Complex cobordism, Hamiltonian loops and global Kuranishi charts

Abstract

Let (X, ω) be a closed symplectic manifold. A loop φ : S 1 → Diff(X) of diffeomorphisms of X defines a fibration π : P φ → 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 Ham(X, ω), then the rational cohomology of P φ splits additively. We prove, with the same assumptions, that the E-generalised cohomology of P φ splits additively for any complex-oriented cohomology theory E, in particular the integral cohomology splits. This class of examples includes all complex projective varieties equipped with a smooth morphism to 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 ∈ H2(X; 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 = Ω, β stable genus zero curves in CP d .

An integral Euler cycle in normally complex orbifolds and Z-valued Gromov-Witten type invariants

“…A topological/smooth global Kuranishi chart is a quadruple (G, D, E, s) where G is a compact Lie group, D is a topological/smooth manifold endowed with an almost free continuous/smooth G-action, E is a Gequivariant vector bundle over D, and s is a G-equivariant continuous/smooth section of E → D. The equivalence relations introduced in the definition of Ω C,der * have counterparts in this context, but there is an additional equivalence relation between global Kuranishi charts, namely identifying (G, D, E, s) with (G ′ × G, P, q * E, q * s) where q : P → D is a G-equivariant principal G ′ -bundle. This change-of-group operation should be interpreted as choosing different global quotient presentation of the orbifold [D/G], so the arguments in [AMS21] indeed prove the statements in the form of Proposition 1.5.…”

“…Motivated by seeking for Z-valued Gromov-Witten type invariants, this paper explains how to construct integral "Euler classes" for an orbifold complex vector bundle over an orbifold endowed with a "normal complex structure" following a proposal of Fukaya-Ono [FO97] and the subsequent development by Parker [Par13]. Combined with recent advances in algebraic topology [Par19,Par20] and regularization of moduli spaces of J-holomorphic maps [AMS21], we can study generalized bordism theories over orbispaces and define Z-valued Gromov-Witten type invariants for general symplectic manifolds.…”

“…Instead, in this paper we appeal to a recent work of Abouzaid-McLean-Smith [AMS21] which ingeniously constructed a global Kuranishi chart with a smoothing on the genus zero Gromov-Witten type moduli space using a geometric perturbation. Using the language of derived orbifold charts, one main result from [AMS21] can be formulated as follows.…”

“…Remark 1.6. The paper [AMS21] uses the notion of global Kuranishi charts instead of derived orbifold charts. A topological/smooth global Kuranishi chart is a quadruple (G, D, E, s) where G is a compact Lie group, D is a topological/smooth manifold endowed with an almost free continuous/smooth G-action, E is a Gequivariant vector bundle over D, and s is a G-equivariant continuous/smooth section of E → D. The equivalence relations introduced in the definition of Ω C,der * have counterparts in this context, but there is an additional equivalence relation between global Kuranishi charts, namely identifying (G, D, E, s) with (G ′ × G, P, q * E, q * s) where q : P → D is a G-equivariant principal G ′ -bundle.…”

“…In the forthcoming appendix written by Z. Zhou, it is shown that the statements from Proposition 1.5 hold for moduli spaces of higher genus stable pseudo-holomorphic maps as well. Unlike the construction in [AMS21] which depends on smoothing theory, the results in the appendix are proved using polyfolds.…”