Title of the course: Spin c structures on complex manifolds
Instructor: Assoc. Prof. Roger Nakad
Institution: Notre Dame University
Dates: 19-23 September 2022
Prerequisites: Linear Algebra, Riemannian Geometry
Level: Graduate
Abstract: These lecture series aim to give an elementary exposition on basic results about Spin^c structures and the Dirac operator on complex manifolds. For this, we select some key ingredients which illustrate the basic objects and some of their properties as Clifford algebras, spin and spin^c groups, connections, covariant derivatives, Dirac and Twistor operators. We then relate Kahler/complex structures to Spin^c structures on manifolds, and the Dirac operator to the Dolbeault operator. We end by giving beautiful geometric application: The Clifford multiplication between an effective harmonic form and a Kählerian Killing spin^c spinor field vanishes. This extends to the spin^c case the result of A. Moroianu stating that, on a compact Kähler-Einstein manifold of complex dimension 4l+3 carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a Kählerian Killing spinor is zero
Language: EN
Textbook:
1. Th. Friedrich, Dirac operator’s in Riemannian Geometry, Graduate studies in mathematics, Volume 25, Americain Mathamatical Society, 2000
2. J. P. Bourguignon, O. Hijazi, J. L. Milhorat and A. Moroianu, A spinorial approach to Riemannian and conformal geometry, EMS Monographs in Mathematics, 2015