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

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example 0.0.1

Another interesting piece of geometry comes from the cotangent bundle of the 2-sphere, which is a subvariety of \(\CC^{3}\), \[T^*S^2=\{(z_0, z_1, z_2)\;|\; z_0^2+z_1^2+z_2^2=1\}\] We check that this has the topology of the tangent bundle. Let \(S^2=\{(x_0, x_1, x_2)\;|\; x_0^2+x_1^2+x_2^2=1\}\subset \RR^3\). The tangent bundle is then described by the pairs \[T^*S^2:=\{(x_0, x_1, x_2, y_0, y_1, y_2)\;|\;x_0^2+x_1^2+x_2^2=1, \sum_{i=0}^2 x_iy_i=0 \}.\] These two constraints can be rephrased in terms of the real and imaginary components of \(z_0^2+z_1^2+z_2^2=1\). The complex structure on this cotangent bundle interchanges the \(x_i\) base directions with the \(y_i\) tangent bundle directions. The symplectic Lefschetz fibration that we consider for the cotangent bundle of the sphere sends \begin{align*} \pi: T^*S^2\to \CC\\ (z_0, z_1, z_2)\mapsto z_2 \end{align*} The fibers of this function are the conics \[\pi^{-1}(z)=\{(z_0, z_1, z_2)\;|\; z_2=z, z_0^1+z_1^2=1-z_2^2\}\] which are regular, provided that \(z_2\neq \pm 1\).
figure 0.0.2:Lefschetz fibration for the cotangent bundle of the sphere