Clément Hyvrier (Uppsala) Weinstein conjecture in Hamiltonian fibrations

\[ \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Symp}{\operatorname{Symp}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\e}{\operatorname{e}} \newcommand{\pt}{\mathrm{pt}} \newcommand{\tensor}{\otimes} \] 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

1. $(P, \pi)$ 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.
2. $(B, \omega_B)$ is symplectically uniruled for some primitive class $\sigma_B \in H_2(B)$ for spherical class $B$.
3. $(F, \omega)$ 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 $.