\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
the mirror flip
This is the elementary origin of the name ``mirror symmetry.'' If \(X\) and \(\check X\) are mirror Calabi--Yau \(n\)-folds, then the expected numerical relation is
\[
h^{p,q}(X)=h^{n-p,q}(\check X).
\]
For Calabi--Yau threefolds satisfying the simplifying vanishing used above, this says in particular that variations of Kähler or symplectic structure on \(X\) should correspond to variations of complex structure on \(\check X\), and conversely. Since the dimensions of these two deformation spaces are governed by Hodge numbers, one expects
\[
h^{1,1}(X)=h^{2,1}(\check X),\qquad h^{2,1}(X)=h^{1,1}(\check X).
\]
Pictorially, for threefolds this mirror relation exchanges the vertical \(h^{1,1}\) entries with the horizontal \(h^{2,1}\) entries:
Many mirror pairs of Calabi--Yau manifolds were discovered by searching for this kind of numerical symmetry, and those examples provided early evidence for mirror symmetry. The equality of Hodge numbers is only a necessary compatibility check; the stronger mirror statement also identifies deformation theory, enumerative invariants, and eventually categories.