1
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.