\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
Hodge diamond
definition 0.0.1 [Mirror symmetry motivation notes]
This definition uses the Hodge decomposition.
The Hodge diamond of a compact Kähler manifold \(X\) of complex dimension \(n\) is the array of Hodge numbers \(h^{p,q}(X)=\dim_{\CC}H^{p,q}(X)\) arranged so that entries with fixed \(p+q\) lie on the same row and reflection across the vertical axis interchanges the two indices. Thus the row indexed by \(k\) contains the numbers corresponding to the summands in
\[
H^k(X;\CC)=\bigoplus_{p+q=k}H^{p,q}(X).
\]
For a threefold, the diamond has rows
\[
\begin{array}{ccccccc}
&&& h^{0,0} &&&\\
&& h^{1,0} && h^{0,1} &&\\
& h^{2,0} && h^{1,1} && h^{0,2} &\\
h^{3,0} && h^{2,1} && h^{1,2} && h^{0,3}\\
& h^{3,1} && h^{2,2} && h^{1,3} &\\
&& h^{3,2} && h^{2,3} &&\\
&&& h^{3,3} &&&
\end{array}
\]
The diamond is not extra structure beyond the Hodge decomposition; it is a compact way of displaying the dimensions of its summands.