On a conjecture of Tian

TL;DR

This paper investigates Tian's conjecture relating two alpha-invariants for smooth degree d surfaces in projective space, confirming it for d=4, disproving it for d≥8, and providing counterexamples for degrees 5, 6, and 7.

Contribution

The paper proves Tian's conjecture for degree 4 surfaces, shows it fails for degrees 6 and 7 with explicit examples, and offers a potential counterexample for degree 5.

Findings

Confirmed Tian's conjecture for d=4.

Disproved the conjecture for d≥8 with general surfaces.

Constructed explicit counterexamples for d=6 and d=7.

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 $S_5$.