arXiv subjects

Tara S. Holm

Publications and source records attributed to Tara S. Holm.

22 records · Page 2Linked to original sources

arXiv2003

Distinguishing the Chambers of the Moment Polytope

Let M be a compact manifold with a Hamiltonian T action and moment map Phi. The restriction map in equivariant cohomology from M to a level set Phi^{-1}(p) is a surjection, and we denote the kernel by I_p. When T has isolated fixed points, we show that I_p distinguishes the chambers of the moment polytope for M. In particular, counting the number of distinct ideals I_p as p varies over different chambers is equivalent to counting the number of chambers.

math.SG↗

arXiv2003

The mod 2 cohomology of fixed point sets of anti-symplectic involutions

Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $G=T^n$. Suppose that $σ$ is an anti-symplectic involution compatible with the $G$-action. The real locus of $M$ is $X$, the fixed point set of $σ$. Duistermaat uses Morse theory to give a description of the ordinary cohomology of $X$ in terms of the cohomology of $M$. There is a residual $\G=(\Zt)^n$ action on $X$, and we can use Duistermaat's result, as well as some general facts about equivariant cohomology, to prove an equivariant analogue to Duistermaat's theorem. In some cases, we can also extend theorems of Goresky-Kottwitz-MacPherson and Goldin-Holm to the real locus.

math.SG↗

arXiv2003

GKM theory for torus actions with non-isolated fixed points

Let $M^{2d}$ be a compact symplectic manifold and $T$ a compact $n$-dimensional torus. A Hamiltonian action, $τ$, of $T$ on $M$ is a GKM action if, for every $p \in M^T$, the isotropy representation of $T$ on $T_pM$ has pair-wise linearly independent weights. For such an action the projection of the set of zero and one-dimensional orbits onto $M/T$ is a regular $d$-valent graph; and Goresky, Kottwitz and MacPherson have proved that the equivariant cohomology of $M$ can be computed from the combinatorics of this graph. (See \cite{GKM:eqcohom}.) In this paper we define a ``GKM action with non-isolated fixed points'' to be an action, $τ$, of $T$ on $M$ with the property that for every connected component, $F$ of $M^T$ and $ p \in F$ the isotropy representation of $T$ on the normal space to $F$ at $p$ has pair-wise linearly independent weights. For such an action, we show that all components of $M^T$ are diffeomorphic and prove an analogue of the theorem above.

math.SG↗

arXiv2001

The equivariant cohomology of Hamiltonian $G$-spaces From Residual $S^1$ Actions

We show that for a Hamiltonian action of a compact torus $G$ on a compact, connected symplectic manifold $M$, the $G$-equivariant cohomology is determined by the residual $S^1$ action on the submanifolds of $M$ fixed by codimension-1 tori. This theorem allows us to compute the equivariant cohomology of certain manifolds, which have pieces that are four-dimensional or smaller. We give several examples of the computations that this allows.

math.SG↗