Characteristic Classes of Real Manifolds Immersed in Complex Manifolds

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: \[ \Omega (M) + \sum \limits _{r = 0}^{n - 1} {\tilde \Omega } {(\iota )^{n - r - 1}}{\iota ^ \ast }{c_r}(N) = 2{t^ \ast }DK,\] 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 $\iota :M \to {{\text {C}}^n}$.