Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

Correa, Eder M.

doi:10.1007/s00220-019-03337-3

Cited by 7 publications

(13 citation statements)

References 60 publications

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“…The result above provides an explicit description for the unique solution of the homogeneous Kähler-Ricci flow associated to any initial data (X P , ω 0 ) purely in terms of Lie theory. Actually, following the ideas of [4], [31], [32], one can compute explicitly any solution as described in item (1) using algebraic tools of representation theory of complex semisimple Lie algebras. Particularly, from item (4) and item (6) of Theorem A, we have the following:…”

Section: Resultsmentioning

confidence: 99%

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Kähler-Ricci flow on rational homogeneous varieties

Correa¹

2021

Preprint

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In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the homogeneous Kähler-Ricci flow on rational homogeneous varieties. This description enables us to compute explicitly the maximal existence time for any homogeneous solution and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between numerical invariants associated to ample divisors and numerical invariants arising from solutions of the homogeneous Kähler-Ricci flow. In the particular setting of full flag varieties, we prove that the numerical invariants obtained from solutions of the homogeneous Kähler-Ricci flow can be related to certain well-known invariants which appear in some different contexts, including the global Seshadri constant of ample line bundles, the maximum possible radius of embeddings of symplectic and Kähler balls, and the log canonical threshold of ample É-divisors.From this, we obtain constraints in terms of the scalar and Ricci curvatures, respectively, for symplectic embeddings of open Euclidean balls, and for the triviality of analytic multiplier ideal sheaves defined by ample É-divisors.

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Section: Resultsmentioning

confidence: 99%

“…Remark 3.12. An important consequence of Theorem 3.10 is that it allows us to describe the local Kähler potential for any homogeneous Kähler metric in a quite concrete way using geometric tools coming from the representation theory of complex semisimple Lie algebras, for some examples of concrete computations we suggest [31], [32].…”

Section: 2mentioning

confidence: 99%

Kähler-Ricci flow on rational homogeneous varieties

Correa¹

2021

Preprint

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“…Next, if M and M satisfy the ϕ-condition, from Equations (75) and (76) we have ϕ∇ * ϕ + (divξ)ξ + ε∇ ξ ξ = 0 andφ trace |H f * ∇ ϕ − trace |H f * ∇ η ξ = 0 .…”

Section: Smentioning

confidence: 99%

“…Recently E.M. Correa [76] gives a new study on compact, (2n + 1)-dimensional, homogeneous contact manifolds. More precisely, this paper contains:…”

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confidence: 99%

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

Perrone

2019

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There is one-to-one correspondence between contact semi-Riemannian structures ( η , ξ , φ , g ) and non-degenerate almost CR structures ( H , ϑ , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X - i J X , X ∈ H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case.

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“…Our main purpose is classify these complex structures by using elements of Lie theory which underlie the geometry of complex flag manifolds, such as representation theory of simple Lie algebras and painted Dynkin diagrams [1]. In order to do so, we develop some techniques introduced in [17] to explicitly compute the Cartan-Ehresmann connections on principal S 1 -bundles over complex flag manifolds. Then, we combine this with the ideas developed in [52], and [46, Proposition 2.9], obtaining a Eder M. Correa was supported by CNPq grant 150899/2017-3.…”

Section: Introductionmentioning

confidence: 99%

Hermitian non-Kähler structures on products of principal $S^{1}$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

Correa

2018

Preprint

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In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal S 1 -bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order to describe and classify these structures. By following [52], we use these normal almost contact structures to explicitly describe a huge class of compact Hermitian non-Kähler manifolds obtained from products of principal S 1 -bundles over complex flag manifolds. Moreover, by following [46], we obtain from our description several concrete examples of 1-parametric families of complex structures on products of principal S 1 -bundles over flag manifolds, these concrete examples generalize the Calabi-Eckmann manifolds [14]. Further, by following [26], as an application of our main results in the setting of KT structures on toric bundles over flag manifolds, we classify a huge class of explicit examples of Calabi-Yau structures with torsion (CYT) on certain Vaisman manifolds (generalized Hopf manifolds [72]). Also as an application of our main results, we provide several new concrete examples of astheno-Kähler structures on products of compact homogeneous Sasaki manifolds. This last construction generalizes, in the homogeneous setting, the results introduced in [47] for Calabi-Eckmann manifolds.

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Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

Cited by 7 publications

References 60 publications

Kähler-Ricci flow on rational homogeneous varieties

Kähler-Ricci flow on rational homogeneous varieties

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

Hermitian non-Kähler structures on products of principal $S^{1}$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

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