Tian's alpha-invariant conjecture for projectively normal polarised varieties

Let $X$ be a smooth variety and let $L$ be an ample Cartier divisor on $X$. For a positive integer $n$, define

$$\alpha_n\big(X,L\big)=\sup\left\{\lambda\in\mathbb{Q}\ \middle|\ \left(X,\frac{\lambda}{n}D\right)\text{ is log canonical for every }D\in|nL|\right\},$$

with $\alpha_n(X,L)=+\infty$ if $|nL|$ is empty, and define

$$\alpha\big(X,L\big)=\sup\left\{\lambda\in\mathbb{Q}\ \middle|\ (X,\lambda D)\text{ is log canonical for every effective }\mathbb{Q}\text{-divisor }D\sim_{\mathbb{Q}}L\right\}.$$

Assume that $L$ is very ample and that the graded algebra

$$\bigoplus_{i\geqslant 0}H^0\Big(X,\mathcal{O}_X\big(iL\big)\Big)$$

is generated by elements of $H^0(X,\mathcal{O}_X(L))$, so that $L$ defines a projectively normal embedding. Tian's conjecture. Then

$$\alpha(X,L)=\alpha_1(X,L).$$

The alpha-invariant is important in the study of Kähler–Einstein metrics and in birational geometry, while the equality would identify the asymptotic invariant with its first approximation in the projectively normal setting. The paper studies this conjecture for smooth surfaces and pairs consisting of a degree-$d$ surface and a hyperplane section; its general status is not resolved by the supplied text.