\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
genus-zero Gromov--Witten invariant
definition 0.0.1 [Mirror symmetry motivation notes]
Let \(X\) be a smooth projective complex variety, or more generally a closed symplectic manifold equipped with an oriented virtual fundamental class theory. Let \(\beta\in H_2(X;\ZZ)\) and let \(\gamma_1,\ldots,\gamma_k\in H^*(X;\QQ)\). Assume that the stable-map moduli space \(\overline{\mathcal M}_{0,k}(X,\beta)\) is defined, for instance \(k\geq 3\) when \(\beta=0\). The primary genus-zero Gromov--Witten invariant is
\[
\langle \gamma_1,\ldots,\gamma_k\rangle_{0,\beta}
:=
\int_{[\overline{\mathcal M}_{0,k}(X,\beta)]^{\operatorname{vir}}}
\operatorname{ev}_1^*(\gamma_1)\cdots \operatorname{ev}_k^*(\gamma_k).
\]
Here \(\overline{\mathcal M}_{0,k}(X,\beta)\) is the moduli space of stable maps from \(k\)-marked genus-zero curves to \(X\) in class \(\beta\), and \(\operatorname{ev}_i\) evaluates a stable map at its \(i\)th marked point. The integral is nonzero only when the product of insertions has top degree on the virtual fundamental class.
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