Exercises for Symplectic Geometry I
LMU M¨unchen, Summer term 09 Frederik Witt / Vincent Humili`ere Problem set 6, due for 18/06/09
Aspects of symplectic manifolds
1. Symplectic diffeomorphisms of the plane
We consider the 2-plane R2, endowed with the standard symplectic form ω0. Let U and V be two diffeomorphic (by an orientation preserving diffeomorphism φ : U → V) bounded open subsets of R2, having same area. Show that there is a symplectic diffeomorphism fromU ontoV. Can one extend this result to higher dimension ?
Hint: setω1=φ∗ω0and consider the restriction toUofωt= (1−t)ω0+tω1.
2. Exact symplectic manifolds
A symplectic manifold is said exact if the symplectic form is exact.
(a) Give examples of exact symplectic manifolds
(b) Show that a symplectic manifold (M, ω) is exact if and only if there exists an isotopy of diffeomor- phisms (φt) :I×M →M defined on some small interval I = [0, ε) such thatφ∗tω =e−tω, for all t∈I.
(c) Show that an exact symplectic manifold is never compact
We consider a compact orientable surface, with non empty boundary, and denote by S the complement of its boundary. Letω be a volume/symplectic form onS. We are going to show that ω is always exact.
(d) Show that there exists a family of open sets Ut⊂B, I= [0, ε) and a smooth family of diffeomor- phismsψt : S → Ut such that volω(Ut) is a decreasing function of t. Show that one can suppose that this function ise−t.
(e) Apply Moser’s trick to a suitable path of symplectic forms and prove that (S, ω) is exact.
3. Blowing up a symplectic manifold
In this exercise we describe a way to construct new symplectic manifolds from old ones.
(a) We consider
C˜n={(z1, . . . , zn; [w1:. . .:wn])∈Cn×CPn−1|wjzk =wkzj,∀j, k},
and denote Φ : ˜Cn→Cn, pr : ˜Cn→CPn−1the obvious projections. What is the preimage by pr of an elementw∈CPn−1? Show that ˜Cn is the total space of a complex line bundle with projection pr.
Note that Φ is bijective when restricted to Φ−1(Cn− {0}) and thatL0= Φ−1(0)'CPn−1, so that one can see ˜Cn as Cn where 0 has been replaced withCPn−1. The space ˜Cn is called thecomplex blow up of Cn at 0.
(b) We set B(δ) ={z∈Cn| |z| ≤δ}, L(δ) = Φ−1(δ) and consider the 2-form on ˜Cn ρλ= Φ∗ω0+λ2pr∗ωF S,
whereλis a positive real parameter,ω0 the standard symplectic form on Cn andωF S the Fubini- Study form onCPn−1. Show thatρλ is a symplectic form on ˜Cn.
The symplectic manifold ( ˜Cn, λ) is called thesymplectic blow up ofCn at 0 with weightλ.
(c) Show that for every λ, δ > 0, the space (L(δ)−L0, ρλ) is symplectomorphic to (B(√
λ2+δ2− B(λ, ω0)))
Hint: the mapF(z) =p
|z|2+λ2|z|z might be useful...
(d) Use the previous question and Darboux theorem to define the symplectic blow up (with weightλ) of a general symplectic manifold at an arbitrary point.
