Understand the moment map

Question

I'm studying moment map via the book Geometry of Four-Manifolds, in this book, the authors give several ways to understand (co-)moment map:

For given symplectic group action $G$ (and correspondent Lie algebra $\mathfrak{g}$) and symplectic manifold $M$.

1. Lifting the Lie algebra homomorphism $\mathfrak{g}\to Vect(M)$ to $\mathfrak{g}\to C^{\infty}(M)$.

2. Lifting the group action $G$ on $M$ to $G$ on $L$, here $L$ the complex line bundle with curvature $2\pi i\Omega$, $\Omega$ the symplectic form of $M$.

3. Differential of Kempf-Ness function.

However, there are some obstructions of existence of moment map, for instance $H^1(M)$. But all constructions mentioned in the book do not show how those obstructions arise, I wonder if there is a explicit way to construct moment map such that we can see how the obstructions arise naturally?

If there is no such way to construct moment map, how can we find the examples of moment map? Are there any other motivations?

Thank you for your answer!

Answer 1

I invite you to read the book of Jean-Louis Koszul "Introduction to Symplectic Geometry": https://link.springer.com/book/10.1007/978-981-13-3987-5 Koszul develops the seminal Jean-Marie Souriau model of "Moment map" (Souriau is the inventor of moment map). Koszul and Souriau consider both the case of non-null cohomology, when the coadjoint operator on the moment map is not equivariant. A (Souriau) cocycle should be taken into account.