Positivity of the global log canonical threshold of a pair

Question

Let $(X,L)$ be a polarized smooth projective variety. Let $D$ be a smooth irreducible divisor in $X$. Let $0<c<1$ be a real number. We denote $cD$ as $\Delta$. We can define the $\alpha$ invariant or the global log canonical threshold of the pair $(X,L,\Delta)$ as:

$$ \alpha(L,\mu_{\Delta}) = \inf_{D_{m}} \mathrm{lct}_{X}(X,D_{m}+\Delta)$$

where $m$ is a positive integer and $D_{m}$ is the zero divisor of some $s_{m}\in H^{0}(X,mL)$. There is also an analytic way to define this invariant as in the last section of Robert J.Berman's paper A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics: https://arxiv.org/abs/1011.3976 .

My question is whether we know that this invariant is always positive as in the case where $\Delta$ is trivial?

Answer 1

I think the definition of the $\alpha$ invariant is slightly off. I think you want to consider $\mathrm{lct}_{(X,\Delta)}(\frac{1}{m}D_m)$. The usual way I've seen this written is $$ \alpha(L) = \inf_{D \sim_\mathbb{Q} L} \mathrm{lct}_{(X,\Delta)}(D) $$ where the infimum runs over all divisors $\mathbb{Q}$-linearly equivalent to $L$.

In any case, this is Theorem 9.14 in this paper of Boucksom, Hisamoto and Jonsson. It is also proved under the assumption that $L$ is big rather than ample by Blum-Jonsson, Theorem A.