1973), pp. 269-280 | DOI | MR | Zbl

[5] Marvin J. Greenberg Lectures on algebraic topology, W. A. Benjamin, Inc., 1972

[6] Alexander Grothendieck Cohomologie locale des faisceaux cohérents et Théorèmes de Lefschetz locaux et globaux. (SGA 2). Augmenté d’un exposé par Michèle Raynaud. Séminaire de géométrie algébrique du Bois-Marie 1962, Advanced Studies in Pure Mathematics (Amsterdam), 2, North-Holland; Masson, 1968 | Zbl

[7] Revêtements étales et groupe fondamental (SGA I) (Alexander Grothendieck, ed.), Lecture Notes in Mathematics, 224, Springer, 1971 | DOI | Zbl

[8] Helmut A. Hamm; Dung Tang Lê On the Picard group for non-complete algebraic varieties, Franco-Japanese singularities. Proceedings of the 2nd Franco-Japanese singularity conference, CIRM, Marseille-Luminy, France, September 9–13, 2002 (Séminaires et Congrès), Volume 10, Société Mathématique de France, 2005, pp. 71-86 | MR | Zbl

[9] Robin Hartshorne Ample subvarieties of Algebraic varieties, Lecture Notes in Mathematics, 156, Springer, 1970 | DOI | Zbl

[10] Robin Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | DOI | Zbl

[11] Alan Howard On the homotopy groups of an affine algebraic hypersurface, Ann. Math., Volume 84 (1966), pp. 197-216 | DOI | MR | Zbl

[12] Mitsuyoshi Kato Topology of k-regular spaces and algebraic sets, Manifolds—Tokyo 1973 (Proc. Internat. Conf. on Manifolds and Related Topics in Topology) (1975), pp. 153-159 | MR | Zbl

[13] Mitsuyoshi Kato Partial Poincare duality for k-regular spaces and complex algebraic sets, Topology, Volume 16 (1977) no. 1, pp. 33-50 | DOI | MR | Zbl

[14] Andy R. Magid The Picard sequence of a fibration, Proc. Am. Math. Soc., Volume 53 (1975), pp. 37-40 | MR | Zbl

[15] José Seade On the topology of isolated singularities in analytic spaces, Progress in Mathematics, 241, Birkhäuser, 2006 | Zbl

[16] R. R. Simha Algebraic varieties bihomorphic to C * ×C * , Tôhoku Math. J., Volume 30 (1978), pp. 455-461 | Zbl

[17] Tan Vo Van On the parallelism between algebraic and analytic Picard groups of algebraic affine hypersurfaces (to appear) | Numdam

[18] Claire Voisin Théorie de Hodge et géométrie algébrique complexe, Contributions in Mathematical and Computational Sciences, 10, Société Mathématique de France, 2002 | Zbl

Cited by Sources:

Comments - Policy