Fano varieties: Kaehler-Einstein metrics, K-stability and moduli spaces

Ph.D. course

Lecturer: Giulio Codogni.

Schedule: 20 hours, every friday, from 6th of March to 22nd of May, 11:00 to 13:00, room Dal Passo (only exception: on Friday 6th of March room 2001).

Abstract: The Yau-Tian-Donaldson conjecture predicts that a Fano variety admits a Keahler- Einstein metric if and only if it is K-stable. K-stability is an algebraic notion modeled on geometric invariant theory. It is also expected that K-stable Fano varities have a good moduli space.

By now, the Yau-Tian-Donaldson has been fully proved for smoothable Fano varieties, and partially for all Fano varities.

Thanks to a surprising and deep relation with the Minimal Model Program, building on the proof of the BAB conjecture by Birkar, the existence of the moduli space of K-stable varieties has been recently established. However, its projectivity is still an open problem. This result also opens the way to the study of specific interesting examples of moduli spaces.

Preliminary program:

  1. (1)  Algebraic definition of K-stability

  2. (2)  K-stable Fano varieties are klt (following Odaka)

  3. (3)  sketch of the variational proof of the Yau-Tian-Donaldson conjecture

  4. (4)  special test configuration (following Li and Xu)

  5. (5)  K-stability via filtrations and valuations (following Fujita, Li and Xu)

  6. (6)  relation between K-stability and Birkar’s theory of complements (following Blum and Xu)

  7. (7)  existence of the good moduli space of K-stable Fano varities (following Alper, Blum, Halpern-Leistner, Liu and Xu )