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 into weight spaces 
$$T\_x M =\oplus\_{k\in\mathbb{Z}} H\_{k}.$$


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 **split** of $T\_{F\_\alpha} M$ into bundles that correspond to weight-spaces?



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


