\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
Hodge decomposition
definition 0.0.1 [Mirror symmetry motivation notes]
Let \(X\) be a compact Kähler manifold of complex dimension \(n\). The complexified cotangent bundle decomposes as
\[
T^*X\otimes_{\RR}\CC=(T^{1,0}X)^*\oplus (T^{0,1}X)^*,
\]
and hence the complex-valued differential forms decompose as
\[
\Omega^k(X;\CC)=\bigoplus_{p+q=k}\Omega^{p,q}(X).
\]
The Hodge decomposition is the canonical decomposition of complexified de Rham cohomology
\[
H^k(X;\CC)=\bigoplus_{p+q=k}H^{p,q}(X),
\]
where \(H^{p,q}(X)\) is naturally identified with Dolbeault cohomology, and hence with sheaf cohomology:
\[
H^{p,q}(X)\simeq H^q(X,\Omega_X^p)
\]
For any choice of Kähler metric, \(H^{p,q}(X)\) is represented by harmonic \((p,q)\)-forms. The Hodge numbers are
\[
h^{p,q}(X):=\dim_{\CC}H^{p,q}(X).
\]
They refine the Betti numbers by the identity
\[
b_k(X)=\sum_{p+q=k}h^{p,q}(X).
\]
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