Suppose I have a torus fibration over the disc with $n$ nodal singular fibers $F_1,\dots,F_n$ over the points $p_1,\dots,p_n$. I was specifically thinking about a Lagrangian fibration, but I'd be happy to know about Lefschetz fibrations, too. I was wondering what happens when you collide all the points together. What fiber lies over the collision point? It is possible to "fill in" this fiber? Symplectically, or holomorphically?

I know that if the vanishing cycles are all equal, then the fiber can be a necklace of $n$ two-spheres. Furthermore, it seems that if the products of the monodromies produce the monodromy of a Kodaira fiber, then one can just put in that fiber. But I am most interested in the case where the vanishing cycles are arbitrary. If the fibers can always be combined in some way, when is the total space smooth?

Sorry if this question isn't too well-defined. Any references would be most welcome.

  • 3
    $\begingroup$ Kodaira gives a classification of singular elliptic fibers, so your guess is, in fact, if and only if: all fibers within the disk can be combined into one (so that the result is still smooth analytic) if and only if the monodromy along the boundary is one of the known ones. $\endgroup$ – Alex Degtyarev Nov 9 '14 at 17:13

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