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Let $X$ be a compact complex manifold of dimension $n$ and $Z$ be a hypersurface. I want to know about its homology class $[Z]\in H_{2n-2}(X)$.

Some methods I know to define $[Z]$:

  1. Find a line bundle $L$ which has a holomorphic section vanishing precisely on $[Z]$, and $c_1(L)\in H^2(X)$. Then its Poincare duality gives $[Z]\in H_{2n-2}(X)$. (This is equivalent to the definition of fundamental class. This is the part I can understand)
  2. The image of $1\in H^0(Z)$ in the composition $$H^0(Z)\xrightarrow[\text{ isomorphism}]{\text{Thom}} H^2(X,X-Z) \to H^2(X)$$ defines a class in $H^2(X)$ and then its Poincare duality gives $[Z]\in H_{2n-2}(X)$.

My questions:

  1. Both methods need to define the cohomology class first and use Poincare duality. Is there a direct way to define $[Z]\in H_{2n-2}(X)$?
  2. Why the two definitions are compatible?
Akatsuki
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    I mean, $Z$ is a $(2n-2)$-dimensional subset without boundary, so by the definition of homology, it gives you a well-defined class. In fact, your methods are usually used in the other direction, to get a geometric representative for an element of $H^2$. An old question of mine gives references to show the methods are equivalent. Basically, $\mathbb{CP}^\infty$ is both a $K(\mathbb{Z},2)$ and a classifying space for complex line bundles. – Steve D Sep 22 '18 at 04:07
  • @SteveD Which definition of homology do you use? For example if we consider singular homology of degree $k$, an element should be a linear combination of maps from $\Delta^k$ to $X$, which cannot be associated to $Z$ in a natural way I think? – Akatsuki Sep 22 '18 at 06:25
  • @SteveD Yeah I see, you mean to take the cycle $[Z]$ to be the sum of all the $\Delta^{2n-2}\to X$ onto some triangulation of $Z$. And via the definition of fundamental class it is related to the other definitions above. – Akatsuki Sep 22 '18 at 08:22
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    Whatever definition of homology you use, for a closed orientable manifold $Z$ of dimension $2n - 2$, we have $H_{2n-2}(Z; \mathbb{Z}) \cong \mathbb{Z}$. Moreover, an orientation on $Z$ defines a generator of $H_{2n-2}(Z; \mathbb{Z})$ called the fundamental class and is denoted $[Z]$. Given any map $f : Z \to X$, we have $f_*[Z] \in H_{2n-2}(X; \mathbb{Z})$. In particular, if $f$ is an embedding, the resulting homology class is the homology class of the submanifold $Z$. – Michael Albanese Sep 22 '18 at 13:25
  • @MichaelAlbanese Yes, this makes sense. Thx :) – Akatsuki Sep 26 '18 at 16:33
  • @SteveD One more question: if $[Z]$ is not a hyperplane but only a submanifold, what is the analogue of the first Chern class $c_1$? Is it $c_n$ (where $n$ is the complex codimension)? – Akatsuki Sep 26 '18 at 16:37

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