\[
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Symp}{\operatorname{Symp}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\e}{\operatorname{e}}
\newcommand{\pt}{\mathrm{pt}}
\newcommand{\tensor}{\otimes}
\]
Defined Weinstein conjecture. Interested in contact-type hypersurfaces
in $(M,\omega)$.
Examples in which known: Viterbo for $(\R^{2n}, \omega_0)$.
Hofer-Viterbo 1992 for symplectically rationally connected manifolds
under semi-positivity assumption. (Through any 2 points of $M$, there
is a holomorphic sphere.)
The actual definition of symplectically rationally connected:
exists a non-vanishing GW invariant $GW_0(\pt, \pt, A, B, \dots)$.
Liu-Tian result with separating hypersurface and classes $A_0$, $A_1$
with support on either side.
Lu: If $<\pt, \beta_1, \dots, \beta_n>^{M, A}_{0, n+1} \ne 0$ then
result holds. (symplectically uniruled)
proof sketch: This non-vanishing implies $\star = <\pt, PD(\omega), \beta_1, \dots,
\beta_n>^{M, A}_{0, n+2} \ne 0$.
Now the trick is to find $\hat \omega = \omega - d(\beta \lambda)$ where
$\lambda$ is a local primitive of $\omega$ near $\Sigma$ and $\beta$ is a cut-off function.
Then, this has support away from $\Sigma$. Now obtain
\[
\star = < \pt, \gamma_+, \beta> + <\pt, \gamma_-, \beta>
\]
since $\star \ne 0$, one of these terms $\ne 0$. The result now follows
by Liu-Tian.
Corollary: shWC true for $M \times N$ with product form and $M$ symp uniruled.
Question: can we generalize to Hamiltonian fibrations?
\[
(F, \omega) \hookrightarrow P \to (B, \omega_B)
\]
s.t.
- $P|_{B_1}$ is sympl trivial, where $B_1$ is the 1-skeleton
- exists a connection $P$ such that the holonomy is hamiltonian
i.e.~exists $\tau \in \Omega^2(P)$ such that
$\tau|_{F} = \omega$ and $d\tau = 0$ with a normalization
condition.
Then, $(P, \epsilon \tau + \pi^* \omega_B)$ is symplectic for $\epsilon > 0$
small enough.
THEOREM:
Suppose
-
$(P, \pi)$ is cohomologically split over $\Q$,
i.e.~
\\[
H^*(P) \sim H^*(B) \otimes H^*(F)
\\]
as vector spaces.
And $i_* : H^*(P) \to H^*(F)$ is surjective.
- $(B, \omega_B)$ is symplectically uniruled for some
primitive class $\sigma_B \in H_2(B)$ for spherical class $B$.
-
$(F, \omega)$ satisfies semi-positivity relative to $P$.
then, $(P, \omega_P)$ is symplectically uniruled and thus shWC
applies there.
THEOREM:
We can drop condition (1) by asking that $(B, \omega_B)$ is symp
rationally connected.
PROOF:
Case 1: the fibre is symplectically uniruled.
Result done by Ruan, Tian and Liu. \\
Basic idea: \[
0 \ne < \pt, \beta>^{F, \sigma}_{0, n} =
<\pt, i_* \beta>_{0,n}^{P, i_* \sigma}
\] where $\sigma \in H_2(F)$.
Case 2: Suppose $(F, \omega)$ not symp uni. consider:
$C$ the image of a curve counted in
$<\pt, \beta_B>_{0,n}^{\beta, \sigma_B}$
Now restrict $P|_C$. Gives a fibration over $S^2$. this is then
described by a loop in $Ham(F, \omega)$.
Since $(F, \omega)$ NOT symp uni, for any symplectic loop,
exists $\sigma' \in H_2^{sph}(P|_C)$ such that
$<\pt_{0,1}^{P|_C, \sigma'} \ne 0$, where $\sigma'$ is a section class.
(i.e.~projects to the fundamental class of the base)
Then, $=<\pt, [F], \dots, [F]>_{0, n+1}^{P|_C, \sigma'}$.
THEOREM:
\[
< \pt, \pi^{-1} \beta_B >_{0,n}^{P, i_* \sigma'}
=
< \pt, \beta_B>_{0, n}^{B, \sigma_B}
< \pt, [F], \dots, [F]>_{0,n}^{P|_C, \sigma'}
\]
Remarkable fact: the Seidel element, when we are in a NOT uniruled case,
looks like
$ S(\gamma) = [F] \tensor \lambda + x $.
