Cohomology with support and Poincare duality

Question

Say we have hypersurface $D \subset \mathbb{P}^2$. I ran into the following sequence of isomorphisms, justified only with "by Poincare duality formulated algebro-geometrically": $$ H^2(\mathbb{P}^2-D,\mathbb{Q}) \cong H^3_D(\mathbb{P}^2,\mathbb{Q}) \cong H_1(D,\mathbb{Q}) $$ I looked up cohomology with supports, and I could only find something like $H^k_c(X) \cong H^{n-k}(X)^*$. Could you help me with either a reference or an explenation of why the above hold? Thank you.

Answer 1

There is two isomorphisms in your question, the second one is indeed a form of Poincaré duality/purity isomorphism. The first one is on the contrary specific to the situation. I will explain both of them.

---

First let us talk about the Poincaré duality. Every homology/cohomology groups will be with rational coefficients, so I won't write them.

If $Z\subset X$ is a closed subset and $U$ is its complement, then there are localization long exact sequences in cohomology with compact support, with Borel-Moore homology and cohomology : $$ ...\to H^i_c(U)\to H^i_c(X)\to H^i_c(Z)\to H^{i+1}_c(U)\to...$$ $$ ...\to H^{BM}_{d-i}(Z) \to H^{BM}_{d-i}(X) \to H^{BM}_{d-i}(U)\to H^{BM}_{d-i-1}(Z)\to...$$ $$ ...\to H^i_Z(X)\to H^i(X)\to H^i(U)\to H^{i+1}_Z(X)\to...$$

A version of Poincaré duality says that for an oriented manifold of (topological) dimension $d$, then $H^i=H^{BM}_{d-i}=(H_c^{d-i})^*$. In the case where $X,U,Z$ are smooth, then the second exact sequence is the dual of the first.

But we also have an identification between the second and the third : there is a relative Poincaré duality $H^i_Z(X)=H^{BM}_{d-i}(Z)$. Together with this relative duality, the second and the third exact sequence are the same.

This relative Poincaré duality gives in your specific case, you get $H_1(D)=H^3_D(\mathbb{P}^2)$.

---

So here is a "derived" variant of the previous isomorphism, and what is usually called the purity isomorphism.

Let $i:Z\rightarrow X$ and $j:U\rightarrow X$ be the closed and open immersion. Let $f:X\to pt$ be the canonical morphism. Then, in the derived category of abelian groups, there are isomorphisms : $$ H^i(X)=Hom_D(\mathbb{Q},Rf_*f^*\mathbb{Q}[i])$$ $$H^{BM}_{d-i}(X)=Hom_D(\mathbb{Q},Rf_*Rf^!\mathbb{Q}[i-d])$$ $$ H^i_c(X)=Hom_D(\mathbb{Q},Rf_!f^*\mathbb{Q}[i])$$

And the long exact sequence above comes from the distinguished triangles $$ i_*Ri^!\to 1\to Rj_*j^*\overset{+1}\to$$ $$ j_!j^*\to 1\to i_*i^*\overset{+1}\to$$ Now the version of Poincaré-Verdier duality states that for an oriented manifold of dimension $d$, we have an isomorphism of functor $Rf^!=f^*[d]$. This gives in particular the isomorphism $H^i=H^{BM}_{d-i}$.

If $X$ and $Z$ are smooth and $Z$ of codimension $c$, then $$i^!\mathbb{Q}_X[d]=i^!f^!\mathbb{Q}=(fi)^!\mathbb{Q}=\mathbb{Q}_Z[d-c]$$ Hence $i^!\mathbb{Q}=\mathbb{Q}[-c]$. This holds isomorphism is called a purity isomorphism in algebraic geometry.

---

Now let us have a look at the first isomorphism $H^2(\mathbb{P}^2-D)\simeq H^3_D(\mathbb{P}^2)$. We will apply the long exact sequences with $X=\mathbb{P}^2$ and $Z=D$ a smooth hypersurface. Since they are proper, $H^i=H^i_c$ and $H^{BM}_i=H_i$. The localization sequence in (say Borel Moore homology) is then :

$$ ...\to H_2(D)\to H_2(\mathbb{P}^2)\to H_2^{BM}(\mathbb{P}-D)\to H_1(D)\to H_1(\mathbb{P}^2)\to... $$ But now, by Poincaré dualiy (with $d=4$) this is : $$ ...\to H^2_D(\mathbb{P}^2)\to H^2(\mathbb{P}^2)\to H^2(\mathbb{P}-D)\to H^3_D(\mathbb{P}^2)\to H^3(\mathbb{P}^2)\to... $$

Now, since we are working with rational coefficient, $[D]\in H_2(D)$ is sent to a generator of $H_2(\mathbb{P}^2)$ so the map $H_2(D)\to H_2(\mathbb{P}^2)$ is onto, and since $H_1(\mathbb{P}^2)=0$, we have the isomorphism $H^2(\mathbb{P}^2-D)\simeq H_1(D)\simeq H^*_D(\mathbb{P}^2)$

---