On A Conjecture of Tian

Hamid Ahmadinezhad; Ivan Cheltsov; Josef Schicho

doi:10.1007/s00209-017-1886-z

On A Conjecture of Tian

Hamid Ahmadinezhad, Ivan Cheltsov, Josef Schicho

School of Mathematics

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Abstract / Description of output

We study Tian's α-invariant in comparison with the α1-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 α(S_d;H) = α₁(S_d;H). We show that this is indeed true for d = 4 (the result is well known for d ≤3), and we show that α(S_d;H) < α₁(S_d;H) for d ≥ 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 S₅.

Original language	English
Pages (from-to)	217-241
Number of pages	21
Journal	Mathematische zeitschrift
Volume	288
Issue number	1-2
Early online date	19 May 2017
DOIs	https://doi.org/10.1007/s00209-017-1886-z
Publication status	Published - Feb 2018

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abstract = "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 generalenough. 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.",

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AU - Cheltsov, Ivan

AU - Schicho, Josef

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AB - 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 generalenough. 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.

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DO - 10.1007/s00209-017-1886-z

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VL - 288

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EP - 241

JO - Mathematische zeitschrift

JF - Mathematische zeitschrift

IS - 1-2

ER -

On A Conjecture of Tian

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