ScalingStacks

Proposition 4.17 . [04JI]

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Proposition 4.17.

Let (B,ฮ”,๐’œ)(B,\Delta,\mathscr{A}) be a given simple affine 3-manifold with singularities. Then we have the following

  • (i)

    if JโІฮ”gJ\subseteq\Delta_{g} is an edge of ฮ”\Delta, then there is a generic-singular fibration โ„ฑ\mathcal{F}, with affine base (Bโ€ฒ,ฮ”โ€ฒ,๐’œโ€ฒ)(B^{\prime},\Delta^{\prime},\mathscr{A}^{\prime}) and neighborhood UโІBU\subseteq B of JJ such that there exists an integral affine isomorphism (Bโ€ฒ,ฮ”โ€ฒ,๐’œโ€ฒ)โ‰…(U,J,๐’œ)(B^{\prime},\Delta^{\prime},\mathscr{A}^{\prime})\cong(U,J,\mathscr{A}) inducing a symplectic conjugation Xโก(B0โ€ฒ,๐’œโ€ฒ)โ‰…Xโก(Uโˆ’J,๐’œ)X(B^{\prime}_{0},\mathscr{A}^{\prime})\cong X(U-J,\mathscr{A});

  • (ii)

    if pโˆˆฮ”dp\in\Delta_{d} is a positive vertex of ฮ”\Delta, then there is positive fibration โ„ฑ\mathcal{F} with base (Bโ€ฒ,ฮ”โ€ฒ,๐’œโ€ฒ)(B^{\prime},\Delta^{\prime},\mathscr{A}^{\prime}) and a neighborhood UโІBU\subseteq B of pp such that there exists an integral affine isomorphism (Bโ€ฒ,ฮ”โ€ฒ,๐’œโ€ฒ)โ‰…(U,Uโˆฉฮ”,๐’œ)(B^{\prime},\Delta^{\prime},\mathscr{A}^{\prime})\cong(U,U\cap\Delta,\mathscr{A}) inducing a symplectic conjugation Xโก(B0โ€ฒ,๐’œโ€ฒ)โ‰…Xโก(Uโˆ’(Uโˆฉฮ”),๐’œ)X(B^{\prime}_{0},\mathscr{A}^{\prime})\cong X(U-(U\cap\Delta),\mathscr{A}).

Moreover, using the symplectic conjugations in (i) and (ii), we can symplectically glue the germ of โ„ฑ\mathcal{F} into Xโก(B0,๐’œ)X(B_{0},\mathscr{A}).

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