Abel-Jacobi Map and Symplectic Topology

Abstract

We develop a harmonic-coordinate approach to the identity component $G_{\omega }({\Sigma }_g)$ of the symplectomorphism group of a closed oriented surface of genus $g\ge 2$. Using an intrinsic decomposition of the Abel-Jacobi displacement into a global flux part and a zero-average harmonic fluctuation, we introduce the harmonic flux norm and prove its non-degeneracy on the full identity component without Floer theory. We also show the norm is continuous in the $C^0$-topology, deduce that $\mathrm{Ham}({\Sigma }_g,\omega )$ is $C^0$-closed inside $G_{\omega }({\Sigma }_g)$, and produce a locally injective harmonic-coordinate chart near the identity. Along the way we derive first-order expansions for the norm, quantitative fixed-point obstructions, and propose a finite-dimensional persistence invariant (the harmonic barcode) associated to the harmonic displacement filtration.