\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
Calabi--Yau manifold
definition 0.0.1 [Mirror symmetry motivation notes]
In these notes, a Calabi--Yau \(n\)-fold is a compact connected Kähler manifold \(X\) of complex dimension \(n\) whose canonical bundle
\[
K_X:=\Lambda^n(T^{1,0}X)^*
\]
is holomorphically trivial. Equivalently, \(X\) admits a nowhere-vanishing holomorphic volume form
\[
\Omega\in H^0(X,K_X).
\]
Thus a Calabi--Yau manifold carries compatible complex and symplectic data:
- an integrable complex structure \(J:TX\to TX\);
- a Kähler form \(\omega\in \Omega^2(X)\), which in particular makes \(X\) a symplectic manifold;
- a holomorphic volume form \(\Omega\) trivializing \(K_X\).
This convention does not impose the stronger condition \(h^{p,0}(X)=0\) for \(0<p<n\), nor does it require \(X\) to be simply connected. When those extra hypotheses are needed, they will be stated explicitly.