Complex cobordism, Hamiltonian loops and global Kuranishi charts

Abstract

Let $(X,ω)$ be a closed symplectic manifold. A loop $φ: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $π: P_φ \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $π$, Lalonde, McDuff and Polterovich proved that if $φ$ lifts to the Hamiltonian group $\mathrm{Ham}(X,ω)$, then the rational cohomology of $P_φ$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_φ$ 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 $π$ 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 $β\in H_2(X;\mathbb{Z})$, depending on a choice of integral symplectic form $Ω$ on $X$ and ample Hermitian line bundle over the moduli space of one-pointed degree $d = \langle Ω,β\rangle$ stable genus zero curves in $\mathbb{CP}^d$.