Spin(7)-structures on complex linear bundles and explicit Riemannian metrics with holonomy group SU(4)

Bazaikin, Y.; Malkovich, Evgeny

doi:10.1070/sm2011v202n04abeh004152

Cited by 5 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The resulting 1-parameter family of AC Calabi-Yau metrics on T * P 2 is originally due to Calabi [12]; these metrics are in fact hyper-Kähler. Conjecture 4.8 was motivated by the fact that such contractions were recently constructed in this case [5,49].…”

Section: 2mentioning

confidence: 99%

Asymptotically conical Calabi–Yau manifolds, I

Conlon¹,

Hein²

2013

Duke Math. J.

162

View full text Add to dashboard Cite

Abstract. This is the first part in a series of articles on complete Calabi-Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition O(r −n−ε ) needed in earlier work to O(r −ε ), relying on some new ideas about harmonic functions. We then look at a few examples: (1) Crepant resolutions of cones. This includes a new class of Ricci-flat small resolutions associated with flag manifolds. (2) Affine deformations of cones. One focus here is the question of the precise rate of decay of the metric to its tangent cone. We prove that the optimal rate for the Stenzel metric on T * S n is −2 n n−1 .

show abstract

Section: 2mentioning

confidence: 99%

Asymptotically conical Calabi–Yau manifolds, I

Conlon¹,

Hein²

2013

Duke Math. J.

162

View full text Add to dashboard Cite

show abstract

“…Our definition of ω is independent of the choice of (e x , e y ) and ω is thus globally defined. The Spin 0 (3, 4)-case is completely analogous, since Spin 0 (3, 4)/U (1,2) is the Grassmannian of all positive oriented planes in R 4,4 .…”

Section: Discussionmentioning

confidence: 99%

“…Conversely, we assume that there exists a line bundle isomorphism η : M 8 → 3,0 T * N 6 and that N 6 carries a U (3)-or U (1, 2)-structure (ω, g, J). We choose local trivializations ϕ α : U α × R 2 → π −1 (U α ) ⊆ M 8 such that the transition functions have values in SO (2). Let x and y be the standard coordinates of R 2 .…”

Section: Discussionmentioning

confidence: 99%

“…In the sixth section, we finally discuss if our result can be generalized to non-trivial bundles over 6-dimensional manifolds. (1,2). In order to prove our theorem we have to introduce several G-structures.…”

Section: Introductionmentioning

confidence: 99%

“…Many of the known complete metrics with holonomy Spin (7) are not defined on a manifold of type M × (−ǫ, ǫ) but on a disc bundle over a lowerdimensional manifold [1,2,5,10,13,14,17]. The reason behind this is that those metrics are of cohomogeneity one and that the cohomogeneity-one manifolds of this type are the only ones that admit complete metrics with holonomy Spin (7) [17].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exceptional Holonomy on Vector Bundles with Two-Dimensional Fibers

Reidegeld

2013

J Geom Anal

View full text Add to dashboard Cite

Abstract. An SU (3)-or SU (1, 2)-structure on a 6-dimensional manifold N 6 can be defined as a pair of a 2-form ω and a 3-form ρ. We prove that any analytic SU (3)-or SU (1, 2)-structure on N 6 with dω ∧ ω = 0 can be extended to a parallel Spin(7)-or Spin 0 (3, 4)-structure Φ that is defined on the trivial disc bundle N 6 × Bǫ(0) for a sufficiently small ǫ > 0. Furthermore, we show by an example that Φ is not uniquely determined by (ω, ρ) and discuss if our result can be generalized to nontrivial bundles.

show abstract

Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

Reidegeld

2011

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

We investigate cohomogeneity-1 metrics whose principal orbit is an Aloff-Wallach space SU(3)/U (1). In particular, we are interested in metrics whose holonomy is contained in Spin (7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metrics on tubular neighbourhoods of various singular orbits. Since the equation for the holonomy reduction has only few explicit solutions, we make use of power series techniques. In order to prove the convergence and the smoothness near the singular orbit, we apply methods developed by Eschenburg and Wang. As a by-product of these methods, we find many new examples of Einstein metrics of cohomogeneity 1.

show abstract

Spin(7)-structures on complex linear bundles and explicit Riemannian metrics with holonomy group SU(4)

Cited by 5 publications

References 10 publications

Asymptotically conical Calabi–Yau manifolds, I

Asymptotically conical Calabi–Yau manifolds, I

Exceptional Holonomy on Vector Bundles with Two-Dimensional Fibers

Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

Contact Info

Product

Resources

About