Abstract
We describe the properties of some foliations which arise in the study of the characteristic variety of \({\mathcal{D}}\) -modules constructed from vector fields of an affine space.
Similar content being viewed by others
References
Arnold V.I.: Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Paris (1996)
Baum P., Bott R.: Singularities of holomorphic foliations. J. Diff. Geo. 7, 279–342 (1972)
Bernstein J., Lunts V.: On non-holonomic irreducible D-modules. Invent. Math. 94, 223–243 (1988)
Brunella M., Mendes L.G.: Bounding the degree of solutions to Pfaff equations. Publ. Mat. 44(2), 593–604 (2000)
Carnicer M.M.: The Poincaré problem in the nondicritical case. Ann. Math. 140, 289–294 (1994)
Cerveau D., Lins A.N.: Holomorphic foliations in CP(2) having an invariant algebraic curve. Ann. Sci. de l’Institute Fourier 41, 883–903 (1991)
Chriss N., Ginzburg V.: Representation Theory and Complex Geometry. Birkhäuser Boston Inc., Boston (1997)
Coutinho S.C.: A Primer of Algebraic D-Modules. London Mathematical Society Student Texts, Vol. 33. Cambridge University Press, Cambridge (1995)
Coutinho S.C.: d-simple rings and simple \({\mathcal{D}}\) -modules. Math. Proc. Cambridge Philos. Soc. 125(3), 405–415 (1999)
Coutinho S.C.: Indecomposable non-holonomic \({\mathcal{D}}\) -modules in dimension 2. Proc. Edinburgh Math. Soc. 46, 341–355 (2003)
Coutinho S.C.: Foliations of multiprojective spaces and a conjecture of Bernstein and Lunts. Trans. Am. Math. Soc. 363(4), 2125–2142 (2011)
Coutinho S.C., Holland M.P., Levcovitz D.: Conormal varieties and characteristic varieties. Proc. Am. Math. Soc. 128(4), 975–980 (2000)
Coutinho S.C., Pereira J.V.: On the density of algebraic foliations without algebraic invariant sets. J. Reine Angew. Math. 594, 117–135 (2006)
Darboux, G.: Mémoire sur les équations différentielles algébriques du Io ordre et du premier degré. Bull. des Sc. Math. (Mélanges), pp. 60–96, 123–144, 151–200 (1878)
Doering A.M.D.S., Lequain Y., Ripoll C.: Differential simplicity and cyclic maximal ideals of the Weyl algebra A 2(K). Glasg. Math. J. 48(2), 269– (2006)
Eisenbud, D.: Commutative Algebra, Graduate Texts in Mathematics, Vol. 150. Springer, New York, With a view toward algebraic geometry (1995)
Esteves E.: The Castelnuovo-Mumford regularity of an integral variety of a vector field on projective space. Math. Res. Let. 9, 1–15 (2002)
Esteves, E., Kleiman, S.L.: Bounding solutions of Pfaff equations. Comm. Algebra 31(8): 3771–3793, Special issue in honor of Steven L. Kleiman (2003)
Esteves, E.: Bounds on leaves of foliations of the plane, Real and complex singularities, Contemporary Mathematics, Vol. 354. American Mathematical Society, Providence, pp. 57–67 (2004)
Gabber O.: The integrability of the characteristic variety. Am. J. Math. 103(3), 445–468 (1981)
Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. no. 24, p. 231 (1965)
Harris, J.: Algebraic geometry, Graduate Texts in Mathematics, A first course, Vol. 133. Springer, New York (1992)
Jouanolou J.P.: Equations de Pfaff algébriques, Lecture Notes in Mathematics, Vol. 708. Springer, Verlag (1979)
Kashiwara, M.: Systems of microdifferential equations, Progress in Mathematics, Vol. 34, Birkhäuser Boston Inc., Boston, MA, Based on lecture notes by Teresa Monteiro Fernandes translated from the French, With an introduction by Jean-Luc Brylinski (1983)
Lins Neto A., Soares M.G.: Algebraic solutions of one-dimensional foliations. J. Differ. Geom. 43(3), 652–673 (1996)
Lunts V.: Algebraic varieties preserved by generic flows. Duke Math. J. 58(3), 531–554 (1989)
Samuel P.: Anneaux factoriels, Rédaction de Artibano Micali. Sociedade de Matemática de São Paulo, São Paulo (1963)
Soares M.G.: On algebraic sets invariant by one-dimensional foliations of cp(3). Ann. Inst. Fourier 43, 143–162 (1993)
Soares M.G.: The Poincaré problem for hypersurfaces invariant by one-dimensional foliations. Invent. Math. 128(3), 495–500 (1997)
Stafford J.T.: Nonholonomic modules over Weyl algebras and enveloping algebras. Invent. Math. 79(3), 619–638 (1985)
Walcher S.: On the Poincaré problem. J. Differ. Equ. 166, 51–78 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
I wish to thank Israel Vainsencher and the referee for pointing out a number of important corrections to the original version of this paper. The work on this paper was partially supported by grants from CNPq and Pronex/Faperj.
Rights and permissions
About this article
Cite this article
Coutinho, S.C. On some foliations arising in \({\mathcal{D}}\) -module theory. Geom Dedicata 164, 27–45 (2013). https://doi.org/10.1007/s10711-012-9757-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-012-9757-6