Given a compact Kahler manifold $M$ with an $S^1$-action by Kahler isometries, we know that

- Its fixed loci $F=M^{S^1}$is a smooth Kahler submanifold.
- It splits $F=\sqcup_{\alpha \in A} F_{\alpha}$ into connected components
- The tangent space of M at fixed point splits as a complex $S^1$-representation $$T_x M =\oplus_{m_i} \mathbb{C}_{m_i},$$ where $m_i$ are the weights.

These are all classical results, e.g. claim 1. follows from the work of Atiyah-Bott and F. Kirwan (the moment map of the $S^1$-action is a Morse-Bott function). Claims 2. and 3. are immediate.

The question is:

Does the tangent space decomposition from claim 3. induces a **trivialization**
of the restriction $T_{F_\alpha} M$ of the tangent bundle TM to $F_\alpha$? Or does it at least yield a **split** of $T_{F_\alpha} M$ into bundles that correspond to homogeneous weight-spaces?

Further:
If these are well-known results, can one provide a reference for it?

If not true in general, can some further conditions on $M$ or $F$ make it to be true?