For every compact almost complex manifold (M, J) equipped with a J-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that M is symplectic and the action is Hamiltonian. If the manifold satisfies an extra "positivity condition" this algorithm determines a family of vector spaces which contain the admissible lattices of weights.When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever dim(M) ≤ 6, and, when dim(M) = 8, whenever the S 1 -action extends to an effective Hamiltonian T 2 -action, or none of the isotropy weights is 1. Moreover there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces [Y]. We run the algorithm for dim(M) ≤ 8, quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for dim(M) = 6 [Ah, T1] and, when dim(M) = 8, we prove that the equivariant cohomology ring, Chern classes and isotropy weights agree with the ones of CP 4 with the standard S 1 -action (thus proving the symplectic Petrie conjecture [T1] in this setting).
We generalize the well-known "12" and "24" Theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a wider category of objects, here called reflexive GKM graphs, that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action.As an application, we provide bounds on the Betti numbers for certain monotone Hamiltonian spaces which depend on the minimal Chern number of the manifold.
We use a version of localization in equivariant cohomology for the norm-square of the moment map, described by Paradan, to give several weighted decompositions for simple polytopes. As an application, we study Euler-Maclaurin formulas.
DEPARTAMENTODE MATEMÁTICA, INSTITUTO SUPERIOR TÉCNICO, AV. ROVIS-CO PAIS, 1049-001 LISBON, PORTUGAL, FAX: (351) 21 841 7035
Abstract. Given an n-tuple of positive real numbers α we consider the hyperpolygon space X(α), the hyperkähler quotient analogue to the Kähler moduli space of polygons in R 3 . We prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable, rank-2, holomorphically trivial parabolic Higgs bundles over CP 1 with fixed determinant and trace-free Higgs field. This isomorphism allows us to prove that hyperpolygon spaces X(α) undergo an elementary transformation in the sense of Mukai as α crosses a wall in the space of its admissible values. We describe the changes in the core of X(α) as a result of this transformation as well as the changes in the nilpotent cone of the corresponding moduli spaces of parabolic Higgs bundles. Moreover, we study the intersection rings of the core components of X(α). In particular, we find generators of these rings, prove a recursion relation in n for their intersection numbers and use it to obtain explicit formulas for the computation of these numbers. Using our isomorphism, we obtain similar formulas for each connected component of the nilpotent cone of the corresponding moduli spaces of parabolic Higgs bundles thus determining their intersection rings. As a final application of our isomorphism we describe the cohomology ring structure of these moduli spaces of parabolic Higgs bundles and of the components of their nilpotent cone.
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