Residues and Gysin long exact for open varieties

Question

I am familiar with the following:

let $X$ be a smooth projective complex variety, $D$ a smooth divisor in $X$ and $U=X \setminus Z$. Then there is on the one hand a residue map $$ \mathrm{Res}_D \colon H^n_{dR}(U) \longrightarrow H^{n-1}_{dR}(D) $$ induced from the exact sequence of sheaves $0 \to \Omega^p_X \to \Omega^p_X(\log D) \to \Omega^{p-1}_D \to 0,$ and on the other hand the tubular map in homology $$ T \colon H_{n-1}(D) \to H_n(U). $$ They are related by $$ \int_\sigma \mathrm{Res}_D(\omega)=\frac{1}{2\pi i} \int_{T(\sigma)} \omega $$ for all $\sigma \in H_{n-1}(D)$ and $\omega \in H^n_{dR}(U)$.

Question: I guess this is still true when $X$ is not supposed projective, because everything seems very local. Do you know a good reference?

Answer 1

The Gysin sequence in the form you need is a purely topological based on the long exact cohomological sequence of a pair, the Thom isomophism and Poincare duality.

The Gysin sequence is obtained from the long exact sequence of the pair $(T(D), \partial T(D))$ where $T(D)$ is a tubular neighborhood of $D$ in $X$. Note that we have a natural projection $\pi: T(D)\to D$. Moreover, by excision

$$ H^\bullet(T(D),\partial T(D)\;)\cong H^\bullet (X, X\setminus D). $$ $\newcommand{\pa}{\partial}$ If $D$ is compact then it carries a homology class $[D]$ and its Poincare dual is the Thom class $\tau\in H^2(T(D),\partial T(D))$.

The Thom isomorphism theorem states that the map $\Psi: H^\bullet (T(D)D)\to H^{\bullet +2}(T(D),\pa T(D))$ given by $$ H^kT(D)\cong H^k(D)\ni \alpha \mapsto \tau \cup \pi^*\alpha\in H^{k+2}(T(D),\pa T(D)) $$ is an isomorphism.

The long exact sequence

$$ \cdots \to H^k(T(D),\pa T(D))\to H^k(T(D))\to H^k (\pa T(D))\to H^{k+1}(T(D),\pa T(D))\to\cdots $$

can be rewritten using the Thom isomorphism in the form $$ \cdots \to H^{k-2}(T(D))\to H^k(T(D))\to H^k (\pa T(D))\to H^{k-1}(T(D))\to\cdots $$ Note that $H^\bullet(T(D))\cong H^\bullet (D)$ so we obtain $$ \cdots \to H^{k-2}(D)\to H^k(D)\to H^k (\pa T(D))\to H^{k-1}(D)\to\cdots $$

We set $N:=\dim_{\mathbb{C}} X$ and $m:=2N-k$.

Using the Poincare dualities $$ H^j(D)\cong H_{2N-2-j}(D),\;\; H^j(\pa T(D))\cong H_{2N-1-j}( \pa T(D)) $$ we obtain a long exact sequence

$$\cdots \to H_m(D)\to H_{m-2}( D) \stackrel{T}{\to} H_{m-1}(\pa T(D))\to H_{m-1}(D)\to\cdots .$$

Above, $T$ is the so called tube morphism. Its Kronecker dual is the Residue morphism. See these notes for more details.

If $D$ is non-compact, then you will have to use Borel-Moore homology $H_\bullet^{BM}$ instead of the usual homology,

$$ H_\bullet^{BM}(X,\mathbb{C})={\rm Hom}\;(\; H^\bullet_{cpt}(X,\mathbb{C}),\mathbb{C}\;). $$