2022
Lotay, Jason D.; Earp, Henrique N. Sá; Saavedra, Julieth
Flows of G2-structures on contact Calabi–Yau 7-manifolds Journal Article
Em: Annals of Global Analysis and Geometry, vol. 62, pp. 367–389, 2022.
Resumo | Links | BibTeX | Tags: Calabi–Yau, G2-Structures
@article{nokey,
title = {Flows of G2-structures on contact Calabi\textendashYau 7-manifolds},
author = {Jason D. Lotay and Henrique N. S\'{a} Earp and Julieth Saavedra },
url = {https://link.springer.com/article/10.1007/s10455-022-09854-0},
doi = {http://dx.doi.org/10.1007/s10455-022-09854-0},
year = {2022},
date = {2022-06-21},
journal = {Annals of Global Analysis and Geometry},
volume = {62},
pages = {367\textendash389},
abstract = {We study the Laplacian flow and coflow on contact Calabi\textendashYau 7-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas it produces an immortal (though neither eternal nor self-similar) solution of the Laplacian coflow which has an infinite time singularity of Type IIb, unless the transverse Calabi\textendashYau geometry is flat. The flows in each case collapse (under normalised volume) to a lower-dimensional limit, which is either R, for the Laplacian flow, or standard C3, for the Laplacian coflow. We also study the Hitchin flow in this setting, which we show coincides with the Laplacian coflow, up to reparametrisation of time, and defines an (incomplete) Calabi\textendashYau structure on the spacetime track of the flow.},
keywords = {Calabi\textendashYau, G2-Structures},
pubstate = {published},
tppubtype = {article}
}
We study the Laplacian flow and coflow on contact Calabi–Yau 7-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas it produces an immortal (though neither eternal nor self-similar) solution of the Laplacian coflow which has an infinite time singularity of Type IIb, unless the transverse Calabi–Yau geometry is flat. The flows in each case collapse (under normalised volume) to a lower-dimensional limit, which is either R, for the Laplacian flow, or standard C3, for the Laplacian coflow. We also study the Hitchin flow in this setting, which we show coincides with the Laplacian coflow, up to reparametrisation of time, and defines an (incomplete) Calabi–Yau structure on the spacetime track of the flow.
Correa, Eder M.
Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones Journal Article
Em: Communications in Mathematical Physics, vol. 367, pp. 1095–1151, 2022.
Resumo | Links | BibTeX | Tags: Calabi–Yau
@article{nokey,
title = {Homogeneous Contact Manifolds and Resolutions of Calabi\textendashYau Cones},
author = {Eder M. Correa},
url = {https://link.springer.com/article/10.1007/s00220-019-03337-3},
doi = {https://doi.org/10.1007/s00220-019-03337-3},
year = {2022},
date = {2022-02-09},
journal = {Communications in Mathematical Physics},
volume = {367},
pages = {1095\textendash1151},
abstract = {In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan\textendashEhresmann connection (gauge field) for principal U(1)-bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby\textendashWang fibration over complex flag manifolds (Boothby and Wang in Ann Math 68:721\textendash734, 1958) as well as their underlying Sasaki structures. By following Conlon and Hein (Duke Math J 162:2855\textendash2902, 2013), Van Coevering (Math Ann, 2009. https://doi.org/10.1007/s00208-009-0446-1) and Goto (J Math Soc Jpn 64:1005\textendash1052, 2012), as an application of our results we use the Cartan\textendashRemmert reduction (Grauert in Math Ann 146:331\textendash368, 1962) and the Calabi Ansatz technique (Calabi in Ann Sci \'{E}cole Norm Sup (4) 12:269\textendash294, 1979) to provide many explicit examples of crepant resolutions of Calabi\textendashYau cones with certain homogeneous Sasaki\textendashEinstein manifolds realized as links of isolated singularities. These concrete examples illustrate the existence part of the conjecture introduced in Martelli and Sparks (Phys Rev D 79(6):065009, 2009).},
keywords = {Calabi\textendashYau},
pubstate = {published},
tppubtype = {article}
}
In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan–Ehresmann connection (gauge field) for principal U(1)-bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby–Wang fibration over complex flag manifolds (Boothby and Wang in Ann Math 68:721–734, 1958) as well as their underlying Sasaki structures. By following Conlon and Hein (Duke Math J 162:2855–2902, 2013), Van Coevering (Math Ann, 2009. https://doi.org/10.1007/s00208-009-0446-1) and Goto (J Math Soc Jpn 64:1005–1052, 2012), as an application of our results we use the Cartan–Remmert reduction (Grauert in Math Ann 146:331–368, 1962) and the Calabi Ansatz technique (Calabi in Ann Sci École Norm Sup (4) 12:269–294, 1979) to provide many explicit examples of crepant resolutions of Calabi–Yau cones with certain homogeneous Sasaki–Einstein manifolds realized as links of isolated singularities. These concrete examples illustrate the existence part of the conjecture introduced in Martelli and Sparks (Phys Rev D 79(6):065009, 2009).
2019
Correa, Eder M.; Grama, Lino
Calabi-Yau metrics on canonical bundles of complex flag manifolds Journal Article
Em: Journal of Algebra, vol. 527, iss. 1, pp. 109-135, 2019.
Resumo | Links | BibTeX | Tags: Calabi–Yau
@article{nokey,
title = {Calabi-Yau metrics on canonical bundles of complex flag manifolds},
author = {Eder M. Correa and Lino Grama},
url = {https://www.sciencedirect.com/science/article/abs/pii/S0021869319301164?via%3Dihub},
doi = {https://doi.org/10.1016/j.jalgebra.2019.02.027},
year = {2019},
date = {2019-03-13},
journal = {Journal of Algebra},
volume = {527},
issue = {1},
pages = {109-135},
abstract = {In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat K\"{a}hler metrics obtained through the Calabi Ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag manifolds).},
keywords = {Calabi\textendashYau},
pubstate = {published},
tppubtype = {article}
}
In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi Ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag manifolds).