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I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration structure.

So I wonder if every symplectic manifold (especially symplectic 4-manifold) admits a Lefschetz fibration instead of just a Lefschetz pencil? If not, are there any examples?

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  • $\begingroup$ I would say $\mathbb{C}p^2$. It obviously cannot have a Lefschetz fibration with a section and, on the other hand, I think any LF has one. $\endgroup$
    – Alex Degtyarev
    Commented Nov 26, 2014 at 6:41
  • $\begingroup$ @AlexDegtyarev Sorry I don't see why $\mathbb{CP}^2$ cannot have a section. Also I am not sure if every LF has a section. Any hints or references on that? $\endgroup$
    – Jie Min
    Commented Nov 26, 2014 at 9:09
  • $\begingroup$ Fibers must be disjoint, hence homologically trivial in $\mathbb{C}p^2$. On the other hand, if there is a section, it intersects fibers at $1$, preventing them from being homologous to $0$. I'm not sure that every LF has a section (in fact, their might be an obstruction), but I can hardly imagine a LF with homologically trivial fibers. Anyway, I don't know a complete proof, so I can only comment. $\endgroup$
    – Alex Degtyarev
    Commented Nov 26, 2014 at 10:47
  • $\begingroup$ @AlexDegtyarev Now I get it. Then if the fiber is homologically trivial, the manifold cannot be symplectic according to Gompf's theorem. Thanks very much for explaining that to me. $\endgroup$
    – Jie Min
    Commented Nov 26, 2014 at 11:13
  • $\begingroup$ OK, then $\mathbb{C}p^2$ is an example :) $\endgroup$
    – Alex Degtyarev
    Commented Nov 26, 2014 at 11:23

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