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The existence of an $S^1$ action sometimes helps us in computing topological invariants. For example we can compute the Euler characteristic looking at the fixed point set (see Euler characteristic expression in terms the number of fixed points of an $\mathbb{S}^1$ action). If we are lucky enough that we have a $T^n$-action on a symplectic $2n$ manifold and the action is Hamiltonian then we can compute the Betti numbers and Chern-classes from the image of the moment map.

I want to understand the normal bundle of an invariant subset. Therefore I would like to compute its characteristic classes. I wonder if there is some way to exploit the existence of a toric action to compute easily these invariants, more precisely: denote as $T^n$ the n-torus $\mathbb{R}^n/\mathbb{Z}^n$.

Let $\nu_S\to S$ be a invariant regular neighbourhood/normal bundle of an invariant subset $S\subset M$ of a manifold $M$ with an $T^n$ action. Then the action of $T^n$ on $\nu_s$ preserves the fibers and it is liner with some weights $w_1,\dots, w_k\in \mathbb{R}^n$. Can we compute characteristic classes of $\nu_S$ from the weights?

More generally

Does the existence of a $T^n$ action on a vector bundle help us in classifying it/computing characteristic classes?

Overflowian
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1 Answers1

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The answer to your first question is no, you cannot compute characteristic classes from the weights. In what follows I assume you are familiar with a little bit of complex/algebraic geometry.

Let $C$ be a Riemann surface of genus $g$, and $L$ any complex line bundle over $C$. (Topologically there will be one complex line bundle for every element in $H^{2}(C,\mathbb{Z}) \cong \mathbb{Z}$.)

Consider $X = \mathbb{P}_{C}(L \oplus \mathcal{O})$ (i.e. the projectivisation of the rank 2 complex vector bundle $L \oplus \mathcal{O}$ over $C$).

There is a natural Hamiltonian $S^{1}$-action on this space given by $z.[v_{1},v_{2}] = [v_{1},zv_{2}]$ (which is well-defined because the second factor is trivial). Furthermore the subset $S$ where $v_{2}=0$ is isomorphic to $C$, and consists of fixed points for the $S^{1}$-action. Furthermore the action has weight $1$ along $S$ (since the action is semi-free). One can see that the normal bundle of $S$ in $X$ is isomorphic to $L$. Since the degree of $L$ can take any value here, this example shows that degree of the normal bundle definitely cannot be recovered from the weight of the action.

A summary of what is true (at least for a fixed submanifold for Hamiltonian $S^{1}$-actions) is:

  1. The class of the normal bundle plus the weights define a neighbourhood of the fixed submanifold up to equivariant symplectomorphism.

  2. There are certain global localisation theorems, almost all coming from the Atiyah-Bott Localisation theorem, which state that the sum of certain combinations of weights and characteristic classes of the normal bundles of fixed submanifolds, summed over ALL of the fixed point data are equal to certain characteristic classes of the original manifold.

Nick L
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  • In the case of a (compact) symplectic toric 4 manifold we can compute the intersection number of an invariant 2-sphere from the Delzant polytope, and hence the Euler class of the normal bundle. Do you know a simple way to explaining this computation? And maybe generalize it? – Overflowian Apr 04 '19 at 15:56
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    Yes that is right for toric manifolds (i.e. when $\dim(T^{n}) = \frac{1}{2} \dim(M)$). basically if we put the edge in standard form by an $SL_{2}(\mathbb{Z})$ transformation so that the angle at one of the end points in $\pi/2$, then the angle at the other end point will uniquely determine the self intersection of the (invariant $\mathbb{P}^{1}$ corresponding to the) edge. Yes this generalises to higher dimensional toric manifolds but not to general Hamiltonian torus actions as the above answer shows. – Nick L Apr 05 '19 at 01:14
  • Thanks, a reference for this generalization would be nice. – Overflowian Apr 05 '19 at 09:32
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    This will be in Cox's big book on toric varieties, but the idea can be summarised quickly. Each edge $E$ will be contained in (n-1)-facets of dimension $2$ say $S_{1},...,S_{n-1}$. The self intersection $k_{i}$ of (the invariant $\mathbb{P}^{1}$ corresponding to) $E$ in each $S_{i}$ can be calculated by using the method of my previous comment (although you have to find an appropriate $SL_{n}(\mathbb{Z})$ transformation). Then the normal bundle of (the invariant $\mathbb{P}^{1}$ corresponding to) $E$ will be isomorphic to $\mathcal{O}(k_{1}) \oplus \cdots \oplus \mathcal{O}(k_{n-1})$. – Nick L Apr 05 '19 at 15:12