ScalingStacks

Proof. [04J7]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Proof.

Consider the coordinates (b1,b2,b3)(b_{1},b_{2},b_{3}) on B=D2×D1B=D^{2}\times D^{1} and the period lattice as in Proposition 4.8. With respect to these coordinates Δ={b1=b2=0}\Delta=\{b_{1}=b_{2}=0\}. Define open subsets of B0=B−ΔB_{0}=B-\Delta:

V1\displaystyle V_{1} =\displaystyle= B−{(b1,0,b3)|b1>0},\displaystyle B-\{(b_{1},0,b_{3})\ |\ b_{1}>0\},
V2\displaystyle V_{2} =\displaystyle= B−{(b1,0,b3)|b1<0}.\displaystyle B-\{(b_{1},0,b_{3})\ |\ b_{1}<0\}.

On VjV_{j} the action coordinates have the form

Aj​(b1,b2,b3)=(ψj​(b1,b2)+H⁡(b1,b2,b3),2​π​b2,b3),A_{j}(b_{1},b_{2},b_{3})=(\psi_{j}(b_{1},b_{2})+H(b_{1},b_{2},b_{3}),2\pi b_{2},b_{3}),

where ψj\psi_{j} is a choice of primitive of λ0\lambda_{0}. Then 𝒜={Uj,Aj}\mathscr{A}=\{U_{j},A_{j}\} gives the integral affine structure on B0B_{0}. As in the focus-focus case, for either j=1,2j=1,2, the map AjA_{j} extends to a homeomorphism, A:B→A⁡(B)⊆ℝ2×ℝA:B\rightarrow A(B)\subseteq\mathbb{R}^{2}\times\mathbb{R} such that A⁡(0)=0A(0)=0. It is easy to show that, if τ⁡(t)=H⁡(0,0,t)\tau(t)=H(0,0,t), then AA is an isomorphism between (B,Δ,𝒜)(B,\Delta,\mathscr{A}) and a neighborhood of Δτ\Delta_{\tau} in the affine manifold with singularities of Example 3.9. ∎

Original mathematics by the credited authors. Source-backed reader collection; mathematical self-containment is not assessed.