3
$\begingroup$

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E (i.e. line bundles). Let $$ n_1:= \langle c_1(L_1), [D]\rangle , ~~n_2 := \langle c_1(L_2), [D]\rangle , ~~n_3:= \langle c_1(E), [D]\rangle . $$ If I know the values of $n_1, n_2$ and $n_3$, is this sufficient information to calculate the topological intersection number $$ N:= \mathbb{P}L_1 \cdot \mathbb{P}L_2 $$ inside $\mathbb{P}E$, where $\mathbb{P}$ denotes projectivization of a bundle. In particular is there a formula for $N$ in terms of $n_1$, $n_2$ and $n_3$? If not, what further information is required to calculate $N$?

$\endgroup$

1 Answer 1

5
$\begingroup$

It looks like $n_3-n_1-n_2$, but double check the computation. Tensor everything by $L_1^{-1}$ to make $L_1$ trivial and recompute the classes to get $0$, $n_2-n_1$, and $n_3-2n_1$. Then project a section of (trivial now) $L_1$ to $E/L_2$: you are interested in the zeroes of this projection, which are counted by $c_1(E/L_2)=(n_3-2n_1)-(n_2-n_1)$.

$\endgroup$
1
  • $\begingroup$ Very neat solution! $\endgroup$
    – Ritwik
    Feb 26, 2015 at 22:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.