\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

variation of Hodge structure

definition 0.0.1 [Mirror symmetry motivation notes]

Let \(S\) be a complex manifold. An integral variation of Hodge structure of weight \(k\) on \(S\) consists of a torsion-free local system \(\mathbb H_{\ZZ}\) of finitely generated \(\ZZ\)-modules, together with a decreasing filtration by holomorphic subbundles \[ \mathcal H=F^0\mathcal H\supset F^1\mathcal H\supset \cdots \] of \[ \mathcal H:=\mathbb H_{\ZZ}\otimes_{\ZZ}\mathcal O_S. \] The local system induces a flat connection \[ \nabla:\mathcal H\to \mathcal H\otimes \Omega_S^1. \] For every \(s\in S\), the filtration \(F^\bullet_s\) is required to define a pure Hodge structure of weight \(k\) on \(\mathbb H_{\ZZ,s}\), equivalently \[ \mathbb H_{\CC,s}=\bigoplus_{p+q=k}H^{p,q}_s, \qquad (F^p\mathcal H)_s=\bigoplus_{r\geq p}H^{r,k-r}_s. \] The filtration is also required to satisfy Griffiths transversality: \[ \nabla F^p\mathcal H\subset F^{p-1}\mathcal H\otimes \Omega_S^1. \] A smooth proper Kähler family \(\pi:\mathcal X\to S\) gives such a variation on \(R^k\pi_*\ZZ/\mathrm{tors}\). For mirror symmetry of Calabi--Yau threefolds, the central example is usually the variation on \(H^3\).

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