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Topics in complex geometry

Lecturer: 
Course Type: 
PhD Course
Anno (LM): 
Second Year
Academic Year: 
2023-2024
Period: 
March–June
Duration: 
40 h
Description: 

The aim of the course is to give an introduction to some topics of current interest in the global geometry of compact Kähler manifols.

Part I – Basic notions. Hermitian and Kähler manifolds, harmonic theory, Kodaira Vanishing and Embedding, Kempf-Ness Theorem, notions of canonical Kähler metrics.

Some textbooks:

  • D. Huybrechts, Introduction to complex geometry, Springer.
  • G. Székelyhidi, Introduction to extremal metrics, AMS. 

Part II – K-stability and applications. Test configurations and Donaldson-Futaki invariant; Donaldson’s lower bound on the Calabi functional; Mabuchi functional and geodesics; sketch of the uniform, Fano case of the Yau-Tian-Donaldson Conjecture.

Some references:

  • S. K. Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), no. 3.Boucksom, Sébastien; Hisamoto, Tomoyuki; Jonsson, Mattias Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841. 

Part III – Further topics (according to the interests of the audience). New equations in Kähler geometry, e.g. Z-critical connections and metrics; problems involving a deformation of the complex structure or a mirror map.

Some references:

  • R. Dervan, J. B. McCarthy, L. M. Sektnan, Z-critical connections and Bridgeland stability conditions, arXiv.
  • Scarpa, Carlo, Scalar curvature and deformations of complex structures. J. Reine Angew. Math. 797 (2023), 255–283.
  • J. Stoppa, K-stability and large complex structure limits, arXiv.

 

Next Lectures: 
Monday, March 4, 2024 - 11:00 to 13:00
Wednesday, March 6, 2024 - 11:00 to 13:00
Monday, March 11, 2024 - 11:00 to 13:00
Wednesday, March 13, 2024 - 11:00 to 13:00
Monday, March 18, 2024 - 11:00 to 13:00
Wednesday, March 20, 2024 - 11:00 to 13:00
Monday, March 25, 2024 - 11:00 to 13:00
Wednesday, March 27, 2024 - 11:00 to 13:00
Monday, April 1, 2024 - 11:00 to 13:00
Wednesday, April 3, 2024 - 11:00 to 13:00
Monday, April 8, 2024 - 11:00 to 13:00
Wednesday, April 10, 2024 - 11:00 to 13:00
Monday, April 15, 2024 - 11:00 to 13:00
Wednesday, April 17, 2024 - 11:00 to 13:00
Monday, April 22, 2024 - 11:00 to 13:00
Wednesday, April 24, 2024 - 11:00 to 13:00
Monday, April 29, 2024 - 11:00 to 13:00

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