I'm trying to work out Remark 11.1.9 in Salamon in J-holomorphic Curves and Symplectic Topology (Second Edition) by Dusa McDuff and Dietmar Salamon.
They say that one should take $L=[\mathbb{C}P^1]\in H^2(\mathbb{C}P^n)$ (I think this is a typo and it should be $H_2$, but anyway) and $c=PD[\mathbb{C}P^{n-1}]\in H^2(\mathbb{C}P^n)$.
Then representing $PD(c^k)$ by a $n-k$ dimensional plane $X\subset \mathbb{C}P^n$ and $PD(c^l)$ by a $n-l$ plane $Y\subset \mathbb{C}P^n$ so that $k+l>n$ and $X\cap Y=\emptyset$. Then the cohomology class $(c^k*c^l)_L$ is Poincaré dual to the homology class represented by the set of points that lie on lines passing through $X$ and $Y$ and so $$ (c^k*c^l)_L = c^{k+l-n-1}\quad(1) $$
Questions:
- If $X\cap Y=\emptyset$, then there is no line though $X$ and $Y$ no? What is "the set of points that lie on lines passing through $X$ and $Y$"?
- We consider here $(c^k*c^l)_L$, as $L$ is a generator of $H_2(\mathbb{C}P^n)$. The classes $(c^k*c^l)_{dL}$ are then $d(c^k*c^l)_{L}$?
- Why are we taking that $c$ actually?
- I would like to picture this with $n=1$ or $n=2$ but didn't manage to do it. Hints are welcome :)
- How does $(1)$ follow form the preceding statement?
