\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
Picard--Fuchs equation
definition 0.0.1 [Mirror symmetry motivation notes]
Let
\[
\pi:\mathcal X\to S
\]
be a smooth proper family of Calabi--Yau \(n\)-folds, and, locally on \(S\), let \(\Omega\) be a holomorphic section of the relative canonical bundle. If \(\gamma_s\in H_n(\mathcal X_s,\ZZ)\) is a locally flat family of middle-dimensional cycles, the period integral
\[
\Pi_\gamma(s):=\int_{\gamma_s}\Omega_s
\]
varies with \(s\). A Picard--Fuchs equation is a linear differential equation, or more generally a system of linear differential operators in the parameters on \(S\), annihilating such period functions. These equations arise from the Gauss--Manin connection on relative de Rham cohomology and the finite rank of the corresponding cohomology bundle.
In mirror-symmetry computations, Picard--Fuchs equations are often used to describe the \(B\)-model variation of Hodge structure, which is then compared with \(A\)-model enumerative data via the mirror map.
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