\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
special Lagrangian torus fibration
definition 0.0.1 [Mirror symmetry motivation notes]
Let \((X,\omega,\Omega)\) be a Calabi--Yau \(n\)-fold with its Ricci-flat Kähler metric, and assume that \(\Omega\) is normalized with respect to this metric. For \(\theta\in \RR/2\pi\ZZ\), an oriented real \(n\)-dimensional submanifold \(L\subset X\) is special Lagrangian of phase \(\theta\) if
\[
\omega|_L=0,\qquad \operatorname{Im}(e^{-i\theta}\Omega)|_L=0,
\]
and
\[
\operatorname{Re}(e^{-i\theta}\Omega)|_L=\operatorname{vol}_L
\]
as positive volume forms on \(L\).
A special Lagrangian torus fibration of phase \(\theta\) is a continuous map
\[
f:X\to B
\]
to a real \(n\)-dimensional base such that, over a dense open subset \(B_0\subset B\), the restriction
\[
f^{-1}(B_0)\to B_0
\]
is a smooth fiber bundle whose fibers are \(n\)-tori that are special Lagrangian submanifolds of phase \(\theta\).
In the SYZ setting, nontrivial compact examples are expected to have singular fibers over a discriminant locus \(\Delta=B\setminus B_0\).
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