Higher dimensional Clifford-Severi equalities

Abstract

Let $X$ be a smooth complex projective variety, $a:X\to A$ 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 $h_a^0(X,L)$ the minimum of $h^0(X,L\otimes a^{\ast }\alpha )$ 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: $$\mathrm{vol}(L)\ge n!h_a^0(X,L),$$ and, if $K_X-L$ is pseudoeffective, $$\mathrm{vol}(L)\ge 2n!h_a^0(X,L).$$ In this paper we characterize varieties and line bundles for which the above Clifford-Severi inequalities are equalities.

14 pages; this is an expanded version of the final section of arXiv:1606.03290v4 [math.AG]