# What is the poinacre dual of the projectivization of a line bundle inside the projectivization of the sum of two line bundles?

Let $S:= \mathbb{CP}^1 \times \mathbb{CP}^1$. Let $TS \approx L_1 \oplus L_2$, where $L_1$ and $L_2$ are the pullback of $T\mathbb{CP}^1$ wrt to the respective projection maps. Note that $\mathbb{P}(L_1)$ is a submanifold of $\mathbb{P}(L_1 \oplus L_2)$. Note that the cohomology of $\mathbb{P}(L_1 \oplus L_2)$ is generated by $a_1$, $a_2$ and $\lambda$, where $a_i$ and $\lambda$ are the fist chern classes of the dual of the tautological line bundles over $\mathbb{CP}^1$ and $\mathbb{P}(L_1 \oplus L_2)$ respectively.

My question is the following: what is the Poincare Dual of the homology class $[\mathbb{P}(L_1)]$ inside $\mathbb{P}(L_1 \oplus L_2)$? It has to be a linear combination of $a_1$, $a_2$ and $\lambda$.

Note that $L_1$ and $L_2$ are both bundles over $\mathbb{CP}^1 \times \mathbb{CP}^1$, i.e. $L_i:= \pi_i^* T\mathbb{CP}^1 \rightarrow \mathbb{CP}^1 \times \mathbb{CP}^1$.

• A comment: The projectivization of a one dimensional vector space is a point, this also works for bundles. Thus $\mathbb{P}(L_1)\cong\mathbb{CP}^1\times \mathbb{CP^1}$. Is the normal bundle of this manifold the tautogical bundle of $\mathbb{P}(L_1\oplus L_2)$? – Thomas Rot Jul 21 '15 at 14:45