Fixed locus of a Kahler $S^1$-action

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

1. Its fixed loci $$F=M^{S^1}$$is a smooth Kahler submanifold.
2. It splits $$F=\sqcup_{\alpha \in A} F_{\alpha}$$ into connected components
3. 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?

It doesn't give a trivialization (think of the trivial action...), but indeed a splitting. For $$y\in F_{\alpha }$$, $$S^1$$ acts on $$T_y(M)$$; denote by $$t_y$$ the action of an element $$t\in S^{1}$$. The coefficients of the characteristic polynomial $$\det(X\cdot \mathrm{I}- t_{y})$$ are holomorphic functions on $$F_{\alpha }$$, hence constant; therefore the eigenvalues of $$t_y$$ and their multiplicities are independent of $$y$$. It follows that the eigenspaces $$E^\chi_y$$ of the action of $$S^1$$ associated to a character $$\chi$$ of $$S^1$$ form a subbundle $$E^{\chi }$$ of $$T(M)_{|F_{\alpha }}$$, and that $$T(M)_{|F_{\alpha }}$$ is the direct sum of these subbundles.