Quotient singularities, eta invariants, and self-dual metrics

Lock, Michael T.; Viaclovsky, Jeff A.

doi:10.2140/gt.2016.20.1773

Cited by 8 publications

(4 citation statements)

References 43 publications

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“…Let us now give topological conditions which will ensure that the Ricci-flat ALE orbifolds appearing as blow ups in our degenerations are Kähler, and therefore that the obstruction det R = 0 holds. All of these topological conditions come from the topological characterization of [Nak90], see also [LV16] for a generalization. Basically, if a desingularization has the topology of a minimal resolution of a SU(2)-singularity (or a quotient for the U(2) singularities) in a neighborhood of a singularity, then, all of the bubbles Kähler and we can apply Theorem 6.13.…”

Section: Obstructions Under Topological Assumptionsmentioning

confidence: 99%

Noncollapsed degeneration of Einstein 4-manifolds II

Ozuch¹

2019

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In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place.The description of Einstein metrics as the result of a gluing-perturbation procedure also sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we extend the obstruction to the desingularization of Einstein orbifolds found by Biquard, and prove that it holds for any desingularization by trees of quotients of gravitational instantons only assuming a mere Gromov-Hausdorff convergence instead of specific weighted Hölder spaces. This is conjecturally the general case, but it can at least be ensured by topological assumptions such as a spin structure on the degenerating manifolds. We also identify an obstruction to desingularizing spherical and hyperbolic orbifolds by general Ricci-flat ALE spaces.

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Section: Obstructions Under Topological Assumptionsmentioning

confidence: 99%

Noncollapsed degeneration of Einstein 4-manifolds II

Ozuch¹

2019

Preprint

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“…Thus we need to check when the bound 2π 2 (2χ(M) + 3τ (M)) is strictly greater than those in the part (i) of Theorem 1.2, (31), and (32). Since…”

Section: Application and Examplesmentioning

confidence: 99%

“…To compute η(S 3 /Z n ) for such a Z n -action, one can use the formula from [3,32,35] η(S (Recall that eta invariant changes sign when reversing the orientation.) In fact this self-dual orbifold turns out to be the weighted projective plane CP 2 (1,1,n) .…”

Section: Application and Examplesmentioning

confidence: 99%

The Weyl functional on 4-manifolds of positive Yamabe invariant

Sung¹

2021

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It is shown that on every closed oriented Riemannian 4-manifold (M, g) with positive scalar curvature,where W + g , χ(M ) and τ (M ) respectively denote the self-dual Weyl tensor of g, the Euler characteristic and the signature of M . This generalizes Gursky's inequality [15] for the case of b 1 (M ) > 0 in a much simpler way.We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gursky's inequalities for the case of b + 2 (M ) > 0 or δ g W + g = 0, and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.

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“…It is conjectured that any simply-connected Ricci-flat ALE 4-manifold must be hyperkähler, the latter of which has been classified by Kronheimer [13,14]. For progress in this direction, see Lock and Viaclovsky [15].…”

Section: 3mentioning

confidence: 99%

Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

Chow,

Freedman,

Shin

et al. 2019

Preprint

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Quotient singularities, eta invariants, and self-dual metrics

Cited by 8 publications

References 43 publications

Noncollapsed degeneration of Einstein 4-manifolds II

Noncollapsed degeneration of Einstein 4-manifolds II

The Weyl functional on 4-manifolds of positive Yamabe invariant

Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

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