ScalingStacks

Corollary 3.35 . [0449]

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

Corollary 3.35.

(Special Lagrangian fibration) There exist moment coordinates μ~1,μ~2\tilde{\mu}_{1},\tilde{\mu}_{2} for the T2T^{2} action on ω+\omega^{+}. The special Lagrangian fibration

(3.14) Mν+→(μ~1,μ~2,Im​η)ℝ3M^{+}_{\nu}\xrightarrow{(\tilde{\mu}_{1},\tilde{\mu}_{2},\text{Im}\eta)}\mathbb{R}^{3}

is proper over {|(μ~1,μ~2,Imη)|a′≤12A1/2eν}⊂ℝ3\{|(\tilde{\mu}_{1},\tilde{\mu}_{2},\text{Im}\eta)|_{a}^{\prime}\leq\frac{1}{2}A^{1/2}e^{\nu}\}\subset\mathbb{R}^{3} where the generic fibre is topologically T3T^{3}. The critical point set is ⋃i,j∈{0,1,2}{Z~i=Z~j=0}\bigcup_{i,j\in\{0,1,2\}}\{\tilde{Z}_{i}=\tilde{Z}_{j}=0\} and the discriminant locus is contained in 𝔇\mathfrak{D}. The monodromy of the fibration and the topology of the central singular fibre agrees with the Gross-Ruan prediction in Section 1.1.3.

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