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

Symplectic snippets

Liouville domains

definition 0.0.1

A Liouville domain is a pair \((X,\lambda)\), where
  • \(X\) is a \(2n\)-manifold with boundary \(\partial X\) and,
  • \(\lambda\in \Omega^1(X, \RR)\) is a one form on \(\Omega^1(X, \RR)\) so that \(\omega=d\lambda\) is a symplectic form for \(X\).
To this data we can associate a Liouville vector field \(Z\) defined by the property \(\iota_Z\omega= \lambda\). We require that this vector field transversely points outward along \(\partial X\). We call \(X\) a Liouville manifold if \(X\) is non-compact and obtained from a Liouville manifold \(X_0\) by attaching the \snap{symplectization of its contact boundary}{def:symplectization} In some ways, Liouville domains are the ``simplest'' symplectic manifolds to study. While the presence of boundary on \(X\) may seem problematic, this is more than compensated by the fact that \(X\) is exact. This is an honest trade-off: by Stoke's theorem, there are no closed exact symplectic manifolds. By (contact hypersurfaces), \(\alpha:= \lambda|_{\partial X}\) is a contact form on the boundary of a Liouville domain \(X\). A Liouville domain can be extended to a Liouville manifold by attaching the symplectization of \(\partial X\) to the boundary of \(X\). The resulting manifold \[\hat X := X\cup_{\partial X} ([0, \infty)\times \partial X)\] is called the symplectic completion of \(X\).

example 0.0.2

Let \(Q\) be a manifold. The cotangent bundle \(X=T^*Q\) is an exact symplectic manifold; call the primitive \(\lambda_{can}=\sum_{i=1}^np_idq_i\). In local coordinates, the Liouville vector field is \(Z= \sum_{i=1}^n p_i \partial_{p_i}\). The flow of this vector field acts by scalar multiplication on the fibers of \(T^*Q\), \begin{align*}\phi^t: T^*Q\to T^*Q && (q, p) \mapsto (q, e^t p)\end{align*} which inflates the symplectic form. Now equip \(Q\) with a metric \(g\). The metric determines a unit sphere bundle \(S^*Q\subset T^*Q\), which is transverse to \(Z\). Letting \(\alpha=\lambda|_{S^*Q}\) makes \((S^*Q, \alpha)\) a contact manifold. The time \(t\)-flow on \((S^*Q, \alpha)\) corresponds to the time \(t\) geodesic flow on the base ((geodesics and symplectic cohohomology of the cotangent bundle)). It is known ([Fet52]) that every closed manifold has at least 1 closed geodesic, so all of these examples of contact manifolds have a Reeb orbit.

example 0.0.3

A Stein manifold is a complex manifold \(X\) which admits a proper embedding into \(\CC^N\) with \(N\) sufficiently large; therefore \(X\) is a symplectic manifold with symplectic form \(\omega_{\CC^N}|_X\). On \(\CC^n\) we have the function \(\phi=|z|^2:\CC^n\to \RR\) which is exhausting and strictly plurisubharmonic. The symplectic form on \(\CC^n\) can be written as \(dd^c\phi\). By restricting \(\phi\) to \(X\), we obtain a primitive \(d^c\phi|_X\) for \(\omega|_X\), which makes \(X\) a Liouville manifold. In particular, every affine variety is an example of a Liouville manifold.

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References

[Fet52]Abram Il'ich Fet. Variational problems on closed manifolds. Matematicheskii Sbornik, 72(2):271--316, 1952.