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

Symplectic snippets

large complex-structure limit

definition 0.0.1 [Mirror symmetry motivation notes]

Let \[ \pi:\mathcal X^*\to \Delta^* \] be a one-parameter family of Calabi--Yau \(n\)-folds over a punctured disc. Parallel transport around the puncture gives a monodromy operator \[ T:H^n(X_t;\QQ)\to H^n(X_t;\QQ). \] After replacing the punctured disc by a finite cover if necessary, assume that the monodromy is unipotent, and write \(T\) for this unipotent monodromy. Then one can define \[ N:=\log T=(T-I)-\frac{(T-I)^2}{2}+\frac{(T-I)^3}{3}-\cdots . \] The series is finite because \(T-I\) is nilpotent. In this introductory convention, a boundary point \(t=0\) is called a large complex-structure limit, or a maximally unipotent monodromy point, if \[ N^n\neq 0 \] on \(H^n(X_t;\QQ)\). Equivalently, \(N\) has nilpotency index \(n+1\), the maximum possible for a weight-\(n\) variation of Hodge structure. More refined definitions also impose conditions on the limiting mixed Hodge structure; the maximally unipotent monodromy criterion is the part used in this introductory discussion.

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