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.

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