Dual Class of a Subvariety
Tatsuo Suwa · Tokyo Journal of Mathematics · 2000
Dedicated to the memory of N. Sasakura Let $M$ be a complex manifold of dimension $n$ and $E$ a holomorphic vector bundle ofcorresponds to the homology class [X] of $X$ under the Poincar\'e duality $P$ : $H^{2k}(M;C)\rightarrow\sim H_{2n-2k}(M;C)$ (in fact this holds with $Z$ coefficients).The nature of the proof of this fact depends on how one defines the class $c_{k}(E)$ (cf.[G] \S 5 for the projective non-singular case, [F] \S 14.1 for the general case in the algebraic category and [GH] Ch. 1, \S 1 for the case $k=1$ in the complex analytic category).In this article, we take up the definition of Chem classes via the Chem-Weil theory and give a relatively elementary proof of a more precise statement in the complex analytic category.Namely, we prove the following.Let $V$ denote the support of $X$ , then there is a canonical localization $c_{k}(E, s)$ , in the relative cohomology $H^{2k}(M, M\backslash V;C)$ , of $c_{k}(E)$ with respect to $s$ and, if $V$ is compact ( $M$ may not be), the class $c_{k}(E, s)$ corresponds to [X] under the Alexander dualitywhere $i$ and $j$ denote the inclusions $V\rightarrow M$ and $(M, \emptyset)\simeq*(M, M\backslash V)$ , respectively.Since $j^{*}(c_{k}(E, s))=c_{k}(E)$ , we recover the result we first mentioned.For an application, see [S2].As related topics, we discuss intersections of analytic subspaces.We also prove a duality theorem when $V$ as above may not be compact, considering $X$ as a relative cycle in $M$ modulo $M\backslash S$ for a compact connected component $S$ of its singular set (Theorem 6.4).This fact is effectively used in [BLSS].The proofs of the above results are done in the framework of \v{C}ech-de Rham cohomology.