Description: In this mini-course, we will discuss recent advances on the existence of Kähler-Einstein metrics on Fano varieties. More precisely, the works of Chen-Donaldson-Sun and Tian imply that a Fano variety X (i.e., a complex smooth projective variety such that det(TX) is ample) possesses a Kähler-Einstein metric if and only if it is K-polystable. This latter condition, introduced by Tian (1997) and Donaldson (2002), allows us to reduce a problem in Partial Differential Equations and Differential Geometry to the language of Algebraic Geometry.

The objective of the mini-course will be to introduce the definition of K-stability and the properties of such varieties, also aiming to learn how to use this language in explicit examples.

Target public: PhD students and advanced Master students in Algebraic Geometry.
Prerequisites: Previous knowledge of algebraic geometry at the level of Hartshorne’s book.

 

References:
ARAUJO, CAROLINA, et al. – The Calabi problem for Fano threefolds. Vol. 485. Cambridge University Press, 2023.
BLUM, HAROLD – Math 7800: K-stability. University of Utah, 2022.
DEVLEMING, KRISTIN – K-moduli of Fano varieties and log Fano pairs. University of Massachusetts Amherst, 2023.
XU, CHENYANG – K-stability of Fano varieties: an algebro-geometric approach. EMS Surveys in Mathematical Sciences, 8(1), 265-354, 2021.