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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
MathSciNet review: 0314066
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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}$.

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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