On a conjecture of Tian
journal contributionposted on 12.04.2017, 08:40 by Hamid AhmadinezhadHamid Ahmadinezhad, Ivan Cheltsov, Josef Schicho
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 6 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.
- Mathematical Sciences