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Given a symplectic 6-manifold $(M,\omega)$ and an embedded symplectic 2-sphere $C\subset M$ whose normal bundle has the first Chern class -2. How to find on $M$ another closed 2-form $\eta$ which only degenerate along the 2-sphere $C$?

Here, the restriction on a neighborhood of $C$ is isomorphic to the pull-back the canonical symplectic form on $\mathbb{C}^{4}$ by the following map:$$L\oplus L\to \{w^{2}+x^{2}+y^{2}+z^{2}=0\}\subset\mathbb{C}^{4}$$ $$w=a+b',x=i(a-b'),y=a'-b,z=i(a'+b)$$

where $L$ is the complex line bundle over $\mathbb{C}P^{1}$ with the first Chern class -1, $a,b$ are the linear functionals on the first $L$, $a',b'$ are the linear functionals on the second $L$, and they satisfy $ab'-a'b=0$(see C. H. Clemens' paper "Double Solids"(Adv. in Math. 47(1983), 107-230))

In I. Smith, R.P. Thomas, S.-T. Yau's paper"Symplectic Conifold Transitions" (J. Differential Geometry 62(2002), 209-242) , it says that "finding families of symplectic forms which degenerate only along such spheres, yielding conifolds, is more subtle". According to their paper, finding such a 2-form $\eta$ is a necessary and sufficient condition which can make the corresponding symplectic surgery (maybe called "inverse symplectic conifold transition") realizable. I wondered if there is any other paper discussion about this topic until now.

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  • $\begingroup$ Actually, you need to specify something about how you want the the degeneration to look near $C$. You can always (easily) find such a closed $\eta$ that is nondegenerate off $C$, drops to rank $2$ along $C$ and is non-vanishing when pulled back to $C$. I suspect that you, instead, want a form that is nondegenerate off $C$, drops to rank $4$ along $C$, and vanishes when pulled back to $C$. Is that, in fact, what you want? $\endgroup$ Sep 19, 2014 at 13:41
  • $\begingroup$ @Robert Bryant The form I wanted near $C$ is just $da\wedge db+da'\wedge db'$. I think this is consistent with the latter one you mentioned. $\endgroup$
    – Jiang
    Sep 19, 2014 at 14:38
  • $\begingroup$ @RobertBryant The form I wanted near $C$ must be $da\wedge d\overline{a}+db\wedge d\overline{b}+da'\wedge d\overline{a'}+db'\wedge d\overline{b'}$. I'm sorry for such a mistake. $\endgroup$
    – Jiang
    Sep 19, 2014 at 19:29

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