\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
dual torus fibration
definition 0.0.1 [Mirror symmetry motivation notes]
Let
\[
f:X_0\to B_0
\]
be a smooth torus fibration with fibers \(F_b\simeq V_b/\Lambda_b\), where \(V_b\) is a real vector space and \(\Lambda_b\subset V_b\) is a lattice. The dual torus fibration is
\[
\check f:\check X_0\to B_0,
\qquad
\check F_b:=\check f^{-1}(b)=\operatorname{Hom}(\Lambda_b,U(1))
\simeq H^1(F_b;\RR)/H^1(F_b;\ZZ).
\]
Fiberwise,
\[
H_1(\check F_b;\ZZ)\simeq H^1(F_b;\ZZ),
\qquad
H^1(\check F_b;\ZZ)\simeq H_1(F_b;\ZZ).
\]
If the three-torus fibers are oriented, then Poincar\'e duality, with respect to the chosen orientations, gives the useful identifications
\[
H^1(\check F_b;\CC)\simeq H^2(F_b;\CC),
\qquad
H^2(\check F_b;\CC)\simeq H^1(F_b;\CC).
\]
These fiberwise identifications induce the corresponding dual local systems over \(B_0\).
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