[09/12/2022] Σεμινάριο Γεωμετρίας – Ραφαήλ Τσιάμης (Harvard University), The hyperkahler structure on the cotangent bundle of SL(2,C)

Αγαπητοί Συνάδελφοι, Φοιτητές και Φίλοι,

Θα θέλαμε να σας ενημερώσουμε ότι τη Παρασκευή 9 Δεκεμβρίου στις 12:00 στην αίθουσα Μ2 και στο πλαίσιο του Σεμιναρίου Γεωμετριας, θα δώσει ομιλία ο Ραφαήλ Τσιάμης (Harvard University).

Τίτλος: The hyperkahler structure on the cotangent bundle of SL(2,C)

Περίληψη: Hyperkahler manifolds have several analytic and algebraic properties making them accessible for description but difficult to construct. A construction of P. Kronheimer [3] gives an example of hyperkahler structure abstractly constructed on the cotangent bundle $T^\ast G_\mathbb{C}$ of the complex Lie group $G_\mathbb{C}$ corresponding to a compact Lie group $G$. This hyperkahler structure emerges through infinite dimensional reduction (hyperk\”ahler reduction) of the space of solutions of the so-called “Nahm equations”, and so its description is particularly difficult even for basic Lie groups. We present a partial description of the family of Kahler structures on $G_\mathbb{C}$ as complex submanifold of $T^\ast G_\mathbb{C}$ in the general case, extending a result of Stenzel [5] for homogeneous spaces. We then focus on $G=SU(2)$, for which $G_\mathbb{C}=SL(2,\mathbb{C})$; our calculation, method and results partially generalize a calculation of Dancer [1]. This is work in collaboration with Richard B. Melrose (MIT) and Michael Singer (UCL).

[1] A. Dancer, Nahm’s equations and hyperkähler geometry. Comm. Math. Phys. 158(3): 545-568 (1993).
[2] F. Kirwan, Symplectic quotients of unstable Morse strata for norm-squares of moment maps. Preprint, arXiv:1802.09237.

[3] P.B. Kronheimer, The hyperkähler structure on the cotangent bundle of a complex Lie group. MSRI preprint (1998).

[4] G. Patrizio & P.M. Wong, Stein manifolds with compact symmetric center. Math. Ann. 289 (3), 355-382 (1991).
[5] M. Stenzel, Ricci-flat metrics on the complexification of a compact rank one symmetric space. Manuscripta Math. (80), 151-163 (1993).

Zoom link:https://authgr.zoom.us/j/94933117785?pwd=VS9hOHMxMXB1bkUvem5HMWpFaTlvUT09

Meeting ID:949 3311 7785

Passcode: 093582

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