Foliations of multiprojective spaces and a conjecture of Bernstein and Lunts
HTML articles powered by AMS MathViewer
- by S. C. Coutinho PDF
- Trans. Amer. Math. Soc. 363 (2011), 2125-2142 Request permission
Abstract:
We use foliations of multiprojective spaces defined by Hamiltonian functions on the underlying affine space to prove the three dimensional case of a conjecture of Bernstein and Lunts, according to which the symbol of a generic first-order differential operator gives rise to a hypersurface of the cotangent bundle which does not contain involutive conical subvarieties apart from the zero section and fibres of the bundle.References
- William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608, DOI 10.1090/gsm/003
- V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295, DOI 10.1007/978-1-4757-2063-1
- Joseph Bernstein and Valery Lunts, On nonholonomic irreducible $D$-modules, Invent. Math. 94 (1988), no. 2, 223–243. MR 958832, DOI 10.1007/BF01394325
- A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic $D$-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
- Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
- S. C. Coutinho, A primer of algebraic $D$-modules, London Mathematical Society Student Texts, vol. 33, Cambridge University Press, Cambridge, 1995. MR 1356713, DOI 10.1017/CBO9780511623653
- S. C. Coutinho and J. V. Pereira, On the density of algebraic foliations without algebraic invariant sets, J. Reine Angew. Math. 594 (2006), 117–135. MR 2248154, DOI 10.1515/CRELLE.2006.037
- Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- Ofer Gabber, The integrability of the characteristic variety, Amer. J. Math. 103 (1981), no. 3, 445–468. MR 618321, DOI 10.2307/2374101
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361, DOI 10.1007/978-0-8176-4523-6
- J. P. Jouanolou, Équations de Pfaff algébriques, Lecture Notes in Mathematics, vol. 708, Springer, Berlin, 1979 (French). MR 537038
- Valery Lunts, Algebraic varieties preserved by generic flows, Duke Math. J. 58 (1989), no. 3, 531–554. MR 1016433, DOI 10.1215/S0012-7094-89-05824-9
- A. Lins Neto, P. Sad, and B. Scárdua, On topological rigidity of projective foliations, Bull. Soc. Math. France 126 (1998), no. 3, 381–406 (English, with English and French summaries). MR 1682801
- P. Samuel, Anneaux factoriels, Sociedade de Matemática de São Paulo, São Paulo, 1963 (French). Rédaction de Artibano Micali. MR 0156867
- J. T. Stafford, Nonholonomic modules over Weyl algebras and enveloping algebras, Invent. Math. 79 (1985), no. 3, 619–638. MR 782240, DOI 10.1007/BF01388528
Additional Information
- S. C. Coutinho
- Affiliation: Departamento de Ciência da Computação, Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, 21945-970 Rio de Janeiro, Rio de Janeiro, Brazil – and – Programa de Engenharia de Sistemas e Computação, COPPE, Universidade Federaldo Rio de Janeiro, PO Box 68511, 21941-972 Rio de Janeiro, Rio de Janeiro, Brazil
- Email: collier@impa.br
- Received by editor(s): July 6, 2009
- Published electronically: October 28, 2010
- Additional Notes: The author wishes to thank Jorge Vitório Pereira for his help with section 4. The work on this paper was partially supported by a grant from CNPq.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 2125-2142
- MSC (2000): Primary 37F75, 16S32; Secondary 37J30, 32C38, 32S65
- DOI: https://doi.org/10.1090/S0002-9947-2010-05230-4
- MathSciNet review: 2746677
Dedicated: To Israel Vainsencher on his 60th birthday