Higher dimensional Clifford-Severi equalities

Let $X$ be a smooth complex projective variety, $a\colon X\rightarrow A$ a morphism to an abelian variety such that $\mathrm{Pic}^0(A)$ injects into $\mathrm{Pic}^0(X)$ and let $L$ be a line bundle on $X$; denote by $h^0_a(X,L)$ the minimum of $h^0(X,L\otimes a^*\alpha)$ for $\alpha\in \mathrm{Pic}^0(A)$. The so-called Clifford-Severi inequalities have been proven in arXiv:1303.3045 [math.AG] and arXiv:1606.03290 [math.AG]}; in particular, for any $L$ there is a lower bound for the volume given by: $$\mathrm{vol}(L)\ge n! h^0_a(X,L),$$ and, if $K_X-L$ is pseudoeffective, $$\mathrm{vol}(L)\ge 2n! h^0_a(X,L).$$ In this paper we characterize varieties and line bundles for which the above Clifford-Severi inequalities are equalities.