The Grothendieck-Lefschetz theorem for normal projective varieties
Authors:
G. V. Ravindra and V. Srinivas
Journal:
J. Algebraic Geom. 15 (2006), 563-590
DOI:
https://doi.org/10.1090/S1056-3911-05-00421-2
Published electronically:
October 25, 2005
MathSciNet review:
2219849
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We prove that for a normal projective variety $X$ in characteristic $0$, and a base-point free ample line bundle $L$ on it, the restriction map of divisor class groups $\operatorname {Cl} (X)\to \operatorname {Cl}(Y)$ is an isomorphism for a general member $Y\in |L|$ provided that $\dim {X}\geq 4$. This is a generalization of the Grothendieck-Lefschetz theorem, for divisor class groups of singular varieties.
- L. Barbieri-Viale, A. Rosenschon, and M. Saito, Deligne’s conjecture on 1-motives, Ann. of Math. (2) 158 (2003), no. 2, 593–633. MR 2018930, DOI https://doi.org/10.4007/annals.2003.158.593
- Luca Barbieri-Viale and Vasudevan Srinivas, Albanese and Picard 1-motives, Mém. Soc. Math. Fr. (N.S.) 87 (2001), vi+104 (English, with English and French summaries). MR 1891270, DOI https://doi.org/10.24033/msmf.400
- J. Biswas and V. Srinivas, Roitman’s theorem for singular projective varieties, Compositio Math. 119 (1999), no. 2, 213–237. MR 1723130, DOI https://doi.org/10.1023/A%3A1001793226084
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551
AG1 A. Grothendieck, J. Dieudonne, Elements de Geometrie Algebrique I, III, Publ. Math. IHES. 4 (1960), 11 (1961).
- Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
- Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR 932724
- Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977
- Serge Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983. Reprint of the 1959 original. MR 713430
Mehta-Srinivas V. B. Mehta and V. Srinivas, A Characterisation of Rational Singularities, Asian J. Math, Vol. 1, No. 2, 249–271.
- Niranjan Ramachandran, One-motives and a conjecture of Deligne, J. Algebraic Geom. 13 (2004), no. 1, 29–80. MR 2008715, DOI https://doi.org/10.1090/S1056-3911-03-00370-9
- Jean-Louis Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36 (1976), 295–312 (French). MR 481096, DOI https://doi.org/10.1007/BF01390015
- André Weil, Sur les critères d’équivalence en géométrie algébrique, Math. Ann. 128 (1954), 95–127 (French). MR 65219, DOI https://doi.org/10.1007/BF01360127
BRS L. Barbieri-Viale, A. Rosenschon, M. Saito, Deligne’s conjecture on 1-motives, Ann. of Math. (2) 158 (2003) 593–633.
BS L. Barbieri-Viale, V. Srinivas, Albanese and Picard 1-motives, Mem. Soc. Math. France 87 (2001).
BiS J. G. Biswas, V. Srinivas, Roitman’s theorem for singular projective varieties, Compositio Math. 119 (1999) 213–237.
D P. Deligne, Théorie de Hodge III, Publ. Math. IHES. 44 (1974) 5–77.
AG1 A. Grothendieck, J. Dieudonne, Elements de Geometrie Algebrique I, III, Publ. Math. IHES. 4 (1960), 11 (1961).
EV H. Esnault, E. Viehweg, Lectures on Vanishing Theorems, DMV Seminar, Band 20, Birkhäuser (1992).
Fulton W. Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3rd Series, 2. Springer-Verlag, Berlin, 1998.
GM M. Goresky and R. MacPherson, Stratified Morse Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 14. Springer-Verlag, Berlin, 1988.
AG2 A. Grothendieck,Cohomologie locale des faisceaus coherents et Theoremes de Lefschetz locaux and globale (SGA 2), Advanced studies in Pure Mathematics, North-Holland, Amsterdam (1968).
Ha Robin Hartshorne, Algebraic Geometry, GTM No.52, Springer-Verlag, New York-Heidelberg, 1977.
H ---, Ample Subvarieties of Algebraic Varieties, Lect. Notes in Math. 156, Springer-Verlag (1970).
Lang S. Lang, Abelian Varieties, Springer-Verlag, New York-Berlin, 1983 (reprint of 1959 original).
Mehta-Srinivas V. B. Mehta and V. Srinivas, A Characterisation of Rational Singularities, Asian J. Math, Vol. 1, No. 2, 249–271.
R N. Ramachandran, One-motives and a conjecture of Deligne, J. Algebraic Geom. 13 (2004) 29–80.
Ve J.-L. Verdier, Stratifications de Whitney et Theoreme de Bertini-Sard, Invent. Math. 36 (1976), 295–312.
Weil A. Weil, Sur les critéres d’équivalence en géométrie algébrique, Math. Ann. 128 (1954) 95–127.
Additional Information
G. V. Ravindra
Affiliation:
Department of Mathematics, Washington University, St. Louis, Missouri 63130
MR Author ID:
705313
Email:
ravindra@math.wustl.edu
V. Srinivas
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India
MR Author ID:
194380
Email:
srinivas@math.tifr.res.in
Received by editor(s):
April 21, 2005
Received by editor(s) in revised form:
May 31, 2005, and June 15, 2005
Published electronically:
October 25, 2005
Additional Notes:
Srinivas was partially supported by a Swarnajayanthi Fellowship of the D.S.T