Symmetric Products of Equivariantly Formal Spaces

Franz, Matthias

doi:10.4153/cmb-2017-032-0

Cited by 12 publications

(14 citation statements)

References 23 publications

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“…In the language of Biswas-D'Mello [7], the means that SP m (X) is an M-variety if and only if X is an M-curve. The if direction was proven in [7] when m = 2, 3 or m ≥ 2g − 1, and extended to all g by Franz [16].…”

Section: We Deduce Surjectionsmentioning

confidence: 93%

Symmetric products of a real curve and the moduli space of Higgs bundles

Baird

2018

Journal of Geometry and Physics

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Consider a Riemann surface X of genus g ≥ 2 equipped with an antiholomorphic involution τ . This induces a natural involution on the moduli space M (r, d) of semistable Higgs bundles of rank r and degree d. If D is a divisor such that τ (D) = D, this restricts to an involution on the moduli space M (r, D) of those Higgs bundles with fixed determinant O(D) and trace-free Higgs field. The fixed point sets of these involutions M (r, d) τ and M (r, D) τ are (A, A, B)-branes introduced by Baraglia-Schaposnik [4].In this paper, we derive formulas for the mod 2 Betti numbers of M (r, d) τ and M (r, D) τ when r = 2 and d is odd. In the course of this calculation, we also compute the mod 2 cohomology ring of SP m (X) τ , the fixed point set of the involution induced by τ on symmetric products of the Riemann surface.

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Section: We Deduce Surjectionsmentioning

confidence: 93%

Symmetric products of a real curve and the moduli space of Higgs bundles

Baird

2018

Journal of Geometry and Physics

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“…). The space S 4 arises from A by attaching a single free cell T 2 × D 2 along an equivariant map T 2 × S 1 → S 4 . In order to extend E from A to S 4 it is sufficient to extend ϕ.…”

Section: The Constructionmentioning

confidence: 99%

GKM manifolds are not rigid

Goertsches¹,

Konstantis²,

Zoller³

2020

Preprint

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We construct effective GKM T 3 -actions with connected stabilizers on the total spaces of the two S 2 -bundles over S 6 such that the GKM graphs are identical. This shows the GKM graph of a simply-connected integer GKM manifold with connected stabilizers does not determine its homotopy type. We complement this by a discussion of the minimality of this example: the homotopy type of integer GKM manifolds with connected stabilizers is indeed encoded in the GKM graph for smaller dimensions, lower complexity, or lower number of fixed points. Regarding geometric structures on the new example, we find an almost complex structure which is invariant under the action of a subtorus. This proves the existence of an almost complex S 1 -manifold with 4 fixed points in dimension 8 apart from S 2 ×S 6 . In addition to the minimal example, we provide an analogous example where the torus actions are Hamiltonian, which disproves symplectic cohomological rigidity for Hamiltonian integer GKM manifolds.

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“…where b is the lowest occurring orbit dimension, see [44] or [35,Section 5]. In particular, the equivariant cohomology algebra, for Cohen-Macaulay actions, is computable as for equivariantly formal actions, by determining the image of the restriction map H * T (M ) → H * T (M b ).…”

Section: Algebraic Generalizations Of Equivariant Formalitymentioning

confidence: 99%