Inhomogeneous Einstein metrics on complex projective spaces

The complex projective space CP^n, which parametrizes the complex lines in C^{n+1}, is one of the most extensively studied objects in algebraic and complex geometry. The first Einstein metric discovered on CP^n was the Fubini–Study metric, and in 1965 Berger asked whether it was the only one. In 1982, Ziller constructed a second homogeneous Einstein metric in odd complex dimensions. Berger's question remained open in even complex dimensions, where Fubini–Study was the only known Einstein metric. Together with Gonzalo Cao Labora (EPFL, Switzerland), we have constructed new Einstein metrics on CP^n for n= 3, 4, 5, 6, 7. These are the first known non-homogeneous examples and, together with the Fubini–Study and Ziller metrics, the only ones known to date. Our construction settles Berger's question in the previously open cases n = 4 and n = 6. This talk is part of the activities of the FIS2 project BeyondCY3 - CUP B53C25001070001. More information available at: https://sites.google.com/uniroma1.it/lfoscolo/fis2-beyondcy3/events