On the GAGA principle for algebraic affine hypersurfaces

Tan, Vo Van

Comptes Rendus. Mathématique, Tome 360 (2022) no. G2, 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 $Pic\left(Y\right)$ and the analytic Picard group $\mathbb{P}ic\left(𝒴\right)$ 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.

Classification : 32E10, 32Q28, 14B07, 14C22, 14J30, 57P10

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

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

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

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

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

[6] Grothendieck, Alexander 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) (Grothendieck, Alexander, ed.), Lecture Notes in Mathematics, 224, Springer, 1971 | DOI | Zbl

[8] Hamm, Helmut A.; Lê, Dung Tang 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] Hartshorne, Robin Ample subvarieties of Algebraic varieties, Lecture Notes in Mathematics, 156, Springer, 1970 | DOI | Zbl

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

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

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

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

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

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

[16] Simha, R. R. Algebraic varieties bihomorphic to ${\mathbf{C}}^{*}\times {\mathbf{C}}^{*}$, Tôhoku Math. J., Volume 30 (1978), pp. 455-461 | Zbl

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

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