2
$\begingroup$

I realise I don't have a good definition for the vertical tangent bundle in the case of a Lefschetz fibration $(E,F)\to (S,\partial S)$ ($F$ is the subbundle given by the tangent bundle of the Lagrangian boundary condition over $\partial S$.

usually the vertical sub bundle is defined as the kernel of the differential of the projection map, and that is a bundle if the rank of such map is locally constant, but the entire point of Lefschetz fibrations is that $\pi$ has critical points, so the set of kernels in each fiber do not amount to a bundle.

Nonetheless we have concepts like the relative Chern class for that "bundle", as in P. Seidel's - Fukaya Categories and Picard-Lefschetz Theory. After Lemma 17.5 he computes the index of the linearised Cauchy-Riemann operator for J-hol sections as some integral of the relative Chern class pulled back via the section $u$.

I feel this might be a stupid question but I cannot find an answer in the classical literature and I'm quite puzzled.

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ I haven't looked at Seidel's book, but here's a thought. Since vertical bundle is defined away from the set $C$ of critical points of $\pi$, you can pull it back if the image of $u$ avoids $C$. Could it be that this is what he's doing? $\endgroup$
    – Marco Golla
    Commented Apr 7 at 13:27
  • $\begingroup$ Hi Marco! I’m pretty sure that’s what he’s doing, I was just wondering if it still makes sense to talk about a chern class for it, or if everything is doing “formally” and then using your reasoning to say “well it exists because we are avoiding $C$” $\endgroup$
    – Riccardo
    Commented Apr 7 at 15:16
  • 1
    $\begingroup$ I guess it would make sense, as long as you only talk about the first Chern class, since removing isolated points from a $2n$-manifold doesn't affect $H^2$ for $n>1$. $\endgroup$
    – Marco Golla
    Commented Apr 7 at 16:35

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .