par Fine, Joel
Référence Journal of symplectic geometry, 12, 1, page (105-123)
Publication Publié, 2014
Article révisé par les pairs
Résumé : Let L → M be a Hermitian line bundle over a compact manifold. Write S for the space of all unitary connections in L whose curvatures define symplectic forms on M and G for the identity component of the group of unitary bundle isometries of L, which acts on S by pullback. The main observation of this note is that S carries a G-invariant symplectic structure, there is a moment map for the G-action and that this embeds the components of S as G-coadjoint orbits. Restricting to the subgroup of G which covers the identity on M, we see that prescribing the volume form of a symplectic structure can be seen as finding a zero of a moment map. When M is a Kähler manifold, this gives a moment-map interpretation of the Calabi conjecture. We also describe some directions for future research based upon the picture outlined here.