# Residues and Gysin long exact for open varieties

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?

• I would look at the paper of Herrera-Lieberman, "Residues and principal values on complex spaces". Although I'd bet it's already in Leray's work in this form. – Dan Petersen Oct 5 '18 at 12:02

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,

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