BPF–psef duality conjecture for nef b-divisor classes

Let $X$ be a projective variety of dimension $d$, and let $\mathcal{X}$ denote its Riemann–Zariski space. A $b$-numerical class of codimension $k$ is an element of $\operatorname{w-N}^k(\mathcal{X})$; $\operatorname{c-N}^k(\mathcal{X})$ denotes the Cartier $b$-classes, $\operatorname{BPF}^k(\mathcal{X})$ the cone of basepoint-free classes, and $\operatorname{c-BPF}^k(\mathcal{X})$ its Cartier subcone. Write $(\alpha\cdot\beta)$ for the natural intersection pairing. A class is psef if it lies in the pseudoeffective cone.

BPF–psef duality conjecture. A $b$-numerical class $\alpha\in\operatorname{w-N}^k(\mathcal{X})$ is psef if and only if, for every $b$-numerical class $\beta\in\operatorname{c-BPF}^{d-k}(\mathcal{X})$, one has $(\alpha\cdot\beta)\ge0$. Moreover, $\alpha$ belongs to $\operatorname{BPF}^k(\mathcal{X})$ if and only if, for every psef Cartier $b$-numerical class $\beta\in\operatorname{c-N}^{d-k}(\mathcal{X})$, one has $(\alpha\cdot\beta)\ge0$.

This conjecture proposes a duality between pseudoeffective and basepoint-free cones. The case $k=1$ was proved by B. Lehmann, and the paper extends the result to $k=d-1$; the full statement remains open.