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)

 
 

 

Characteristic classes of real manifolds immersed in complex manifolds


Author: Hon Fei Lai
Journal: Trans. Amer. Math. Soc. 172 (1972), 1-33
MSC: Primary 57D20; Secondary 32C10
DOI: https://doi.org/10.1090/S0002-9947-1972-0314066-8
MathSciNet review: 0314066
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a compact, orientable, $ k$-dimensional real differentiaable manifold and $ N$ an $ n$-dimensional complex manifold, where $ k \geq n$. Given an immersion $ \iota :M \to N$, a point $ x \in M$ is called an RC-singular point of the immersion if the tangent space to $ \iota (M)$ at $ \iota (x)$ contains a complex subspace of dimension $ > k - n$. This paper is devoted to the study of the cohomological properties of the set of RC-singular points of an immersion.

When $ k = 2n - 2$, the following formula is obtained:

$\displaystyle \Omega (M) + \sum\limits_{r = 0}^{n - 1} {\tilde \Omega } {(\iota )^{n - r - 1}}{\iota ^ \ast }{c_r}(N) = 2{t^ \ast }DK,$

where $ \Omega (M)$ is the Euler class of $ M,\widetilde\Omega (\iota )$ is the Euler class of the normal bundle of the immersion, $ {c_r}(N)$ are the Chern classes of $ N$, and $ {t^ \ast }DK$ is a cohomology class of degree $ 2n - 2$ in $ M$ whose value on the fundamental class of $ M$ gives the algebraic number of RC-singular points of $ \iota $. Various applications are discussed.

For $ n \leq k \leq 2n - 2$, it is shown that, as long as dimensions allow, all Pontrjagin classes and the Euler class of $ M$ are carried by subsets of the set of RC-singularities of an immersion $ \iota :M \to {{\text{C}}^n}$.


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

  • [1] E. Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1-21. MR 34 #369. MR 0200476 (34:369)
  • [2] E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes. I, Ann. Mat. 11 (1932), 17-90.
  • [3] S. S. Chern, Topics in differential geometry, The Institute for Advanced Study, Princeton, N. J., 1951 (mimeographed). MR 19, 764. MR 0090080 (19:764e)
  • [4] S. S. Chern and E. Spanier, A theorem on orientable surfaces in four-dimensional space, Comment. Math. Helv. 25 (1951), 205-209. MR 13, 492. MR 0044883 (13:492d)
  • [5] W. Gysin, Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten, Comment. Math. Helv. 14 (1942), 61-122. MR 3, 317. MR 0006511 (3:317a)
  • [6] L. R. Hunt, The envelope of holomorphy of an $ n$-manifold in $ {{\text{C}}^n}$, Boll. Un. Mat. Ital. 4 (1971), 12-35. MR 0294690 (45:3758)
  • [7] R. Lashof and S. Smale, On the immersion of manifolds in Euclidean space, Ann. of Math. (2) 68 (1958), 562-583. MR 21 #2246. MR 0103478 (21:2246)
  • [8] L. S. Pontrjagin, Characteristic cycles on differentiable manifolds, Mat. Sb. 21 (63) (1947), 233-284; English transl., Amer. Math. Soc. Transl. (1) 7 (1962), 149-219. MR 9, 243. MR 0022667 (9:243f)
  • [9] H. Seifert, Algebraische Approximation von Mannigfaltigkeiten, Math. Z 41 (1936), 1-17. MR 1545601
  • [10] E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
  • [11] N. Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of $ n$ complex variables, J. Math. Soc. Japan 14 (1962), 397-429. MR 26 #3086. MR 0145555 (26:3086)
  • [12] -, On generalized graded Lie algebras and geometric structures. I, J. Math. Soc. Japan 19 (1967), 215-254. MR 36 #4470. MR 0221418 (36:4470)
  • [13] R. Thom, Un lemme sur les applications différentiables, Bol. Soc. Mat. Mexicana (2) 1 (1956), 59-71. MR 21 #910. MR 0102115 (21:910)
  • [14] R. O. Wells, Jr., Holomorphic hulls and holomorphic convexity, Complex Analysis (Proc. Conf. Rice Univ., Houston, Tex., 1967), Rice Univ. Studies 54 (1968), no. 4, 75-84. MR 39 #3029. MR 0241690 (39:3029)
  • [15] -, Compact real submanifolds of a complex manifold with nondegenerate holomorphic tangent bundles, Math. Ann. 179 (1969), 123-129. MR 38 #6104. MR 0237823 (38:6104)
  • [16] R. O. Wells, Jr., Concerning the envelope of holomorphy of a compact differentiable submanifold of a complex manifold, Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 347-361. MR 39 #7141. MR 0245835 (39:7141)
  • [17] W. T. Wu, Sur les classes caractéristiques des structures fibrées sphériques, Actualités Sci. Indust., no. 1183 = Publ. Inst. Math. Univ. Strasbourg 11 pp. 5-89, 155-156, Hermann, Paris, 1952. MR 14, 1112. MR 0055691 (14:1112d)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57D20, 32C10

Retrieve articles in all journals with MSC: 57D20, 32C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0314066-8
Keywords: Complex manifold, real submanifold, Poincaré duality, intersection, Schubert varieties, vector bundles, characteristic classes, cohomological extension of fibres
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society