Исключительные поверхности дель Пеццо

The only known sufficient condition for the existence of a Kahler-Einstein metric on a Fano manifold can be formulated in terms of so-called alpha-invariant introduced by Tian and Yau more than 20 years ago. This invariant can be naturally defined for log Fano varieties with log terminal singularities using purely algebraic language. Using a global-to-local results of Shokurov, one can define a similar invariants for a germ of log terminal singularity. We describe the role played by alpha-invariants in birational geometry and singularity theory. We prove the existence of Kahler-Einstein metrics on many quasismooth well-formed weighted del Pezzo hypersurface and compare this result with new obstructions found by J. Gauntlett, D. Martelli, J. Sparks and S.-T. Yau. We apply our technique to classify weakly-exceptional quasismooth well-formed weighted del Pezzo hypersurface using the classification of isolated rational quasihomogeneous three-dimensional singularities obtained by S. S. T. Yau and Y. Yu.