Normally Complex Polynomial Perturbations and Arnold Conjecture

Moduli spaces of J-holomorphic curves usually carry a "normal" complex structure coming from the nature of Cauchy-Riemann-type equations. In recent work joint with Guangbo Xu, we take advantage of such a structure to construct abstract perturbations of J-holomorphic curve equations to define integral Gromov-Witten type invariants and integral Hamiltonian Floer homology for general symplectic manifolds. The latter leads to a solution of the Arnold conjecture over the integers, realizing a conjectural picture of Fukaya-Ono proposed in the 90s. I will survey this circle of ideas and perhaps tell you some possibilities going beyond integer-valued invariants.

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