On a conjecture of Tian

Abban, Hamid; Cheltsov, Ivan; Schicho, Josef

On a conjecture of Tian

journal contribution

posted on 2017-04-12, 08:40 authored by Hamid Abban, 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.

History

School

Science

Department

Mathematical Sciences

Published in

Mathematische Zeitschrift

Citation

AHMADINEZHAD, H., CHELTSOV, I. and SCHICHO, J., 2017. On a conjecture of Tian. Mathematische Zeitschrift, 288 (1-2), pp.217–241.

Publisher

Version

NA (Not Applicable or Unknown)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution 4.0 International (CC BY 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by/4.0/

Acceptance date

2017-03-20

Publication date

2017

Notes

This is an Open Access Article. It is published by Springer under the Creative Commons Attribution 4.0 Unported Licence (CC BY). Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/