# Intersection pairing of weighted projective plane

Let $$a,b,c$$ be mutually relatively prime positive integers. The weighted projective plane $$X:=\Bbb CP^2(a,b,c)$$ is the quotient space $$\Bbb C^3-\{0\}/(z_1,z_2,z_3)\sim (\lambda^a z_1,\lambda^b z_2,\lambda^c z_3)$$ for $$\lambda\in \Bbb C-\{0\}$$. $$X$$ is a singular 4-manifold with three singular points $$[1,0,0], [0,1,0], [0,0,1]$$, and near these singular points, $$X$$ is a cone on lens space. Let $$Y$$ be the complement of small open neighborhoods of the singular points. Then $$Y$$ is a smooth 4-manifold with boundary. How can we compute the intersection form of $$Y$$?

Since $$H_2(Y)=\Bbb Z$$, it suffices to compute $$\alpha^2$$ for a generator $$\alpha \in H_2(Y)$$. Such a generator $$\alpha$$ may be represented by an embedded surface in $$H_2(Y)$$ which I can't find.

According to Weighted projective plane as a quotient of $\Bbb CP^2$, $$X$$ can be regarded as a quotient space of $$\Bbb CP^2$$. So we have a quotient map $$f:\Bbb CP^2\to X$$ and a covering map $$f:f^{-1}(Y)\to Y$$. A generator $$H$$ of $$H_2(\Bbb CP^2)=\Bbb Z$$ is represented by a line in $$\Bbb CP^2$$ and its self-intersection is $$1$$. Using these, I tried to compute the self-intersection of $$f($$line$$)$$ but I got stuck in the simple case $$(a,b,c)=(2,3,5)$$ (Intersection of two lines in weighted projective plane).

• @hm2020 Can we get the information of cohomology ring of $X/G$ from its Chow group? Commented Dec 4, 2021 at 19:07

Question: "How can we compute the intersection form of Y?"

Answer: There is another construction of the weighted projective plane $$X:=\mathbb{P}(a,b,c):=\mathbb{P}^2/G:=X/G$$ where $$G:=\mathbb{Z}/a×\mathbb{Z}/b×\mathbb{Z}/c$$ which is a finite group (Harris "Algebraic geometry - a first course"). There is an explicit formula for its Chow group with rational coefficients:

$$CH^∗(X/G)≅CH^∗(X)^G.$$

@hm2020 Can we get the information of cohomology ring of $$X/G$$ from its Chow group?

Response: I do not have a reference but I believe there are similar formulas on the form

$$\text{ I1. }H^∗(X/G,\mathbb{Q})≅H^∗(X,\mathbb{Q})^G$$

when $$H^∗(−)$$ is singular cohomology. Since there is an isomorphism

$$CH^*(\mathbb{P}^2)_{\mathbb{Q}} \cong H^*(\mathbb{P}^2,\mathbb{Q})$$

it may be this induce an isomorphism

$$CH^*(\mathbb{P}(a,b,c))_{\mathbb{Q}}\cong CH^*(\mathbb{P}^2/G)_{\mathbb{Q}} \cong H^*(\mathbb{P}^2/G, \mathbb{Q}) \cong H^*(\mathbb{P}(a,b,c) ,\mathbb{Q})$$

but this must be checked. If this is the case it may help in the calculation of the intersection form of $$Y$$.

Question: "So we have a quotient map $$f:\Bbb CP^2\to X$$ and a covering map $$f:f^{-1}(Y)\to Y$$. A generator $$H$$ of $$H_2(\Bbb CP^2)=\Bbb Z$$ is represented by a line in $$\Bbb CP^2$$ and its self-intersection is $$1$$. Using these, I tried to compute the self-intersection of $$f($$line$$)$$ but I got stuck in the simple case $$(a,b,c)=(2,3,5)$$".

Whenever there is a quotient map $$\pi: Y \rightarrow Y/G$$ (this holds if $$Y$$ is quasi-proejctive of finite type over a field) this induce a sequence (with rational coefficients $$k$$)

$$CH^*(Y/G)_k \rightarrow^{\pi^*} CH^*(Y)_k^G \rightarrow CH^*(Y)_k$$

and an isomorphism $$CH^*(Y/G)_k \cong CH^*(Y)_k^G$$ of abelian groups, induced by $$\pi^*$$.

Note: The Chow group of the projective plane is $$CH^*(\mathbb{P}^2) \cong \mathbb{Z}[t]/(t^3)$$ by the "projective bundle formula". From this you get a formula for the Chow group of the weighted projective plane with rational coefficients. There are similar "projective bundle formulas" for singular cohomology, hence it should be possible to calculate

$$\phi:H^*(\mathbb{P}(a,b,c),\mathbb{Q}) \cong H^*(\mathbb{P}^2,\mathbb{Q})^G$$

if $$I1$$ holds. With such an isomorphism $$\phi$$ it is easy to calculate any product in $$H^*(\mathbb{P}^2,\mathbb{Q})^G$$ or $$CH^*(\mathbb{P}^2)_\mathbb{Q}^G$$. Take any pair of cycles

$$\alpha, \beta \in CH^*(Y/G)_k \cong CH^*(Y)_k^G$$

and calculate the product $$\alpha \beta$$ in $$CH^*(Y)_k$$ which by the above construction is well known. A similar construction holds for singular cohomology.

Note: The intersection product is defined geometrically for smooth schemes and the weighted projective plane is not smooth. Hence care must be taken when defining the "product": It is not completely clear how to define the product on $$CH^*(X/G)_k$$. You may for any subvarieties $$V,W \subseteq X/G$$ define

$$[V]\bullet[W]:=\frac{1}{\# G} \eta_*(\pi^*[V]\pi^*[W])$$

where $$\eta:\pi^{-1}(V \cap W) \rightarrow V \cap W$$

is the canonical map. Here $$\pi^*[V]\pi^*[W]$$ is the product in $$CH^*(X)_k$$ and this product is well known by the above calculation. This product is "intrinsic" in a certain sense.

If $$G$$ acts on $$H^*(X/G, \mathbb{Q})$$ as ring-automorphisms, it follows the invariant ring has canonically a ring structure.