Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



These are the differentials of order $n$

Authors: Dan Laksov and Anders Thorup
Journal: Trans. Amer. Math. Soc. 351 (1999), 1293-1353
MSC (1991): Primary 13N05, 14F10; Secondary 16Sxx
MathSciNet review: 1458328
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We answer P.-A. Meyer's question ``Qu'est ce qu'une différentielle d'ordre $n$?''. In fact, we present a general theory of higher order differentials based upon a construction of universal objects for higher order differentials. Applied to successive tangent spaces on a differentiable manifold, our theory gives the higher order differentials of Meyer as well as several new results on differentials on differentiable manifolds. In addition our approach gives a natural explanation of the quite mysterious multiplicative structure on higher order differentials observed by Meyer. Applied to iterations of the first order Kähler differentials our theory gives an algebra of higher order differentials for any smooth scheme. We also observe that much of the recent work on higher order osculation spaces of varieties fits well into the framework of our theory.

References [Enhancements On Off] (What's this?)

  • 1. A. Buium, Intersections in jet spaces and a conjecture of S. Lang, Ann. of Math. 136 (1992), 557-567. MR 93j:14055
  • 2. A. Buium, Geometry of differential polynomial functions, I: algebraic groups, Amer. J. Math. 115 (1993), 1385-1444. MR 95c:12011
  • 3. A. Buium, Geometry of differential polynomial functions, II: algebraic curves, Amer. J. Math. 116 (1994), 785-818. MR 96a:14039
  • 4. A. Buium, Geometry of differential polynomial functions, III: moduli spaces, Amer. J. Math. 117 (1995), 1-73. MR 96b:14059
  • 5. S. J. Colley and G. Kennedy, A higher-order contact formula for plane curves, Comm. Algebra 19(2) (1991), 479-508. MR 92f:14056
  • 6. S. J. Colley and G. Kennedy, The enumeration of simultaneous higher-order contacts between plane curves, Compositio Math. 93(2) (1994), 171-206. MR 95f:14097
  • 7. A. Collino, Evidence for a conjecture of Ellingsrud and Strömme on the Chow ring of $\operatorname{Hilb}_{d}(\mathbf{P}^{2})$, Illinois J. Math. 32 (1988), 171-210. MR 89d:14004
  • 8. J.-P. Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Lecture Notes, AMS Summer Research Institute Santa Cruz 1995, PSPM, vol. 62, part 2, 1997. CMP 98:07
  • 9. A. Grothendieck, with J. Dieudonné, Eléments de Géométrie Algébrique IV$_{4}$, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 1-361. MR 39:220
  • 10. S. Iitaka, Symmetric forms and Weierstrass cycles, Proc. Japan Acad., Ser. A 54 (1978), 101-103. MR 58:10893
  • 11. S. Iitaka, Duality theorems for symmetric differential forms, Proc. Jap. Acad., Ser. A 55 (1979), 53-58. MR 80j:14018
  • 12. S. Iitaka, Symmetric forms and Weierstrass semigroups, Algebraic geometry, Proc. Summer Meet., Copenh. 1978, Lect. Notes Math. 732, 1979, pp. 157-170. MR 81e:14012
  • 13. S. Iitaka, Weierstrass forms associated with linear systems, Adv. Math. 33 (1979), 14-30. MR 81c:14003
  • 14. J. Johnson, Order for systems of differential equations and a generalization of the notion of differential ring, J. Algebra 78 (1982), 91-119. MR 85i:12010
  • 15. J. Johnson, Prolongations of integral domains, J. Algebra 94 (1985), 173-210. MR 86i:12008
  • 16. E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, 1973. MR 58:27929
  • 17. E. R. Kolchin, Differential algebraic groups, Academic Press, New York, 1985. MR 87i:12016
  • 18. D. Laksov and A. Thorup, The Brill-Segre formula for families of curves, Enumerative Algebraic Geometry: Proceedings of the 1989 Zeuthen Symposium in Copenhagen, S. L. Kleiman and A. Thorup (Eds.), Contemporary Mathematics 123, AMS, Providence, 1991, pp. 131-148. MR 92k:14029
  • 19. D. Laksov and A. Thorup, Weierstrass points and gap sequences for families of curves, Ark. Mat. 32 (1994), 393-422. MR 96b:14041
  • 20. D. Laksov and A. Thorup, Weierstrass points on schemes, J. Reine Angew. Math. 460 (1995), 127-164. MR 96b:14016
  • 21. P.-A. Meyer, Formes differentielles d'ordre $n>1$, Publication IRMA, Université Louis Pasteur, Strasbourg, 1979/80.
  • 22. P.-A. Meyer, Qu'est ce qu'une différentielle d'ordre $n$?, Exposition. Math. 7 (1989), 249-264. MR 91d:60125
  • 23. J. F. Ritt, Differential Algebra, Amer. Math. Soc. Coll. Publ. 33, Amer. Math. Soc., New York, 1950. MR 12:7c
  • 24. R. Speiser, Derived triangles and differential systems, Projective geometry with applications, Lecture Notes in Pure and Applied Mathematics, 166, Dekker, New York, 1994, pp. 97-109. MR 95k:14073

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13N05, 14F10, 16Sxx

Retrieve articles in all journals with MSC (1991): 13N05, 14F10, 16Sxx

Additional Information

Dan Laksov
Affiliation: KTH, S–100 44 Stockholm, Sweden

Anders Thorup
Affiliation: Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, DK–2100 København Ø, Denmark

Received by editor(s): January 30, 1997
Additional Notes: The first author was partially supported by The Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.
The second author was supported in part by the Danish Natural Science Research Council, grant 11–7428.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society