On a conjecture of Tian

Ahmadinezhad, Hamid; Cheltsov, Ivan; Schicho, Josef

doi:10.1007/s00209-017-1886-z

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On a conjecture of Tian

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Ahmadinezhad, Hamid
Max Planck Institute for Mathematics, Max Planck Society;

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Cheltsov, Ivan
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Ahmadinezhad, H., Cheltsov, I., & Schicho, J. (2018). On a conjecture of Tian. Mathematische Zeitschrift, 288(1-2), 217-241. doi:10.1007/s00209-017-1886-z.

Cite as: https://hdl.handle.net/21.11116/0000-0004-621F-C

Abstract

We study Tian's $\alpha$-invariant in comparison with the $\alpha_1$-invariant for pairs $(S_d,H)$ consisting of a smooth surface $S_d$ of degree $d$ in the projective three-dimensional space and a hyperplane section $H$. A conjecture of Tian asserts that $\alpha(S_d,H)=\alpha_1(S_d,H)$.
We show that this is indeed true for $d=4$ (the result is well known for $d\leqslant 3$), and we show that $\alpha(S_d,H)<\alpha_1(S_d,H)$ for $d\geqslant 8$ provided that $S_d$ is general enough. We also construct examples of $S_d$, for $d=6$ and $d=7$, for which Tian's conjecture fails. We provide a candidate counterexample for $S_5$.