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Regular version of the site

Article

On a conjecture of Tian

Mathematische Zeitschrift. 2018. Vol. 288. No. 1-2. P. 217-241.
Cheltsov I., Ahmadinezhad H.

We study Tian’s α-invariant in comparison with the α1-invariant for pairs (Sd,H) consisting of a smooth surface Sd of degree d in the projective three-dimensional space and a hyperplane section H. A conjecture of Tian asserts that α(Sd,H)=α1(Sd,H). We show that this is indeed true for d=4 (the result is well known for d⩽3), and we show that α(Sd,H)<α1(Sd,H) for d⩾8 provided that Sd is general enough. We also construct examples of Sd, for d=6 and d=7, for which Tian’s conjecture fails. We provide a candidate counterexample for S5.