Complex cobordism, Hamiltonian loops and global Kuranishi charts

Abstract

Let $(X,\omega)$ be a closed symplectic manifold. A loop $\phi: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $\pi: P_{\phi} \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $\pi$, Lalonde, McDuff and Polterovich proved that if $\phi$ lifts to the Hamiltonian group $\mathrm{Ham}(X,\omega)$, then the rational cohomology of $P_{\phi}$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_{\phi}$ splits additively for any complex-oriented cohomology theory $\mathbb{E}$, in particular the integral cohomology splits. This class of examples includes all complex projective varieties equipped with a smooth morphism to $\mathbb{CP}^1$, in which case the analogous rational result was proved by Deligne using Hodge theory. The argument employs virtual fundamental cycles of moduli spaces of sections of $\pi$ in Morava $K$-theory and results from chromatic homotopy theory. Our proof involves a construction of independent interest: we build global Kuranishi charts for moduli spaces of pseudo-holomorphic spheres in $X$ in a class $\beta \in H_2(X;\mathbb{Z})$, depending on a choice of integral symplectic form $\Omega$ on $X$ and ample Hermitian line bundle over the moduli space of one-pointed degree $d = \langle \Omega,\beta\rangle$ stable genus zero curves in $\mathbb{CP}^d$.